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+{"text":"\\section{Introduction}\n\n\n    \n      \n      \n    \n  \n\n\n\n\n\n\n\n\n\n\n    \n      \n      \n    \n  \n\n\nA random walk on a graph is a walk that begins at a particular starting vertex in which each successive step is determined by randomly choosing an edge adjacent to the previous vertex and traveling to the vertex at the other endpoint of the edge.  This random choice is distributed equally over all edges adjacent to the vertex.  For our purposes, we will consider only random walks on unweighted and undirected graphs.  Although much of the historical work on random walks considers infinite graphs, recent work has dealt more with finite graphs.  When dealing with random walks on finite graphs, the focus turns to less qualitative questions; rather than asking whether or not a random walk will return to its starting vertex, it may be interesting to ask how long the random walk would take to return to the starting vertex.\n\nA property that arises from analyzing random walks on finite graphs is the hitting time.  Given a finite graph $G$, the hitting time, $h_{uv}$,  from a vertex $u$ to a vertex $v$, is the expected number of steps it takes for a random walk that starts at vertex $u$ to reach vertex $v$.  Note that when dealing with finite graphs, the hitting time from $u$ to $v$ is finite if and only if the vertices $u$ and $v$ are connected.\n\nThe hitting time of a random walk has many useful properties.  For example, the cover time, the expected time it takes for a random walk to visit all vertices of a graph, can be both bounded above by a function of the largest hitting time from one vertex to another, and below by a function of the smallest hitting time from one vertex to another~\\cite{randomwalk}.  However, the hitting times between vertices of various graphs can be hard to analyze, and finding their values is not intuitive.  There exist a few known bounds on hitting times, as found in~\\cite{c, b, Aldous, a}.  We will focus on finding exact formulas for the hitting time for certain types of graphs.\n\nWe will consider two different methods of calculating the hitting time of a graph and demonstrate their applications to various classes of graphs.  Section 2 gives an explicit formula for finding the hitting time, but only in a few specific cases.  Specifically, the formula can only be used to find the hitting time from one vertex to a neighbor and only if the graph exhibits a symmetry about the starting vertex.  This can then be applied to random walks on a variety of classes of graphs, including grids, hypercubes, and trees.\n\nSection 3 uses the method of calculating hitting times through a system of linear equations first shown in~\\cite{Aldous}.  Although this is not as convenient as a formula, this method can be applied to random walks on any graph.  We can use this to derive formulas for hitting times of random walks on graphs for which the method of Section 2 does not apply, such as for the complete $d$-ary tree, or the tadpole graph.\n\n\\section{Hitting times in graphs with symmetry}\n\n\\subsection{Proof of theorem}\n\n\nThe following theorem gives a formula that can, in certain cases, be used to find hitting times.\n\n\\begin{thm}\\label{recurrence}\nLet $v$ be a vertex of a connected graph $G$ with neighbor $u$.  If for every other neighbor of $v$, there exists an automorphism of $G$ that maps $u$ to that neighbor of $v$, then the hitting time from $u$ to $v$ is $\\frac{2e}{k}-1$, where $e$ is the number of the edges in the graph, and $k$ is the number of neighbors of $v$.\n\\end{thm}\n\n\\begin{proof}\nBecause of the symmetry of the graph, the hitting time is equal from any neighbor of $v$ to $v$.  Let this value be $x$.\n\nNow consider the random walk on $G$ that starts at vertex $v$ and moves to vertex $u$ in the next step, and the next such $t$ in which the $t$th vertex visited is $v$ and the $(t+1)$th vertex visited is $u$.  We can find the expected value of the next such $t$, both in terms of both $k$ and $x$, and in terms of $e$, allowing us to solve for $x$.  This can also be thought of as finding the recurrence time of a random walk along the edges of the directed graph $G'$, formed by replacing each edge in $G$ with two, one in each direction.\n\nConsider the structure of the walk more closely.  After starting at $v$ and then going to $u$, in order to again return back to $v$ and then $u$, the walk must first return to $v$.  This takes an expected $x$ steps.  At this point, the walk can continue in two different ways; the walk may go to $u$, with a probability of $\\frac{1}{k}$ and $1$ additional step.  It may also continue on to another vertex, in which case the walk must again return back to $v$. This adds on average, another $x+1$ to the number of steps the random walk has taken.  Continuing in this manner, we find that that the expected recurrence time is\n\n\\[\\frac{1}{k}\\left(x+1\\right)+\\frac{1}{k}\\frac{k-1}{k}2\\left(x+1\\right)+\\frac{1}{k}\\frac{k-1}{k}\\frac{k-1}{k}3\\left(x+1\\right)+\\cdots\\]\n\nwhich simplifies to $k\\left(x+1\\right)$.  \n\nIt is well known that the expected recurrence time in this walk is $2e$, as there are $2e$ edges in $G'$~\\cite{randomwalk}.  Setting these two values equal and solving for $x$ gives $x = \\frac{2e}{k}-1$.\n\\end{proof}\n\n\n\\subsection{Applications to the grid, hypercube, and trees}\n\nThis technique can be used to find the hitting time in the following graphs. \n\n\\begin{cor}\nIn a $d$-dimensional grid whose dimensions all have length $m$, the hitting time from a corner to one of its neighbors is $2\\left(m-1\\right)m^{d-1}-1$\n\\end{cor}\n\n\\begin{proof}\nThere are $d\\left(m-1\\right)m^{d-1}$ edges in a $d$-dimensional grid whose dimensions all have length $m$.  Applying Theorem \\ref{recurrence} gives the hitting time given above.\n\\end{proof}\n\nIn particular, the application of this technique to the hypercube gives a more general result, as every vertex is a corner.\n\n\\begin{cor}\nIn a $d$-dimensional hypercube, the hitting time from any vertex to one of its neighbors is $2^d-1$.\n\\end{cor}\n\nAdditionally, this technique can be used to find the hitting times from one vertex to a neighbor in trees.  \n\n\\begin{cor}\\label{tree}\nThe hitting time from a vertex $v$ to a neighbor $u$ in a tree is $2e-1$, where $e$ is the number of edges in the connected component containing $u$ after the edges to all other neighbors of $u$ are removed.\n\\end{cor}\n\n\\begin{proof}\nRemoving the edges to all other neighbors of $u$ does not change the hitting time from $v$ to $u$, as it is not possible for a random walk starting at $v$ to reach those vertices without having already reached $u$.  However, this allows the hypothesis of Theorem \\ref{recurrence} to hold for the connected component containing $u$, as $u$ now only has $1$ neighbor.  Finally, we can just apply the result to this modified graph.\n\\end{proof}\n\n\\section{Hitting time via a system of linear equations}\nThe second technique we will use to find hitting times can be applied to any graph, rather than graphs with some property of symmetry, but does not yield formulas as easily.  The following theorem, first shown in~\\cite{Aldous} gives a set of linear equations whose solution gives the hitting times of a random walk on a graph.  If we know the general structure of the graph, then it may be possible to find the hitting times in terms of certain properties of the graph.\n\n\\begin{thm}\\label{eq}\nIn a graph $G$, let $h_{ij}$ be the hitting time from a vertex $i$ to a vertex $j$ and let $\\Gamma\\left(i\\right)$ be the set of neighbors of $i$.  The following set of equations, for a fixed vertex $j$ and all vertices $i$ in $G$\n\n\\[h_{ij} = \\left\\{\n     \\begin{array}{lr}\n       0 &  i=j\\\\\n       1+\\frac{1}{|\\Gamma\\left(i\\right)|}\\displaystyle\\sum_{k\\in\\Gamma\\left(i\\right)}{h_{kj}} &  i \\neq j\n     \\end{array}\n   \\right.\\]\n\nyields a unique solution for all $h_{ij}$.\n\\end{thm}\n\n\nNote that we can use this theorem to both construct hitting times by solving the system of equations, or to show that a given formula for the all the hitting times to a certain vertex satisfy the set of equations, as they describe exactly one solution.\n\n\\subsection{Application to trees}\nWe first use this technique to give an alternate proof to Corollary \\ref{tree}.\n\n\\begin{thm}\nLet $v$ and $u$ be neighbors of a tree rooted at $u$.  Let $n$ be the number of vertices in the subtree rooted at $v$.  Then the hitting time from $v$ to $u$ is $2n-1$.\n\\end{thm}\n\n\\begin{proof}\nThis can be proven using induction.  The base case where $n$ is $1$ can be checked by calculation.\n\nBy Theorem \\ref{eq}, the following must hold\n\n\\[h_{vu} = 1+\\frac{1}{|\\Gamma\\left(i\\right)|}\\displaystyle\\sum_{j\\in\\Gamma\\left(i\\right)}{h_{ju}}.\\]\n\nOne of the neighbors of $v$ is $u$, in which case the hitting time to $u$ is $0$.  For every other neighbor of $v$, $j$, the hitting time from $j$ to $v$ can also be expressed as $h_{jv}+h_{vu}$.  Letting $n_j$ be the number of nodes in the subtree rooted at $j$, by the inductive hypothesis, $h_{jv} = 2n_j-1$.  We are left with the following equation\n\n\\[h_{vu} = 1+\\frac{1}{|\\Gamma\\left(i\\right)|}\\left(\\displaystyle\\sum_{j\\in\\Gamma\\left(i\\right)-\\{u\\}}{2n_j-1}\\right)+\\frac{|\\Gamma\\left(i\\right)|-1}{|\\Gamma\\left(i\\right)|}h_{vu}.\\]\n\nSolving for $h_{vu}$ gives\n\n\\[h_{vu} = 1+\\displaystyle\\sum_{j\\in\\Gamma\\left(i\\right)-\\{u\\}}{2n_j}.\\]\n\nBecause $1+\\displaystyle\\sum_{j\\in\\Gamma\\left(i\\right)-\\{u\\}}{n_j}$ is the number of vertices in the subtree rooted at $v$, we can conclude that $h_{vu} = 2n-1$, completing the inductive step.\n\\end{proof}\n\n\\subsection{Application to the tadpole graph}\nWe can also use these ideas to find the hitting time in a \"tadpole\" graph, a graph which consists of a cycle attached to a line.  \n\n\\begin{thm}\nLet $v$ be vertex with the largest distance from the cycle.  The hitting time from $v$'s neighbor $u$ to $v$ is $2l+2k-1$, where $k$ is the number of vertices in the cycle, and $l$ is the number of vertices in the line, not including $a$.\n\\end{thm}\n\n\\begin{proof}\n\nWe start by characterizing the hitting times of the vertices on the line.\n\n\\begin{lem}\n\nLet the distance from a vertex $u'$ on the line to $v$ be $l$.  Then the hitting from $u'$ to $v$ is $lh_{uv}-l\\left(l-1\\right)$.\n\n\\end{lem}\n\n\\begin{proof}\n\nThis can be shown using induction.  The base cases, $l=0, 1$ are can be checked by calculation.  Assume the above holds true for all $l$, up to $i$.  Using Theorem \\ref{eq}, we know that\n\n\\[h_{iv} = \\frac{1}{2}\\left(h_{\\left(i+1\\right)v}+h_{\\left(i-1\\right)v}\\right)+1.\\]\n\nReplacing the values for $h_{iv}$ and $h_{\\left(i-1\\right)v}$, we have that\n\n\\[h_{uv}-l\\left(l-1\\right) = \\frac{1}{2}\\left(h_{\\left(i+1\\right)v}\\left(l-1\\right)lh_{uv}-\\left(l-1\\right)\\left(l-2\\right)\\right)+1.\\]\n\nSolving for $h_{\\left(i+1\\right)v}$ gives the desired result.\n\n\\end{proof}\n\nWe can then continue by characterizing the hitting times of the vertices on the cycle\n\n\\begin{lem}\nIf $k$ is even, then let $v'$ be the vertex on the cycle farthest away from $v$.  Then, for any vertex on the cycle, $w$, $h_{wv} = h_{v'v}-l^2$, where $l$ is the distance from $w$ to $v'$.\n\nIf $k$ is odd, then let $v'$ and $v''$, be the vertices on the cycle farthest away from $v$.  Then, for any vertex on the cycle, $w$, $h_{wv} = h_{v'v}-l\\left(l+1\\right)$, where $l$  is the smaller of the two distances from $w$ to $v'$ and $v''$.\n\\end{lem}\n\n\\begin{proof}\nBoth cases can be shown using induction.  When $k$ is even, the base case where $l=0$ can be checked by calculation.  Assume the lemma holds true for all $l$ up to $i$.  Then by Theorem \\ref{eq}, we have that\n\n\\[h_{iv} = \\frac{1}{2}\\left(h_{\\left(i+1\\right)v}+h_{\\left(i-1\\right)v}\\right)+1.\\]\n\nReplacing the values for $h_{iv}$ and $h_{\\left(i-1\\right)v}$, we have that\n\n\\[h_{v'v}-i^2 = \\frac{1}{2}\\left(h_{\\left(i+1\\right)v} + h_{v'v}-\\left(i-1\\right)^2\\right)+1.\\]\n\nSolving for $h_{\\left(i+1\\right)v}$ gives the desired result.\n\nWhen $k$ is odd, by a symmetry argument, $h_{v'v} = h_{v''v}$.  Therefore, by Theorem \\ref{eq}, if $l = 1$, then we have that\n\n\\[h_{v'v} = \\frac{1}{2}\\left(h_{v''v}+h_{wv}\\right)+1,\\]\n\nand therefore $h_{wv} = h_{v'v}-2$.\n\nThe rest of the proof uses an induction argument identical to that for when $k$ is even.\n\\end{proof}\n\nFinally, we can take advantage of the two characterizations to solve for $h_{uv}$.  First, consider the vertex on both the cycle, and the line, $c$.  The above lemmas show that the neighbors of $c$ have hitting times of $h_{cv}+k-1, h_{cv}+k-1$, and $h_{cv}-h_{uv}+2\\left(l-1\\right)$.  Again, by Theorem \\ref{eq}, we have that\n\n\\[h_{cv} = \\frac{1}{3}\\left(h_{cv}+k-1\\right)+\\frac{1}{3}\\left(h_{cv}+k-1\\right)+\\frac{1}{3}\\left(h_{cv}-h_{uv}+2\\left(l-1\\right)\\right)+1.\\]\n\nSolving for $h_{uv}$ gives $h_{uv} = 2k+2l-1$, as desired.\n\n\\end{proof}\n\n\n\\subsection{Application to complete $d$-ary trees}\nIn other cases, we can use Theorem \\ref{eq}, to find hitting times between vertices that are not neighbors.  Consider the complete $d$-ary tree.  \n\nTo describe the hitting times between two vertices in a complete $d$-ary tree, we start by describing the hitting time from any vertex to the root.  By symmetry arguments, it follows that the hitting time depends only on the distance away from the root.  To aid in describing these hitting times, we define the following polynomial\n\n\\[f_n(d) =  \\left\\{\n     \\begin{array}{lr}\n       0 & n=0\\\\\n       \\left(\\displaystyle\\sum_{i=0}^{n-1}\\left(2n-2i\\right)d^i\\right)-n & \\text{o.w.}\n     \\end{array}\n   \\right.\\]\n\nfor nonnegative integers $n$.\n\nNow, we claim the following\n\n\\begin{lem}\\label{f}\nThe hitting time for a vertex of distance $l$ away from the root to the root in a complete $d$-ary tree of height $h$ is\n\n\\[f_h\\left(d\\right)-f_{h-l}\\left(d\\right).\\]\n\\end{lem}\n\n\\begin{proof}\n\nFor convenience we will let $f_h$ denote $f_h(d)$.  By Theorem \\ref{eq}, it is sufficient to show both\n\n\\[f_h-f_{0} = 1+f_h-f_1\\]\n\nand\n\n\\[f_h-f_{h-l} = 1+\\frac{1}{d+1}\\left(f_h-f_{h-l+1}\\right) + \\frac{d}{d+1}\\left(f_h-f_{h-l-1}\\right)\\]\n\nwhen $l < h$, as these describes all possible equations.  Because $f_1 = 1$, the first equation holds.\n\nSimplifying the second equation, we see that\n\n\\[\\left(d+1\\right)\\left(f_{h-l}+1\\right) = d\\left(f_{h-l-1}\\right)+f_{h-l+1}.\\]\n\nThe following holds for $n$ greater than $1$\n\n\\[f_n = df_{n-1}+\\left(n-1\\right)d+n.\\]\n\nWe can use this to see that\n\n\\[\\left(d+1\\right)\\left(f_{h-l}+1\\right) = f_{h-l}-\\left(h-l-1\\right)d+h-l-1+df_{h-1}+\\left(h-l\\right)d+h-1.\\]\n\n\\end{proof}\n\nTo continue finding hitting times we calculate the hitting time from each ancestor of a leaf, to that leaf.  To do so, we define the following polynomial\n\n\\[g_{k, m}\\left(d\\right) =  \\left\\{\n     \\begin{array}{lr}\n       0 & m=0\\\\\n       \\left(\\displaystyle\\sum_{i=0}^{m-1}\\left(2m-2i\\right)d^{k-i}\\right)-m & \\text{o.w.}\n\n     \\end{array}\n   \\right.\\]\n\nfor positive integers $k$ and nonnegative integers $m$.  We claim the following\n\n\\begin{lem}\\label{g}\nLet $v$ be a leaf in a complete $d$-ary tree of height $h$.  The hitting time from the ancestor of distance $l$ from $v$, to $v$, is\n\n\\[g_{h, h}\\left(d\\right)-g_{h, h-l}\\left(d\\right).\\]\n\\end{lem}\n\n\\begin{proof}\nIf a vertex is not an ancestor of $v$, then any random walk starting at that vertex must pass through an ancestor of $v$ before reaching $v$.  The hitting time can be calculated by adding the relevant hitting times given by Lemma \\ref{f} and this theorem.  Therefore, it is sufficient to only consider the equations that describe the hitting time from an ancestor to $v$.\n\nBecause the degree of the root of a complete $d$-ary tree is different from the degree of all other non-leaf vertices, we treat this case separately.\n\nThe neighbors of the root include one vertex that is an ancestor of $v$, and $d-1$ other vertices.  For the latter, any random walk originating at any of these vertices must pass through to the root in order to reach $v$.  Therefore by Theorem~\\ref{eq} and the previous lemma, it is necessary that the following holds\n\n\\[g_{h, h}-g_{h, 0} = 1+\\frac{1}{d}\\left(g_{h, h}-g_{h, 1}\\right)+\\frac{d-1}{d}\\left(g_{h, h}+f_h-f_{h-1}\\right).\\]\n\nBecause $g_{h, 0} =0$, this simplifies to\n\n\\[0 = d-g_{h, 1}+\\left(d-1\\right)\\left(f_h-f_{h-1}\\right).\\]\n\nIt can be seen that $\\left(d-1\\right)\\left(f_h-f_{h-1}\\right)+d = 2d^h-1$ for all positive $h$, and therefore we have that\n\n\\[g_{h, 1} = 2d^h-1,\\]\n\nwhich is true by our definition of $g$.\n\nWe continue for all other ancestors of $v$. Let the distance from an ancestor of $v$ to $v$ be $k$.  For these vertices, there are $d+1$ neighbors to consider: $1$ ancestor of $v$ closer to $v$, $1$ ancestor of $v$ farther away, and $d-1$ additional neighbors.  Again, any random walk starting from any vertex in the latter group must pass through this ancestor to reach $v$.  It is necessary that the following holds\n\n\\[g_{h, h}-g_{h, k} = 1+\\frac{1}{d+1}\\left(g_{h, h}-g_{h, k+1}\\right)+\\frac{1}{d+1}\\left(g_{h, h}-g_{h, k-1}\\right)+\\frac{d-1}{d+1}\\left(g_{h, h}-g_{h, k}+f_{h-k}-f_{h-k-1}\\right)\\]\n\n\\[g_{h, k+1}+g_{h, k-1}-2g_{h, k} = d+1+\\left(d-1\\right)\\left(f_{h-k}-f_{h-k-1}\\right).\\]\n\nNote that $g_{h, k+1}+g_{h, k-1}-2g_{h, k} = 2d^{h-k}$, as each term in $g_{h, k}$ is the average of the corresponding terms in $g_{h, k+1}$ and $g_{h, k-1}$, except for the $d^{h-k}$ term.  The right hand side can be simplified using the same identity used in the previous lemma.  Therefore, we get\n\n\\[2d^{h-k} = 2d^{h-k}-1+1,\\]\n\nwhich holds true.\n\nWe have shown that all equations in Theorem \\ref{eq} referring to the hitting time from an ancestor of $v$ to $v$ hold.  As stated previously, this is enough to prove the lemma.\n\\end{proof}\n\nThis gives all the tools needed to find the hitting time from any vertex to any other vertex in a complete $d$-ary tree.\n\n\\begin{thm}\nLet $u$ and $v$ be two vertices in a complete $d$-ary tree, with a least common ancestor of $c$.  Let $u'$, $v'$ and $c'$ be the distances of $u$, $v$, and $c$ respectively, to the root, and let $h$ be the height of the tree.  Then the hitting time from $u$ to $v$ is\n\n\\[f_{h-c'}\\left(d\\right)-f_{h-u'}\\left(d\\right)+g_{h, v'}\\left(d\\right)-g_{h, c'}\\left(d\\right).\\]\n\\end{thm}\n\n\\begin{proof}\nLet $l$ be an arbitrary leaf descended from $v$.  Then any random walk from $u$ to $l$ must also go through $c$ and $v$.  Therefore,\n\n\\[h_{ul} = h_{uc}+h_{cl} = h_{uv}+h_{vl}.\\]\n\nThen, we can express $h_{uv}$ as follows\n\n\\[h_{uv} = h_{uc}+h_{cl}-h_{vl}.\\]\n\nLemma \\ref{f} gives the value of $h_{uc}$, and Lemma \\ref{g}, gives the values of both $h_{cl}$ and $h_{vl}$.  Replacing these values into the above equation gives the formula for hitting times as stated.\n\\end{proof}\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nOver the past three decades, successful video compression technologies have been following the similar {\\it hybrid block-based transform and motion-compensation} framework with handcrafted coding tools, such as recursive block-size, directional intra prediction, discrete cosine transform (DCT), interpolation, context-adaptive entropy coding, etc, resulting in several well-known international standards, e.g., HEVC~\\cite{HEVC} and emerging versatile video coding (VVC)~\\cite{VVC_spec}. All of these and other technical explorations in video compression are trying to exploit and remove signal redundancy using ``causal priors'', e.g, reconstructed neighbor pixels, previous frames, context probability of neighbors, in video content, spatially, temporally and statistically~\\cite{Gary_VideoCompressionBasics}, for better compact representation at the same quality.\n\nMotivated by the recent advances in deep learning, a variety of deep neural network (DNN) based image\/video compression methods were developed via end-to-end learned (not handcrafted) coding tools~\\cite{balle2018variational,Liu_2018_CVPR_Workshops, rippel2017real,mentzer2018conditional,chen2017deepcoder,lu2018dvc, rippel2018learned}. Either conventional or recent emerging learning based video compressions are primarily trying to exploit the correlations between existing priors and ``pixels-to-be-coded''. In addition to previously reconstructed pixels and context probabilities used in traditional video coding methods,  DNN solutions could also generate hyperpriors in feature domain for better prediction.\n\n{In this work, we have presented a {\\it neural video compression} (NVC) framework in Fig.~\\ref{fig:arch}, leveraging the end-to-end learning to generate latent features for compact representation of spatial intra texture, temporal motion and statistical context probability. Joint spatio-temporal priors have been used extensively to improve the compression efficiency, for example, 1) spatial priors for both conditional probability modeling and reconstruction of intra texture (cf. Fig.~\\ref{sfig:intra_residual});  2) temporal priors for frame reconstruction (cf. Fig.~\\ref{sfig:inter_coding}); and 3) joint spatio-temporal priors for temporal predictive residual encoding (cf. Fig.~\\ref{sfig:intra_residual}) (e.g., context probability) and reconstruction.}\n\n{Spatial priors are generated using the low-resolution (e.g., via aggregated downscaling) representations from the same image content, while temporal priors are provided using the ConvLSTM~\\cite{gers2001long,xingjian2015convolutional} to capture the long-short dependency of previously processed frames.\nTemporal motion representation often plays an important role for video compression. Traditional methods adopt straightforward but effective variable block size based motion estimation to exploit the temporal correlations. But, in this work, we have turned to more fundamental {\\it optical flow} for motion description instead.}\n\n\\begin{figure*}[t]\n     \\centering\n    \n     \\subfigure[]{\\includegraphics[scale=0.36]{Fig1a-architecture.pdf} \\label{sfig:overall}}\\\\\n     \\subfigure[]{\\includegraphics[scale=0.34]{Fig1b-intra-residual.pdf} \\label{sfig:intra_residual}}\n     \\subfigure[]{\\includegraphics[scale=0.34]{Fig1c-inter.pdf} \\label{sfig:inter_coding}}\n     \\caption{Neural video compression framework: (a) overall architecture with neural intra, inter and residual coding; (b)  intra and residual coding; (c) inter coding with flow processing and temporal recurrent update.}\n     \\label{fig:arch}\n\\end{figure*}\n\n\n\nWe have evaluated our NVC for a low-delay, e.g., IPPP, coding structure, where except the first frame is encoded as an intra frame, all the rest frames are inter-coded with unidirectional forward prediction\\footnote{Bidirectional prediction is deferred as our future study.}. Performance comparisons have been carried out with well-known H.264\/AVC High Profile (HP)~\\cite{H264AVC} and HEVC MP~\\cite{HEVC}, using industry leading x264 (\\url{https:\/\/git.videolan.org\/git\/x264.git}) and x265 (\\url{http:\/\/x265.org\/}). For fairness, both x264 and x265 are constrained with IPPP low-delay encoding configuration with the other parameters remained as default.  Among those standard common test video sequences, our NVC has demonstrated superior coding efficiency over both H.264\/AVC HP and HEVC MP, e.g., $\\approx$38\\% BD-Rate~\\cite{bjontegaard2001calculation} improvements against the HEVC MP. Note that distortion measurement used in evaluation is the MS-SSIM~\\cite{wang2003multiscale} presented in decibel (dB) scale.\n\n{}\n\n\n\n\n\n\n\n\n\n\n\\section{Neural Video Compression: From Model-Driven to Data-Driven Solution}\n\\label{sec:nvc}\n\nOur NVC has attempted to define a way for efficient video compression through data-driven learning, rather than the traditional model-driven coding tool (e.g., transform model, motion model, etc) development. Details will be unfolded in subsequent paragraphs.\n\n\n\n\\subsection{Neural Intra Coding}\nNeural intra (texture) coding has tried to exploit the correlation within current video frame.\nAs shown in Fig.~\\ref{sfig:intra_residual}, we utilize a variational autoencoder with embedded hyperpriors for high-efficiency image coding. Similar architecture is also used in~\\cite{balle2018variational}.  We use deep residual learning (ResNet)~\\cite{he2016deep} with generalized divisive normalization (GDN) transform based activation, but not conventional rectified linear unit (ReLU), for fast convergence in training and more compact latent features, in both encoder network $\\mathbb E$ and decoder network $\\mathbb D$. Convolution with parametric ReLU is used for hyper encoder and decoder. Quantization $Q$ is approximated by adding uniform noise in training, but carried out with {\\tt ROUND}($\\cdot$) operation in inference.\n\n\nAn accurate probability distribution model of quantized features is the key for high-efficiency compression, not only for the arithmetic encoding (AE), but also for the rate estimation of rate-distortion optimization~\\cite{Gary_RDO} and bit allocation.  We jointly leverage the autoregressive information (i.e., distribution of quantized features) and hyperpriors (i.e., distribution of decoded hyper-features) for conditional context probability $P$ modeling with accurate mean ($\\mu$)  and variance ($\\sigma$) prediction assuming a Gaussian distributed probability mass function, i.e.,\n\\begin{align}\n  (\\mu,\\sigma) = {\\prod_i} p\\left(\\hat{x}_i|\\hat{x}_1,..., \\hat{x}_{i-1},{\\hat{\\bf z}_t}\\right).\n  \\label{eq:intra_feature}\n\\end{align}\nHere,  $\\hat{x}_1, {\\hat {x}}_2,..., \\hat{x}_{i-1}$ denote the causal (and possibly reconstructed) pixels prior to current pixel $\\hat{x}_i$ and $\\hat{\\bf z}_t$ are the hyperpriors, for image $\\hat{\\bf X}_t$. Probability of each pixel symbol can be simply derived using\n\\begin{align}\n  p_{\\hat{x}|\\hat{x}_1,..., \\hat{x}_{i-1},\\hat{\\bf z}_t}&(\\hat{x}|\\hat{x}_1,..., \\hat{x}_{i-1},\\hat{\\bf z}_t) \\nonumber \\\\\n  &\\mbox{~~~~}  = {\\prod_i} (\\mathcal{N(\\mu,\\sigma)} *\\mathcal{U}(-\\frac{1}{2},\\frac{1}{2})) (\\hat{x}_i).\n  \\label{eq:intra_texture_dist}\n \\end{align}\n\nIn addition to use hyperpriors $\\hat{\\bf z}_t$ for conditional probability improvement, we have developed an Information Compensation Network (ICN) to fuse and concatenate ({\\bf cat} operation in Fig.~\\ref{sfig:intra_residual}) decoded hyperpriors with latent features for better reconstruction.\n\n\n\n\n\n\n\n\n\n\\subsection{Neural Inter Coding} \\label{ssec:inter_coding}\nThe key issues for improving the efficiency of temporal inter coding are two folds. One is to accurately represent the motion between consecutive frames, and the other is to have high-fidelity reconstructions for compensation.\n\n\n\nFirst, we use optical flow $f_t$ for accurate motion representation that are learned between consecutive frames, e.g., ${\\bf X}_t$ and $\\hat{\\bf X}_{t-1}$, as shown in Fig.~\\ref{sfig:inter_coding}. A compressed flow representation $\\hat{f}_t$ is encoded into the bitstream for delivery.  For compensation, $\\hat{f}_t$ is then decoded into $\\hat{f}^d_t$ for warping with reference frame to have ${\\bf X}^w_t$, i.e.,\n\\begin{align}\n  \\hat{f}^d_t &= {\\mathbb F}_d\\left({\\mathbb F}_e\\left(\\hat{\\bf X}_{t-1},{\\bf X}_t\\right)\\right), \\label{eq:decoded_flow}\\\\\n  {\\hat{\\bf X}}^w_t &= {\\bf warp}\\left(\\hat{\\bf X}_{t-1},\\hat{f}^d_t \\right). \\label{eq:warped_frame}\n\\end{align}\nHere ${\\mathbb F}_e$ and ${\\mathbb F}_d$ represent the cascaded optical flow encoder and decoder network~\\cite{ilg2017flownet}.\nTo avoid quantization induced motion noise, our flow network is first pre-trained with uncompressed frames ${\\bf X}_{t-1}$ and ${\\bf X}_t$. Then we replace the former one ${\\bf X}_{t-1}$  using the decoded reference frame ${\\hat{\\bf X}}_{t-1}$ as described in Eq.~\\eqref{eq:decoded_flow} and \\eqref{eq:warped_frame}. Note that we have directly utilized the decoded  flow $\\hat{f}^d_t$ for end-to-end training.\n\nOftentimes, warped frame ${\\hat{\\bf X}}^w_t$ suffers from poor quality due to noisy flow estimation,  unexpected object occlusion, etc.  To improve the quality of warped frames, we propose to apply a {\\it processing network} using ten residual blocks with embedded re-sampling to enlarge the receptive field, resulting in ${\\hat{\\bf X}}^{wp}_t$. Such methods have been used in denoising and deblurring applications to improve the quality of reconstruction.\n\nEven with {\\it processing network} included, we have observed that high frequency components are generally missing in ${\\hat{\\bf X}}^{wp}_t$. Motivated by~\\cite{jo2018deep} that uses learned multi-frame residual information to improve the super resolution quality, we have attempted to apply the temporal recurrent network that is based on the ConvLSTM to capture and augment the high frequency priors to derive the $\\tilde{\\bf X}_t$ for temporal prediction, i.e.,\n\\begin{align}\n  ({\\bf c}_t, {\\bf h}_t) = {\\rm ConvLSTM}({\\bf \\Delta}_t, {\\bf c}_{t-1}, {\\bf h}_{t-1}), \\label{eq:trn_priors}\n\\end{align} where ${\\bf c}_t, {\\bf h}_t$ is updated state  at $t$ slot with ${\\bf c}_{t-1}$ used as a memory gate, and ${\\bf h}_{t-1}$ as aggregated prior (i.e., probability, high frequency component, etc) update. ${\\bf \\Delta}_t$ is generally referred as the input feature vector. Here it is extracted features from reference frame $\\hat{\\bf X}_{t-1}$ in Fig.~\\ref{sfig:inter_coding}.\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[{KristenAndSara 1080p}]\n{\\includegraphics[scale=0.39]{Kristen1.eps}}\n\\subfigure[vidyo1 720p]{\\includegraphics[scale=0.39]{vidyo11.eps}}\n\\subfigure[BasketBallDrive 1080p]{\\includegraphics[scale=0.39]{Basketball1.eps}}\n\\subfigure[vidyo3 720p]{\\includegraphics[scale=0.39]{vidyo31.eps}}\n\\subfigure[Cactus 1080p]{\\includegraphics[scale=0.39]{Cactus1.eps}}\n\\subfigure[FourPeople 720p]{\\includegraphics[scale=0.39]{FourPeople1.eps}}\n\\caption{Illustration of the rate-distortion performance for six different sequences. Here we use $-10\\log_{10}(1-d)$ to represent raw MS-SSIM ($d$) in dB scale.\n}\n\\label{rd_curve}\n\\end{figure}\n\n\n\\subsection{Neural Residual Coding} \\label{ssec:residual_coding}\n\nTemporal residual coding shares the similar architecture as the intra coding shown in Fig.~\\ref{sfig:intra_residual}, but with an augmented ConvLSTM to capture the aggregated temporal priors $\\hat{\\bf h}_t$ additionally for residual probability model improvement. Here, ${\\bf \\Delta}_t$ in Eq.~\\eqref{eq:trn_priors} refers to the concatenated features in current frame $\\hat{\\bf Y}_t = {\\bf CAT}(\\hat{\\bf X}_t, \\rm{ICN}(\\hat{\\bf z}_t))$ in Fig.~\\ref{sfig:intra_residual}.\n\nWe assume the same Gaussian distribution for residuals. Thus, Eq.~\\eqref{eq:intra_feature} can be rewritten as\n\\begin{align}\n  (\\mu_r,\\sigma_r) = {\\prod_i} p(\\hat{r}_i| \\hat{r}_1,..., \\hat{r}_{i-1}, \\hat{\\bf z}_t, {\\bf h}_{t-1}), \\label{eq:residual_probability}\n\\end{align} with aggregated temporal priors for residual probability modeling. $\\hat{r}_i, i = 0, 1, 2, ...$ are pixels of residual frame $\\hat{\\bf r}_t$. We then can simply use the cumulative distribution function (CDF) to calculate the probability of each pixel:\n\\begin{align}\n  &p_{\\hat{r}|(\\hat{r}_1,..., \\hat{r}_{i-1},\\hat{\\bf z}_t, {\\bf h}_{t-1})}(\\hat{r}|\\hat{r}_1,..., \\hat{r}_{i-1},\\hat{\\bf z}_t, {\\bf h}_{t-1})  \\nonumber \\\\\n  &\\mbox{~~~~}= {\\prod_i} (\\mathcal{N}{(\\mu_r,\\sigma_r)} *\\mathcal{U}(-\\frac{1}{2},\\frac{1}{2})) (\\hat{r}_i).\n  \\label{eq:residual_dist}\n \\end{align}\n\n\n\n\n\n\n\n\\section{End-to-End Training Strategy}\n\\label{sec:end_to_end_training}\n\nIt is difficult to train multiple networks jointly on-the-fly for our NVC. Thus, we pre-train the intra coding and flow coding networks first, followed by the jointly training with pre-trained nets for an overall rate-distortion optimization~\\cite{Gary_RDO}, i.e.,\n\\begin{align}\n  L =  {\\frac{\\lambda_1}{n}{\\sum_{t=0}^{n}}{\\mathbb D}_1(\\hat{{\\bf X}}_t,{{\\bf X}_t})}\n    & + {\\frac{\\lambda_2}{n}{\\sum_{t=0}^{n}}{\\mathbb D}_2(\\hat{\\bf X}_t^{wp},{{\\bf X}_t})} \\nonumber\\\\\n    & +  R_s+{\\frac{1}{n-1}{\\sum_{t=1}^{n}}R_t}, \\label{eq:rdo}\n\\end{align} where ${\\mathbb D}_1$ is measured using MS-SSIM, and ${\\mathbb D}_2$ is the warp loss evaluated using $L$1 norm and total variation loss. $R_s$ represents the bit rate of intra frame and $R_t$ is the bit rate of inter frames including bits for residual and flow. Currently, $\\lambda_1$ and $\\lambda_2$ will be adapted according to the specified overall bit rate and bit consumption percentage of flow information in inter coding.\n\n\n\n\\begin{table}[t]\n  \\centering\n  \\caption{BD-Rate Gain of our NVC versus HEVC MP}\n  \\label{tab:BDRate}\n  \\begin{tabular}{|c|c|c|c|}\n    \\hline\n       Sequence & Resolution & FPS &  BD-Rate Gain \\\\\n    \\hline\n     KristenAndSara & 1920x1080 &  60 & -96.96\\%\\\\\n     \\hline\n     vidyo1 & 1280x720 & 60 & -54.32\\%\\\\\n    \\hline\n     BasketBallDrive & 1920x1080 & 50 & -12.07\\%  \\\\\n    \\hline\n     vidyo3 & 1280x720 & 60 & -49.15\\% \\\\\n    \\hline\n     Cactus & 1920x1080 & 50&  -2.27\\%\\\\\n     \\hline\n     FourPeople & 1280x720 & 60&  -13.92\\%\\\\\n    \\hline\n    Ave. & & & -38.12\\%\\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\nWe start at a learning rate (LR) of $10^{-4}$ and reduce it by half every 30 epochs.\nWe choose vimeo dataset~\\cite{xue2017video}  and randomly crop the data into 192{$\\times$}192 spatially, as our training set. To well balance the efficiency of temporal information learning and training memory consumption, we have enrolled 5 frames to train the video compression framework and shared the weights for the rest in the time domain.\n\n\\section{Experimental Studies}\nWe have evaluated our NVC framework using BD-Rate but with distortion measured using MS-SSIM in dB scale. All simulation candidates, i.e., NVC, H.264\/AVC HP, HEVC MP, use IPPP mode to encode the video data with the same GOP of 8 for fair comparison.\nWell recognized x264 and x265 softwares are used for H.264\/AVC HP and HEVC MP, respectively.\nSeveral standard test sequences in different content classes are tested, and results have shown that our NVC presents a noticeable BD-Rate improvement compared with traditional H.264\/AVC HP and HEVC MP as shown in Fig.~\\ref{rd_curve}. In the meantime, BD-Rate reduction compared with HEVC MP is calculated\n and shown in Table~\\ref{tab:BDRate}, where $\\approx$ 38.12\\% BD-Rate gain is reported on average.\n\n\nWe have also provided snapshots for original raw, NVC, H.264\/AVC HP and HEVC MP encoded frames, respectively in Fig.~\\ref{subjective_test}. At the similar quality with MS-SSIM close to 0.988, our NVC has demonstrated significant bit rate reduction compared with the H.264\/AVC HP and HEVC MP.\n\n\n\n\\begin{figure}[t]\n     \\centering\n     \\includegraphics[scale=0.23]{subjective_test1.pdf}\n     \\caption{Snapshots of performance evaluation  at the same quality ($r$ for bit rate at bits per pixel, and $d$ for MS-SSIM).}\n     \\label{subjective_test}\n\\end{figure}\n\n\n\\section{Conclusion}\n\\label{sec:print}\nWe proposed a neural video compression framework leveraging the spatio-temporal priors jointly, which outperforms the well-known H.264\/AVC and HEVC with noticeable BD-Rate gains (i.e., 38\\% on average), resulting in the state-of-the-art coding performance. Spatial priors are derived from the downscaled image representations and temporal priors are captured using a recurrent network. As for future studies, bidirectional temporal prediction is one of the primary focuses to further improve the compression efficiency. In the meantime, adaptive bit rate allocation among intra texture, flow motion and residual with optimal rate-distortion efficiency will be another exploration avenue.\n\n\n\n\n\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Dual of the original problem}\nHere, we construct the dual function $g(\\textbf{V}, \\textbf{S})$ corresponding to the Lagrangian in Eq.~4 and show that it provides a trivial bound on the performance metric $f[\\boldsymbol{\\Phi}]$ in problem~3. From Eq.~4, it follows that:\n\\begin{align}\\label{eq:min_lag_phi_orig}\n\\inf_{\\boldsymbol{\\Phi}} \\mathcal{L} = -f^\\star[\\textbf{S} - \\hat{\\mathrm{G}}_b^\\dagger \\textbf{V}] - 2\\text{Re}[\\langle \\textbf{V}, \\textbf{E}_\\text{inc}\\rangle] + 2\\text{Re}[\\langle \\textbf{V} - \\chi \\textbf{S}, \\textbf{E}\\rangle],\n\\end{align}\nwhere $f^\\star$ is the Fenchel dual of $f$ given by~\\cite{bertsekas1997nonlinear}\n\\begin{align}\\label{eq:frenchel}\nf^\\star[\\textbf{N}] = \\sup_{\\boldsymbol{\\Phi}} \\bigg(2\\text{Re}[\\langle \\textbf{N}, \\boldsymbol{\\Phi}\\rangle] - f[\\boldsymbol{\\Phi}]\\bigg).\n\\end{align}\nMinimizing Eq.~\\ref{eq:min_lag_phi_orig} over $\\textbf{E}$, we obtain:\n\\begin{align}\\label{eq:min_lag_phi_e_orig}\n\\inf_{\\textbf{E}, \\boldsymbol{\\Phi}} \\mathcal{L} =\n\\begin{cases}\n-f^\\star[\\textbf{S} - \\hat{\\mathrm{G}}_b^\\dagger \\textbf{V}] - 2\\text{Re}[\\langle \\textbf{V}, \\textbf{E}_\\text{inc}\\rangle] & \\text{if } \\textbf{V}(\\textbf{x}) = \\chi(\\textbf{x})\\textbf{S}(\\textbf{x}), \\ \\forall \\textbf{x} \\in \\Gamma \\\\\n-\\infty & \\text{otherwise}.\n\\end{cases}\n\\end{align}\nFinally, $g[\\textbf{V}, \\textbf{S}]$ can be obtained by minimizing Eq.~\\ref{eq:min_lag_phi_e_orig} over $\\chi \\in [\\chi_-, \\chi_+]$. Note that unless $\\textbf{V}(\\textbf{x}) = 0$ and $\\textbf{S}(\\textbf{x}) = 0, \\ \\forall \\textbf{x} \\in \\Gamma$, $\\chi(\\textbf{x})$ can always be chosen so as to give $g[\\textbf{V}, \\textbf{S}] = -\\infty$. Therefore:\n\\begin{align}\\label{eq:dual_orig}\ng[\\textbf{V}, \\textbf{S}] = \\begin{cases}\n-f^\\star[0] & \\text{if } \\textbf{V}(\\textbf{x}) = 0 \\text{ and } \\textbf{S}(\\textbf{x}) = 0, \\ \\forall \\textbf{x} \\in \\Gamma \\\\\n-\\infty & \\text{otherwise}.\n\\end{cases}\n\\end{align}\nAs described in the main text, $g[\\textbf{V}, \\textbf{S}]$ is a lower bound on the optimal solution $f_\\text{opt}$ of problem 3 for any choice of $\\textbf{V}$ and $\\textbf{S}$. Consequently, maximizing $g[\\textbf{V}, \\textbf{S}]$ over $\\textbf{V}$ and $\\textbf{S}$ gives the best lower bound on $f_\\text{opt}$. It immediately follows from Eqs.~\\ref{eq:dual_orig} and \\ref{eq:frenchel} that this is given by:\n\\begin{align}\\label{eq:sca_cs}\n\\sup_{\\textbf{V}, \\textbf{S}} g[\\textbf{V}, \\textbf{S}] = -f^\\star[0] = \\inf_{\\boldsymbol{\\Phi}} f[\\boldsymbol{\\Phi}].\n\\end{align}\nThis bound is simply the smallest value of the performance metric in problem~3 without accounting for its constraints --- it is consequently a trivial bound on $f_\\text{opt}$. It can be noted that for the problem of calculating an upper limit on the scattering cross-section that is considered in the main text, Eq.~\\ref{eq:sca_cs} indicates that the scattering cross-section is unbounded irrespective of the choice of design-region $\\Gamma$ and the contrast limits $\\chi_\\pm$.\n\\section{Dual of the field-constrained problem}\nHere, we outline a brief derivation of the dual problem of the field constrained problem. Minimizing the Lagrangian in Eq.~9 over $\\boldsymbol{\\Phi}(\\textbf{x})$, we obtain:\n\\begin{align}\\label{eq:lap_inf_phi}\n\\inf_{\\boldsymbol{\\Phi}}\\mathcal{L} = -f^\\star[\\textbf{S} - \\hat{\\textrm{G}}_b^\\dagger \\textbf{V}] + 2\\text{Re}\\big[\\langle \\textbf{V},  \\textbf{E} - \\textbf{E}_\\text{inc}\\rangle \\big] - 2\\text{Re}\\big[\\langle \\textbf{S}, \\chi\\textbf{E}\\rangle\\big]\n+ \\lambda(\\|\\textbf{E} - \\textbf{E}_\\text{ref}\\|^2 - \\alpha^2 \\|\\textbf{E}_\\text{ref}\\|^2).\n\\end{align}\nMinimizing Eq.~\\ref{eq:lap_inf_phi} over $\\textbf{E}(\\textbf{x})$, we obtain:\n\\begin{align}\\label{eq:lap_inf_phi_e}\n\\inf_{\\textbf{E}, \\boldsymbol{\\Phi}}\\mathcal{L} = 2\\text{Re}\\big[\\langle \\textbf{V}, \\textbf{E}_\\text{ref} - \\textbf{E}_\\text{inc}\\rangle \\big] -f^\\star[\\textbf{S} - \\hat{\\textrm{G}}_b^\\dagger \\textbf{V}]\n- \\int_\\Gamma \\frac{|\\textbf{V}(\\textbf{x}) - \\chi(\\textbf{x}) \\textbf{S}(\\textbf{x})|^2}{\\lambda}\\,d^3\\textbf{x}- 2\\text{Re}\\big[\\langle \\textbf{S}, \\chi\\textbf{E}_\\text{ref}\\rangle \\big].\n\\end{align}\nFinally, to calculate the dual function $g[\\textbf{V}, \\textbf{S}, \\lambda]$ by minimizing Eq.~\\ref{eq:lap_inf_phi_e} over $\\chi \\in [\\chi_-, \\chi_+]$, note that $\\inf_{\\textbf{E}, \\boldsymbol{\\Phi}}\\mathcal{L}$ is a concave function in $\\chi(\\textbf{x})$ and is separable at each point in the domain $\\Gamma$, so it will acquire its minimum value with respect to $\\chi(\\textbf{x})$ at either $\\chi_-$ or $\\chi_+ \\ \\forall \\ \\textbf{x}\\in \\Gamma$. Consequently,\n\\begin{align}\ng[\\textbf{V}, \\textbf{S}, \\lambda] = 2\\text{Re}\\big[\\langle \\textbf{V}, \\textbf{E}_\\text{ref} - \\textbf{E}_\\text{inc}\\rangle \\big] -f^\\star[\\textbf{S} - \\hat{\\textrm{G}}_b^\\dagger \\textbf{V}] - \\int_\\Gamma \\beta(\\textbf{x})\\,d^3\\textbf{x},\n\\end{align}\nwhere\n\\begin{align}\n\\beta(\\textbf{x}) = \\max_{\\chi \\in \\{\\chi_+, \\chi_-\\}}  \\bigg[\\frac{|\\textbf{V}(\\textbf{x}) - \\chi\\textbf{S}(\\textbf{x})|^2}{\\lambda} - 2\\chi\\text{Re}\\big(\\textbf{S}^*(\\textbf{x}) \\cdot \\textbf{E}_\\text{ref}(\\textbf{x})\\big)\\bigg].\n\\end{align}\nSince $g[\\textbf{V}, \\textbf{S}, \\lambda]$ provides a bound on $f_\\text{opt}$ for any choice of $\\textbf{V}, \\textbf{S}$ and $\\lambda \\geq 0$~\\cite{boyd2004convex, bertsekas1997nonlinear}, the best lower bound is then obtained by maximizing it over all the dual variables. This immediately yields the convex optimization problem in the main text (problem~10). Except for an explicit introduction of the polarization current $\n\\boldsymbol{\\Phi}(\\textbf{x})$ and the field constraint (Eq.~8), the calculation of the dual problem outlined in this section is similar to that presented in~\\cite{angeris2019computational}.\n\\section{Calculation of the field constraint}\n\\noindent Here we show that if $\\textbf{E}_\\text{ref}(\\textbf{x}) = (\\hat{\\text{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\textbf{E}_\\text{inc}(\\textbf{x})$, then $\\alpha_\\text{ub}$ provided in Eq.~12 provides an upper bound on $\\|\\textbf{E} - \\textbf{E}_\\text{ref}\\|$ for all $\\textbf{E}(\\textbf{x})$ that are solutions of Eq.~1with $\\chi_- \\leq \\chi(\\textbf{x}) \\leq \\chi_+, \\ \\forall \\ \\textbf{x}\\in\\Gamma$.\n\nNote that Eq.~1 can be rewritten to obtain:\n\\begin{align}\\label{eq:vol_int_rewrite}\n\\textbf{E}(\\textbf{x}) - (\\hat{\\textrm{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\hat{\\textrm{G}}_b (\\chi(\\textbf{x}) - \\bar{\\chi}) \\textbf{E}(\\textbf{x}) = \\textbf{E}_\\text{ref}(\\textbf{x}).\n\\end{align}\nIt immediately follows from the reverse triangle inequality and Eq.~\\ref{eq:vol_int_rewrite} that:\n\\begin{align}\\label{eq:rev_trian_ineq}\n\\|\\textbf{E}\\| - \\|(\\hat{\\textrm{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\hat{\\textrm{G}}_b (\\chi - \\bar{\\chi}) \\textbf{E}\\| \\leq \\|\\textbf{E}_\\text{ref}\\|.\n\\end{align}\nFurthermore, $\\|(\\hat{\\textrm{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\hat{\\textrm{G}}_b (\\chi - \\bar{\\chi}) \\textbf{E}\\| \\geq \\|(\\hat{\\textrm{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\hat{\\textrm{G}}_b \\|\\cdot \\|(\\chi - \\bar{\\chi}) \\textbf{E}\\|$, where $\\|(\\hat{\\textrm{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\hat{\\textrm{G}}_b \\|$ is the operator norm of $(\\hat{\\textrm{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\hat{\\textrm{G}}_b $. Furthermore, since $\\chi_- \\leq \\chi(\\textbf{x})\\leq \\chi_+ \\ \\forall \\ \\textbf{x}\\in \\Gamma$, it follows that $|\\chi(\\textbf{x}) - \\bar{\\chi}| \\leq \\delta \\chi \\ \\forall \\ \\textbf{x}\\in \\Gamma$ where $\\delta \\chi = |\\chi_+ - \\chi_-|\/2$. Consequently, from Eq.~\\ref{eq:rev_trian_ineq}, we immediately obtain:\n\\begin{align}\n(1 - \\|(\\hat{\\textrm{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\hat{\\textrm{G}}_b \\| \\delta\\chi ) \\|\\textbf{E}\\| \\leq \\|\\textbf{E}_\\text{ref}\\|.\n\\end{align}\nClearly, if $\\|(\\hat{\\textrm{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\hat{\\textrm{G}}_b \\| \\delta\\chi  < 1$, it follows that $\\|\\textbf{E}\\|$ is bounded above by $\\|\\textbf{E}_\\text{ref}\\| \/ (1 - \\|(\\hat{\\textrm{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\hat{\\textrm{G}}_b \\| \\delta\\chi )$. From Eq.~\\ref{eq:vol_int_rewrite}, this bound can easily be translated to a bound on $\\|\\textbf{E} - \\textbf{E}_\\text{ref}\\|$:\n\\begin{align}\n\\|\\textbf{E} - \\textbf{E}_\\text{ref}\\| = \\|(\\hat{\\textrm{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\hat{\\textrm{G}}_b  (\\chi - \\bar{\\chi}) \\textbf{E} \\| \\leq \\frac{\\|(\\hat{\\textrm{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\hat{\\textrm{G}}_b  \\|\\delta\\chi \\|\\textbf{E}_\\text{ref}\\|}{1 - \\|(\\hat{\\textrm{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\hat{\\textrm{G}}_b  \\|\\delta\\chi }.\n\\end{align}\nIf $\\|(\\hat{\\textrm{I}} - \\bar{\\chi}\\hat{\\textrm{G}}_b)^{-1}\\hat{\\textrm{G}}_b \\| \\delta\\chi  \\geq 1$, then the procedure outlined above cannot be used to generate a bound on $\\|\\textbf{E} - \\textbf{E}_\\text{ref}\\|$.\n\\begin{figure}[htpb]\n\\centering\n\\includegraphics[scale=0.4]{conic_prog_convergence.pdf}\n\\caption{Solution of problem 9 with the objective function being the 2D transverse electric and transverse magnetic scattering cross-section for different discretization parameter $\\delta x$. The design region is assumed to be a circle of radius $0.1\\lambda$ embedded in vacuum and the contrast $\\chi_0 = 2.5$.}\n\\label{fig:conv_conic_program}\n\\end{figure}\n\n\\section{Numerical convergence of the dual problem}\n\\noindent In order to solve problem~10, all the involved vector fields need to be discretized within the device region $\\Gamma$. Furthermore, to ensure that the bounds produced by this problem are not numerical artifacts of the discretization, it is necessary to ensure that the bounds converge with respect to the discretization parameter. This is shown in Fig.~\\ref{fig:conv_conic_program} --- the discretization parameter $\\delta x$ refers to the length of the pixels used for representing the design region $\\Gamma$ for the 2D scattering problems using pulse basis and delta testing functions \\cite{peterson1998computational}.\n\n\n\\section{Divergence of field bound}\n\\noindent Figure~\\ref{fig:alpha_loc_conv} shows locally optimized solutions of problem~11, $\\alpha_\\text{loc}$, for a transverse magnetic scattering problem as a function of the discretization parameter $\\delta x$. We clearly see that for $\\chi_0 < -1$, $\\alpha_\\text{loc}$ does not converge on reducing $\\delta x$, while for $\\chi_0 > -1$ it converges.\n\n\\begin{figure}[htpb]\n\\centering\n\\includegraphics[scale=0.4]{convergence_tm_alpha_loc.pdf}\n\\caption{The field constraint $\\alpha_\\text{loc}$ obtained by locally solving problem 11 as a function of the discretization parameter $\\delta x$. The shaded regions indicate the range of local-optimas obtained by solving the non-convex problem with 50 different initial conditions. A circular design region with radius $0.1\\lambda$ embedded in vacuum is assumed in all calculations.}\n\\label{fig:alpha_loc_conv}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:1}\n\nIn this paper we develop and analyse a numerical scheme for  the following non-local interaction system with cross-diffusion and self-diffusion\n\\begin{align}\n\\label{eq:crossdiffsystem}\n\t\\left\\{\n\t\\begin{array}{r}\n\t\\ds\\frac{\\partial \\rho}{\\partial t} = \\ds\\frac{\\partial}{\\partial x}\\left(\\rho \\frac{\\partial}{\\partial x}\n        \\left(W_{11}\\star\\rho + W_{12}\\star \\eta + \\nu(\\rho+ \\eta)\\right)\\,+\\,\n        \\dfrac\\epsilon2 \\frac{\\partial \\rho^2}{\\partial x} \\right),\n\\\\[1.75em]\n\t\\ds\\frac{\\partial \\eta}{\\partial t}= \\ds\\frac{\\partial}{\\partial x}\\left(\\eta \\frac{\\partial}{\\partial x}\n       \\left( W_{22}\\star\\eta + W_{21}\\star \\rho + \\nu(\\rho+ \\eta)\n                                            \\right)  \\,+\\,\n        \\dfrac\\epsilon2 \\frac{\\partial\\eta^2}{\\partial x} \\right),\n\\end{array}\\right.\n\\end{align}\t\ngoverning the evolution of two species $\\rho$ and $\\eta$ on an\ninterval $(a,b) \\subset \\mathbb{R}$ for $t\\in[0,T)$. The system is equipped\nwith nonnegative initial data $\\rho^0,\\eta^0 \\in L_+^1(a,b) \\cap\nL_+^\\infty(a,b)$. We denote by $m_1$ the mass of $\\rho_0$ and by $m_2$ the mass of $\\eta_0$, respectively,\n$$\nm_1\\,=\\, \\int_a^b \\rho_0(x)\\,\\d x, \\quad \\text{and} \\quad\nm_2 \\,=\\, \\int_a^b \\eta_0(x) \\,\\d x.\n$$\n\nOn the boundary $x=a$ and $b$, we prescribe no-flux boundary conditions\n\\begin{align*}\n\\left\\{\n\\begin{array}{r}\n\\ds\\rho\\,\\frac{\\partial}{\\partial x} \\left(W_{11}\\star\\rho + W_{12}\\star \\eta +\n  \\nu(\\rho+ \\eta)+ \\epsilon \\rho\\right) = 0,\n\\\\\n\\,\\\\\n\\ds\\eta\\, \\frac{\\partial}{\\partial x} \\left(W_{22}\\star\\eta + W_{21}\\star \\rho + \\nu(\\rho+ \\eta) + \\epsilon \\eta\\right) = 0,\n\\end{array}\\right.\n\\end{align*}\nsuch that the total mass of each species is conserved with respect to\ntime $t\\geq 0$. While the self-interaction potentials $W_{11}, W_{22}\\in C_b^2(\\mathbb{R})$ \nmodel the interactions among individuals of the same species (also\nreferred to as \\emph{intraspecific} interactions), the\ncross-interaction potentials $W_{12},W_{21}\\in C_b^2(\\mathbb{R})$ encode the\ninteractions between individuals belonging to  different species,\n\\emph{i.e.} \\emph{interspecific} interactions. Here $C_b^2(\\mathbb{R})$ denotes the set of twice continuously differentiable functions on $\\mathbb{R}$ with bounded derivatives. Notice that the convolutions $W_{ij}\\star \\psi$, with $\\psi$ a density function defined on $[a,b]$, are defined by extending the density $\\psi$ by zero outside the interval $[a,b]$. The two positive parameters $\\epsilon, \\nu > 0$ determine the strengths of the\nself-diffusion and the cross-diffusion of both species, respectively. Nonlinear diffusion, be it self-diffusion or cross-diffusion, is biologically relevant. As a matter of fact, around the second half of the $20^{\\mathrm{th}}$ century biologists found that the dispersal rate of certain insects depend on the density itself, leading to the nonlinear diffusion terms we incorporated in the model, cf. \\cite{Mor50, Mor54, Ito52, Car71, GN75}. At the same time we would like to stress that the self-diffusion terms are relevant for the convergence analysis below.\n\n\n\n\n\n\n\n\n\nIt is the\ninterplay between the non-local interactions of both species and their\nindividual and joint size-exclusion, modelled by the non-linear\ndiffusion \\cite{BGHP85, BGH87, BGH87a, HPVolume, CCVolume, BBRW17},\nthat leads to a large variety of behaviours including complete phase\nseparation or mixing of both densities in both stationary\nconfigurations and travelling pulses \\cite{BDFS, CHS17}.\n\nWhile their single species counterparts have been studied quite\nintensively \\cite{MEK99, CMcCV03, TBL06, CCH14} and references therein,  related two-species models like the system of our interest, Eq. \\eqref{eq:crossdiffsystem}, have only recently gained considerable attention \\cite{DFF13, BDFS, CHS17, DFEF17, CFSS17}. One of the most striking phenomena of these interaction models with cross-diffusion is the possibility of phase separation. Since the seminal papers \\cite{GP84, BGHP85} established segregation effects for the first time for the purely diffusive system corresponding to \\eqref{eq:crossdiffsystem} for $W_{ij}\\equiv 0$, $i,j\\in\\{1,2\\}$ and $\\epsilon=0$, many generalisations were presented. They include reaction-(cross-)diffusion systems \\cite{BDPM10, BHIM12, CFSS17} and references therein, and by adding non-local interactions \\cite{BDFS, CHS17, DFEF17, BBP17} and references therein. Ref. \\cite{DFEF17} have established the existence of weak solutions to a class of non-local systems under a strong coercivity assumption on the cross-diffusion also satisfied by system \\eqref{eq:crossdiffsystem}.\n\nTypical applications of these non-local models comprise many\nbiological contexts such cell-cell adhesion  \\cite{PBSG, MT15,CCS17}, for\ninstance, as well as tumour models \\cite{GerCha, DTGC},\nbut also the  formation of the characteristic stripe patterns of\nzebrafish can be modelled by these non-local models \\cite{VS}. Systems of this kind are truly ubiquitous in nature and we remark that `species' may not only refer to biological species but also to a much wider class of (possibly inanimate) agents such as planets, physical or chemical particles, just to name a few.  \n\nSince system \\eqref{eq:crossdiffsystem} is in conservative form a finite volume scheme is a natural choice as a numerical method. This is owing to the fact that, by construction, finite volume schemes are locally conservative: due the divergence theorem the change in density on a test cell has to equal the sum of the in-flux and the out-flux of the same cell. There is a huge literature on finite volume schemes, first and foremost  \\cite{EGH00}. Therein, the authors give a detailed description of the construction of such methods and address convergence issues. Schemes similar to the one proposed in Section \\ref{sec:numerical_scheme} have been studied in \\cite{BCF12} in the case of nonlinear degenerate diffusion equations in any dimension. A similar scheme for a  system of two coupled PDEs was proposed in \\cite{CHF07}. Later, the authors in \\cite{CCH15} generalised the scheme proposed in \\cite{BCF12} including both local and non-local drifts. The scheme was then extended to two species in \\cite{CHS17}. All the aforementioned schemes have in common that they preserve nonnegativity -- a property that is also crucial for our analysis.\n\nBefore we define the finite volume scheme we shall present a formal energy estimate for the continuous system. The main difficulty in this paper is to establish positivity  and reproducing the continuous energy estimate at the discrete level.\nThe remainder of the introduction is dedicated to presenting the aforementioned energy estimate. Let us consider\n\\begin{align*}\n\\ds\t\\frac{\\d}{\\d t} \\int_a^b \\rho \\log \\rho \\,\\d x &= \\ds\\int_a^b\n                                                         \\log \\rho\n                                                         \\,\\frac{\\partial\n                                                         \\rho}{\\partial t}  \\,\\d x\\\\\n\t&=\\ds\\int_a^b \\log \\rho \\,\\frac{\\partial}{\\partial x}\n            \\left(\\rho \\frac{\\partial}{\\partial x}\\left( W_{11}\\star\\rho + W_{12}\\star\n            \\eta + \\nu(\\rho+ \\eta)+ \\epsilon \\rho\\right)\\right) \\d x\\\\\n\t&= \\ds- \\int_a^b \\rho\\, \\frac{\\partial}{\\partial x}\\left(W_{11}\\star\\rho + W_{12}\\star \\eta + \\nu(\\rho+ \\eta)+ \\epsilon \\rho\\right)\\, \\frac{\\partial}{\\partial x}(\\log \\rho) \\,\\d x.\n\\end{align*}\nUpon rearranging we get\n\\begin{align*}\n\t\\frac{\\d }{\\d t} \\int_a^b \\rho\\log\\rho \\,\\d x \\,+\\, \\nu \\, \\int_a^b \\frac{\\partial}{\\partial x} (\\rho +\\eta)\\frac{\\partial\\rho}{\\partial x}\\,\\d x  \\,+\\, \\epsilon \\int_a^b \\left|\\frac{\\partial\\rho}{\\partial x}\\right|^2\\,\\d x \\,=\\, \\int_a^b (W_{11}'\\star\\rho + W_{12}'\\star \\eta)  \\, \\frac{\\partial\\rho}{\\partial x}\\,\\d x.\n\\end{align*}\nA similar computation for $\\eta$ yields\n\\begin{align*}\n\t\\frac{\\d }{\\d t} \\int_a^b \\eta\\log\\eta \\,\\d x \\,+\\, \\nu \\, \\int_a^b \\frac{\\partial}{\\partial x} (\\rho +\\eta)\\, \\frac{\\partial\\eta}{\\partial x}\\,\\d x  \\,+\\, \\epsilon \\, \\int_a^b \\left|\\frac{\\partial\\eta}{\\partial x} \\right|^2\\,\\d x \\,=\\, \\int_a^b (W_{22}'\\star\\eta + W_{21}'\\star \\rho) \\, \\frac{\\partial\\eta}{\\partial x}  \\,\\d x,\n\\end{align*}\nwhence, upon adding both, we obtain\n$$\n\\begin{array}{rcl}\n\\ds\t\\frac{\\d }{\\d t} \\int_a^b  \\left[\\rho\\log\\rho+\\eta\\log\\eta\\right] \\,\\d x\n  \\,+\\,\\ds \\nu\\int_a^b\\left|\\frac{\\partial \\sigma}{\\partial x} \\right|^2 \\d\n  x \\,+\\,\n  \\epsilon\\,\\int_a^b \\left( \\left|\\frac{\\partial\\rho}{\\partial x} \\right|^2 \\,+\\, \\left|\\frac{\\partial\\eta}{\\partial x} \\right|^2 \\right)\\,\\d\n  x \\,=\\,\\ds \\mathcal{D}_\\rho + \\mathcal{D}_\\eta,\n\\end{array}\n$$\nwhere $\\sigma = \\rho + \\eta$ and\n$$\n\\left\\{\n\t\\begin{array}{l}\n\t\t\\mathcal{D}_\\rho := \\ds\\int_a^b (W_{11}'\\star\\rho + W_{12}'\\star\n  \\eta) \\,\\frac{\\partial\\rho}{\\partial x}  \\, \\d x, \\\\[1em]\n  \t\t\\mathcal{D}_{\\eta} :=\\ds \\int_a^b (W_{22}'\\star\\eta + W_{21}'\\star \\rho) \\, \\frac{\\partial\\eta}{\\partial x} \\,  \\d x,\n\t\\end{array}\n\\right.\n$$\ndenote the advective parts associated to $\\rho$ and $\\eta$, respectively. The advective parts can be controlled by using the weighted Young inequality to get\n\\begin{align*}\n\\ds\t|\\mathcal{D}_\\rho|&= \\ds\\left|\\int_a^b (W_{11}'\\star\\rho + W_{12}'\\star \\eta) \\frac{\\partial\\rho}{\\partial x}  \\, \\d x\\right|\\\\[1em]\n&\\leq \\ds\\frac{1}{2\\alpha}\\int_a^b {|W_{11}'\\star\\rho + W_{12}'\\star \\eta|^2} \\, \\d x\\,+\\,\\frac{\\alpha}{2}\\int_a^b\\left| \\frac{\\partial\\rho}{\\partial x}  \\right|^2 \\, \\d x,\t\t\n\\end{align*}\nfor some $\\alpha>0$. In choosing $0<\\alpha<\\epsilon$ we obtain\n\\begin{align}\n\t\\label{eq:cts_inequality}\t\n\t\\frac{\\d }{\\d t} \\int_a^b  \\left[\\rho\\log\\rho+\\eta\\log\\eta\\right] \\,\\d x\n  \\,+\\, \\nu \\int_a^b \\left|\\frac{\\partial \\sigma}{\\partial x} \\right|^2\\,\\d x \\,+\\, \\left(\\epsilon-\\frac\\alpha2\\right)\\,\\int_a^b \\left(\\left|\\frac{\\partial\\rho}{\\partial x}\\right|^2\\,+\\,\\left|\\frac{\\partial\\eta}{\\partial x}\\right|^2\\right)\\,\\d x \\,\\leq\\, \\frac{C_\\rho + C_\\eta}{2\\alpha},\n\\end{align}\nwhere $C_\\rho = \\|W_{11}'\\star\\rho + W_{12}'\\star \\eta\\|_{L^2}^2$ and $C_\\eta = \\|W_{22}'\\star\\eta + W_{21}'\\star \\rho\\|_{L^2}^2$.\nFrom the last line, Eq. \\eqref{eq:cts_inequality}, we may deduce bounds on the gradient of each species as well as on their sum. As mentioned above the crucial ingredient for this estimate is the positivity of solutions.\\\\\n\n\nThe rest of this paper is organised as follows. In the subsequent section we present a semi-discrete finite volume approximation of system \\eqref{eq:crossdiffsystem} and we present the main result, Theorem \\ref{thm:convergence}. Section \\ref{sec:apriori_estimates} is dedicated to establishing positivity and to the derivation of a priori estimates. In Section \\ref{sec:convergence} we obtain compactness, pass to the limit, and identify the limiting functions as weak solutions to system \\eqref{eq:crossdiffsystem}. We conclude the paper with a numerical exploration in Section \\ref{sec:numerical_results}. We study the numerical order of accuracy and discuss stationary states and phase segregation phenomena.\n\n\n\\section{Numerical scheme and main result}\n\\label{sec:numerical_scheme}\nIn this section we introduce the semi-discrete finite volume scheme for system \\eqref{eq:crossdiffsystem}. To begin with, let us introduce our notion of weak solutions.\n\\begin{definition}[Weak solutions.] \n\\label{def:WeakSolution}\nA couple of functions $(\\rho,\\eta)$ $\\in $ $L^2(0,T;H^1(a,b))^2$ is a\nweak solution to system \\eqref{eq:crossdiffsystem} if it satisfies\n\\begin{subequations}\n\t\\begin{align}\n\t\\begin{split}\n\t\t-\\int_a^b \\rho_0 \\,\\varphi(0,\\cdot) \\,\\d x \\,=\\! \\int_0^T\n                \\int_a^b \\left[\\rho\\, \\left(\\frac{\\partial \\varphi}{\\partial t}  \\,+\\, \\left(-\\nu \\frac{\\partial \\sigma}{\\partial x}\n\t\t\\,+\\,\\frac{\\partial V_1}{\\partial\n                  x} \\right)\\,\\frac{\\partial \\varphi}{\\partial x} \\right) \\,+\\,\\frac{\\epsilon}{2}\\, \\rho^2\\, \\frac{\\partial^2 \\varphi}{\\partial x^2} \\right]\\,\\d x \\,\\d t,\n\t\\end{split}\n\t\\end{align}\n\tand\n\t\\begin{align}\n\t\\begin{split}\n\t\t-\\int_a^b \\eta_0 \\,\\varphi(0,\\cdot) \\, \\d x \\,=\\,\\! \\int_0^T\n                \\int_a^b \\left[\\eta \\left(\\frac{\\partial \\varphi}{\\partial t}\n                  + \\left(-\\nu \\frac{\\partial \\sigma}{\\partial x}  \\,+\\,\n                    \\frac{\\partial V_2}{\\partial x} \\right)\\,\n                  \\frac{\\partial \\varphi}{\\partial x} \\right) \\,+\\, \\frac{\\epsilon}{2}\\,\\eta^2 \\, \\frac{\\partial^2 \\varphi}{\\partial x^2} \\right]\\,\\d x \\d t,\n\t\\end{split}\n\t\\end{align}\n\\end{subequations}\nrespectively, for any $\\varphi\\in C_c^\\infty([0,T)\\times(a,b); \\mathbb{R})$. Here we have set $V_k = - W_{k\\, 1} \\star \\rho - W_{k \\, 2} \\star \\eta$, for $k\\in\\{1,2\\}$, and $\\sigma = \\rho + \\eta$, as above.\n\\end{definition}\n\nNotice that the existence of weak solutions to system\n\\eqref{eq:crossdiffsystem} will follow directly from the convergence\nof the numerical solution. Indeed, our analysis relies on a compactness\nargument which does not suppose a priori existence of solution to system \\eqref{eq:crossdiffsystem}.\n\n\nTo this end we first define the following space discretisation of the domain.\n\\begin{definition}[\\label{def:spacediscretisation}Space discretisation]\n\tTo discretise space, we introduce the mesh \n\t\\begin{align*}\n\t\t\\mathcal{T}:\t= \\bigcup_{i\\in I} C_i,\n\t\\end{align*}\n\twhere the control volumes are given by $C_i=[x_\\imhalf, x_\\ihalf)$ for all $i \\in I:=\\{1,\\ldots, N\\}$. We assume that the measure of the control volumes are given by $|C_i| = \\Delta x_i = x_{i+1\/2} - x_{i-1\/2}>0$, for all $i\\in I$. Note that $x_{1\/2}=a$, and $x_{N+1\/2}=b$.\n\\end{definition}\n\\begin{figure}[ht!]\n\t\\centering\n\t\\begin{tikzpicture}\n\t\n\t\t\\draw[-] (1,0) -- (9,0);\n\t\n\t\t\\foreach \\x in {1,...,4}\n     \t\t\\draw[-, thick] (2*\\x,-0.2) -- (2*\\x,0.2);\n\n\t\n     \t\\draw (2,-0.1) node[below] {$x_{i-3\/2}$};\n     \t\\draw (4,-0.1) node[below] {$x_{i-1\/2}$};\n     \t\\draw (6,-0.1) node[below] {$x_{i+1\/2}$};\n     \t\\draw (8,-0.1) node[below] {$x_{i+3\/2}$};\n\n\n\n\t\n\t\t\\foreach \\x in {1,...,3}\n     \t\t\\draw[-] (2*\\x+1,-0.1) -- (2*\\x+1,0.1);\n\n\t\n     \t\\draw (3,-0.05) node[below] {$x_{i-1}$};\n     \t\\draw (5,-0.05) node[below] {$x_{i}$};\n     \t\\draw (7,-0.05) node[below] {$x_{i+1}$};\n\n\t\t\\draw[decorate,decoration={brace}] ($(4+0.05,0.4)$) -- node[above]{$C_i$} ++(1.9,0);\n\t\t\n\t\t\\draw[decorate,decoration={brace}] ($(2+0.05,0.4)$) -- node[above]{$C_{i-1}$} ++(1.9,0);\n\t\t\n\t\t\\draw[decorate,decoration={brace}] ($(6.05,0.4)$) -- node[above]{$C_{i+1}$} ++(1.9,0);\n\t\t\n\t\n\t\t\\draw[-] (-0.5,0) -- (0.5,0);\n\t\t\\draw[-, dotted, thick] (0.5,0) -- (1,0);\n\t\t\\draw [fill=black] (0,0) circle [radius=0.05];\n\t\t\\draw (0,-0.05) node[below] {$x_{1\/2} = a$};\n\t\t\n\t\t\n\t\t\\draw[-, dotted, thick] (9,0) -- (9.5,0);\n\t\t\\draw[->] (9.5,0) -- (10.5,0);\n\t\t\\draw [fill=black] (10,0) circle [radius=0.05];\n\t\t\\draw (10,-0.05) node[below] {$x_{N+1\/2} = b$};\n\t\\end{tikzpicture}\n\t\\caption{Space discretisation according to Definition \\ref{def:spacediscretisation}.}\n\\end{figure}\n\n\n\n\\noindent\nWe also define $x_i = (x_{i+1\/2} + x_{i-1\/2})\/2$ the centre of cell $C_i$ and set $\\Delta\nx_{i+1\/2}=x_{i+1}-x_i$ for  $i=1,\\ldots,N-1$. We assume that the mesh is\nregular in the sense that there exists $\\xi\\in (0,1)$\nsuch that for $h := \\max_{1\\leq i \\leq N} \\{\\Delta x_i\\}$\n\\begin{equation}\n\\label{admissible}\n\t\\xi\\,h \\,\\leq \\,\\Delta x_{i} \\,\\leq\\, h,\n\\end{equation}\nand, as a consequence, $\\xi\\,h \\,\\leq \\,\\Delta x_{\\ihalf} \\,\\leq\\, h$, as well.\n\n\n\nOn this mesh we shall now define the semi-discrete finite volume approximation of system \\eqref{eq:crossdiffsystem}.  The discretised initial data are given by the cell averages of the continuous initial data, \\emph{i.e.}\n\\begin{align}\n\\label{eq:initial_data}\n\t\\rho_i^0 := \\frac{1}{\\Delta x_i}\\int_{C_i} \\rho_0(x)\\,\\d x, \\qquad \\text{and}\\qquad \t\\eta_i^0 := \\frac{1}{\\Delta x_i} \\int_{C_i} \\eta_0(x)\\,\\d x,\n\\end{align}\nfor all $i\\in I$. Next, we introduce the discrete versions of the cross-diffusion and the interaction terms. We set\n\\begin{align}\n\t\\label{eq:nonlocal_part}\n\t\\left\\{\n\t\\begin{array}{ll}\n\t\\displaystyle\n\t(V_1)_i:=-\\sum_{j=1}^N \\Delta x_j \\,\\big( W_{11}^{i-j}\\rho_j +  W_{12}^{i-j} \\eta_j \\big),\\\\[1.5em]\n\t\\displaystyle\t\n\t(V_2)_i:=-\\sum_{j=1}^N  \\Delta x_j \\,\\big( W_{22}^{i-j}\\eta_j +  W_{21}^{i-j} \\rho_j \\big),\n\t\\end{array}\n\t\\right.\n\\end{align}\nwhere \n\\begin{align}\n\t\\label{eq:discrete_interaction_potential}\n\tW_{kl}^{i-j} \\,=\\,\\frac{1}{\\Delta x_j}\\, \\int_{C_j} W_{kl}(|x_i-s|)\\d s,\n\\end{align}\nfor $k,l=1,2$, and \n\\begin{align*}\n\tU_i := -\\big(\\rho_i + \\eta_i),\n\\end{align*}\nfor the cross-diffusion term, respectively. Then the scheme reads\n\\begin{subequations}\n\\label{eq:scheme}\n\\begin{align}\n\t\\label{eq:scheme_evol}\n\t\\left\\{\n\t\\begin{array}{l}\n\t\\displaystyle\n\t\\frac{\\d \\rho_i}{\\d t}(t)=\\displaystyle- \\frac{\\mathcal{F}_{\\ihalf} - \\mathcal{F}_{i-1\/2}}{\\Delta x_i},\\\\[1em]\n\t\\displaystyle\n\t\\frac{\\d \\eta_i}{\\d t}(t)=\\displaystyle - \\frac{\\mathcal{G}_{\\ihalf} - \\mathcal{G}_{i-1\/2}}{\\Delta x_i},\n\t\\end{array}\n\t\\right.\n\\end{align}\nfor $i\\in I$. Here the numerical fluxes are given by\n\\begin{align}\n\t\\label{eq:numerical_fluxes}\n\t\\left\\{\n\t\\begin{array}{l}\n\t\\displaystyle\n\t\\mathcal{F}_{\\ihalf}=\t\\displaystyle \\left[\\nu\\,(\\d U)_{\\ihalf}^+ +\n                       (\\d V_1)_{\\ihalf}^+\\right] \\,\\rho_i \\,+\\,\n                       \\left[\\nu\\,(\\d U)_{\\ihalf}^- + (\\d\n                       V_1)_{\\ihalf}^-\\right]\\,\\rho_{i+1}\\\\ [1.5em]\n\t\\phantom{\\mathcal{F}_{\\ihalf}=\t}- \\ds\\frac{\\epsilon}{2} \\,\\frac{\\rho_{i+1}^2 - \\rho_i^2}{\\Delta x_{i+1\/2}},\\\\[1.5em]\n\t\\displaystyle\n\t\\mathcal{G}_{\\ihalf} =\t\\displaystyle \\left[ \\nu\\,(\\d U)_{\\ihalf}^+\n                       +(\\d V_2)_{\\ihalf}^+\\right]\\,   \\eta_i \\,+\\,\n                       \\left[\\nu\\, (\\d U)_{\\ihalf}^- + (\\d V_2)_{\\ihalf}^-\\right]\\,\\eta_{i+1}\\\\[1.5em]\n\t\\phantom{\\mathcal{G}_{\\ihalf} =\t}- \\ds\\,\\frac{\\epsilon}{2}\\, \\frac{\\eta_{i+1}^2 - \\eta_i^2}{\\Delta x_{i+1\/2}},\n\t\\end{array}\n\t\\right.\n\\end{align}\nfor $i = 1,\\ldots, N-1$, with the numerical no-flux boundary condition\n\\begin{align}\n\t\\label{eq:num_noflux}\n\t\\mathcal{F}_{1\/2} = \\mathcal{F}_{N+1\/2} &= 0, \\quad \\text{and} \\quad \\mathcal{G}_{1\/2} = \\mathcal{G}_{N+1\/2} = 0,\t\n\\end{align}\n\\end{subequations}\nwhere we introduced the  discrete gradient $\\d u_\\ihalf$ as\n$$\n\\d u_\\ihalf\\,:=\\,\\frac{u_{i+1}-u_i}{\\Delta x_{i+1\/2}}.\n$$ \nAs usual, we use $(z)^\\pm$ to denote the positive (resp. negative) part of $z$, \\textit{i.e.}\n\\begin{align*}\n\t(z)^+ := \\mathrm{max}(z,0), \\qquad \\text{and} \\qquad (z)^-:=\\mathrm{min}(z,0).\n\\end{align*}\n\nAt this stage, the numerical flux \\eqref{eq:numerical_fluxes} may look\nstrange since \n\\begin{itemize}\n\\item the cross-diffusion term is approximated as a convective term\n  using that \n$$\n\\frac{\\partial}{\\partial x}\\left( \\rho \\, \\frac{\\partial}{\\partial\n    x}\\left( \\rho + \\eta \\right) \\right) \\,=\\,\n\\frac{\\partial}{\\partial x}\\left( \\rho \\, \\frac{\\partial \\sigma}{\\partial\n    x} \\right)\n$$\nwith $\\sigma=\\rho+\\eta$ and $\\frac{\\partial \\sigma}{\\partial x}$ is considered\nas a velocity field. This treatment has already been used in\n\\cite{BCF12} and allows to preserve the positivity of both discrete\ndensities $(\\rho,\\eta)$ (see Lemma \\ref{lem:nonnegative}), which is crucial for the convergence\nanalysis. \n\\item In this new formulation, the velocity field is split in two\n  parts both treated by an upwind scheme. One part comes from the\n  cross-diffusion part, and the second one comes from the non-local\n  interaction fields. This splitting is crucial to recovering a consistent\n  dissipative term for the discrete energy estimate corresponding to Eq.  \\eqref{eq:cts_inequality}.\n\\end{itemize}\n\n\\begin{definition}[\\label{def:ApproximateSolutions}Piecewise constant approximation]\n\tFor a given mesh $\\mathcal{T}_{h}$ we define the approximate solution to system \\eqref{eq:crossdiffsystem} by\n\t\\begin{align*}\n\t\t\\rho_{h}(t,x):= \\rho_i(t),\\qquad \\text{and} \\qquad \\eta_{h}(t,x):=\\eta_i(t),\n\t\\end{align*}\n\tfor all $(t,x) \\in [0,T]\\times C_i$, with $i=1,\\ldots,N$. Moreover, we define the following approximations of the gradients\n\t\\begin{align*}\n\t\t\\d \\rho_{h}(t,x) = \\frac{\\rho_{i+1}-\\rho_i}{\\Delta x_{i+1\/2}}, \\qquad \\mbox{and}\\qquad \\d \\eta_{h}(t,x) = \\frac{\\eta_{i+1}-\\eta_i}{\\Delta x_{i+1\/2}}\n\t\\end{align*}\n\tfor $(t,x)\\in[0,T)\\times [x_i,x_{i+1})$, for\n        $i=1,\\ldots,N-1$. Furthermore, in order to define $\\d \\rho_h$\n        and $\\d \\eta_h$ on the whole interval $(a,b)$ we set them to zero\n         on $(a,x_1)$ and  $(x_N,b$). \n\\end{definition}\nNotice that the discrete gradients $(\\d \\rho_h,\\d \\eta_h)$ are\npiecewise constant just like $(\\rho_h,\\eta_h)$ however not on the same partition\nof the interval  $(a,b)$. We have set out all definitions necessary to formulate the convergence of the numerical scheme \\eqref{eq:scheme}.\n\n\\begin{theorem}[\\label{thm:convergence}Convergence to a weak solution.]\n\tLet $\\rho_0, \\eta_0 \\in L_+^1(a,b) \\cap L_+^\\infty(a,b)$ be\n        some initial data  and $Q_T := (0,T)\\times (a,b)$. Then, \n\\begin{itemize}\n\\item [$(i)$] there exists  a nonnegative approximate solution\n        $(\\rho_{h}, \\eta_{h})$ in the sense of Definition\n        \\ref{def:ApproximateSolutions};\n\n\\item[$(ii)$]  up to a subsequence, this\n        approximate solution converges strongly in $L^2(Q_T)$ to\n        $(\\rho,\\eta) \\in L^2(Q_T)$, where $(\\rho,\\eta)$ is  a weak solution as in\n        Definition \\ref{def:WeakSolution}. Furthermore we have $\\rho$, $\\eta\n        \\in L^2(0,T; H^1(a,b))$;\n\\item[$(iii)$] as a consequence system \\eqref{eq:crossdiffsystem} has a weak solution.\n\\end{itemize}\n\\end{theorem}\n\n\\section{\\label{sec:apriori_estimates}A priori estimates}\nThis section is dedicated to deriving a priori estimates for our system. In order to do so we require the positivity of  approximate solutions and their conservation of mass, respectively. The following lemma guarantees these properties.\n\\begin{lemma}[Existence of nonnegative solutions and conservation of mass]\n\\label{lem:nonnegative}\n\\label{lem:conservation_of_m}\nAssume that the initial data $(\\rho_0, \\eta_0)$ in non-negative. Then\nthere  exists a unique  nonnegative approximate\n        solution $(\\rho_{h}, \\eta_{h})_{h>0}$  to the  scheme (\\ref{eq:scheme_evol})-(\\ref{eq:numerical_fluxes}). Furthermore, the finite volume scheme conserves the initial mass of both densities.\n\\end{lemma}\n\\begin{proof}\n\tOn the one hand we notice that the right-hand side of  (\\ref{eq:scheme_evol})-(\\ref{eq:numerical_fluxes})\n\tis  locally Lipschitz with respect to $(\\rho_i,\\eta_i)_{1\\leq\n          i\\leq N}$. Hence, we may apply the Cauchy-Lipschitz theorem to obtain a unique continuously differentiable local-in-time solution.\n \n\tOn the other hand to prove that this solution is global in\n        time, we show the nonnegativity of the solution together with\n        the conservation of mass and argue by contradiction. \n\nOn a given mesh, let some initial data, $\\rho_i(0), \\eta_i(0)\\geq 0$,\nbe given for $i=1, \\ldots N$. We rewrite the scheme in the following way.\n\t\\begin{align}\n\t\\label{eq:PosSchemeRewritten}\n\t\t\\frac{\\d \\rho_i}{\\d t}(t) &= -\\frac{\\mathcal{F}_\\ihalf - \\mathcal{F}_\\imhalf}{\\Delta x_i} \\,=\\,\\frac{1}{\\Delta x_i}\\big(A_i \\,\\rho_i \\,+\\, B_i \\,\\rho_{i+1} \\,+\\, C_i\\, \\rho_{i-1}\\big),\n\t\\end{align}\n\twhere \n\t\\begin{align*}\n\t\t\\left\\{\n\t\t\\begin{array}{l}\n\t\t\tA_i=\\displaystyle \\nu \\,(\\d U)_\\imhalf^- \\,+\\,\n                              (\\d V_1)_\\imhalf^-\\,-\\,\\nu\\,(\\d\n                              U)_\\ihalf^+ \\,-\\, (\\d V_1)_\\ihalf^+ \\,-\\, \n                              \\,\\frac\\epsilon  2 \\left(\\frac{\\rho_i}{\\Delta x_{i+1\/2}} + \\frac{\\rho_i}{\\Delta x_{i-1\/2}}\\right),\n\\\\[1em]\n\t\tB_i =\\displaystyle -\\,\\nu\\,(\\d U)_\\ihalf^- \\,-\\, (\\d\n                      V_1)_\\ihalf^- \\,+\\, \\epsilon\\,\n                      \\frac{\\rho_{i+1}}{2\\Delta x_{i+1\/2}},\n\\\\[1em]\n\t\tC_i =\\displaystyle \\nu \\,(\\d U)_\\imhalf^+ \\,+\\, (\\d V_1)_\\imhalf^+ \\,+\\, \\epsilon \\frac{\\rho_{i-1}}{2\\Delta x_{i-1\/2}}.\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{align*}\n\n\tThen let $t^\\star\\geq 0$ be the maximal time for all densities\n        to remain nonnegative, \\emph{i.e.}\n$$\n\tt^\\star = \\sup\\left\\{ t \\geq 0 \\, |, \\rho_i(s) \\geq 0,\\,\n          \\mbox{for all } s \\in [0,t], \\mbox{ and } i=1,\\ldots,N\n\\right\\}.\n$$\n\n\nIf $t^\\star < \\infty$, then there exists a nonincreasing sequence $(t_k)_{k\\in\\N}$\nsuch that $t_k>t^\\star$,  $t_k\\rightarrow t^\\star$ as $k\\rightarrow \\infty$ and there\nexists $i_k\\in \\{1,\\ldots, N\\}$ verifying \n$$\n\\rho_{i_k}(t_k)<0,\\quad \\forall k\\in\\N.\n$$\nSince the index $i_k$ takes a finite number of  integer values, we can\nextract a nonincreasing subsequence  of $(t_k)_{k\\in\\N}$\nstill labeled in the same manner such that there exists an index  $j_0\\in\n\\{1,\\ldots,N\\}$ and \n$$\n\\rho_{j_0}(t_k)<0,\\quad \\forall k\\in\\N,\n$$\nwhere $t_k\\rightarrow t^\\star$, as $k$ goes to infinity.\n\nAlso note by continuity of $(\\rho_i)_{1\\leq i\\leq N}$, we have that\n$\\rho_{i}(t^\\star)\\geq 0$ for any $i\\in\\{1,\\ldots, N\\}$. \n\nBy the above computation, Eq. \\eqref{eq:PosSchemeRewritten}, we see\nthat, if  $\\rho_{j_0+1}(t^\\star)>0$ or respectively $\\rho_{j_0-1}(t^\\star)>0$, then\neither $B_{j_0}(t^\\star)>0$ or  respectively $ C_{j_0}(t^\\star)>0$ and\n\\begin{align*}\n\\frac{\\d \\rho_{j_0}}{\\d t}(t^\\star) & =\\,\\frac{1}{\\Delta\n                                      x_{j_0}}\\left(A_{j_0}\n                                      \\,\\rho_{j_0}(t^\\star) \\,+\\,\n                                      B_{j_0}\\, \\rho_{j_0+1}(t^\\star)\n                                      \\,+\\, C_{j_0}\n                                      \\,\\rho_{j_0-1}(t^\\star)\\right)\n\\\\\n\t\t                            &=\\,\\frac{1}{\\Delta x_{j_0}}\\left(B_{j_0} \\,\\rho_{j_0+1}(t^\\star) \\,+\\, C_{j_0}\\, \\rho_{j_0-1}(t^\\star)\\right)\n\t\t\\,>\\, 0,\n\\end{align*}\nhence there exists $\\tau>0$ such that for any $t\\in\n[t^\\star,t^\\star+\\tau)$, we have $\\rho_{j_0}(t) > \\rho_{j_0}(t^\\star)=0$,\nwhich cannot occur since $\\rho_{j_0}$ is continuous and for $t_k>t^\\star$,\n$\\rho_{j_0}(t_k) < 0$ for any $k\\in\\N$ with $t_k\\rightarrow t^\\star$\nwhen $k$ goes to infinity.\n\nIf $\\rho_{j_0-1}(t^\\star)=\\rho_{j_0}(t^\\star)=\\rho_{j_0+1}(t^\\star)=0$\nthen by uniqueness of the solution, we have that $\\rho_{j_0}\\equiv 0$\nfor $t\\geq t^\\star$, which contradicts again that $\\rho_{j_0}(t_k) < 0$ for any $k\\in\\N$ large enough.\n\n\n\n\n\n\n\n\n\n\n\n\nFinally we get  the conservation of mass, \n\\begin{align*}\n\t\t\\ds\\frac{\\d}{\\d t}\\int_a^b \\rho_{h}(t,x)\\d x \n\t\t&= \\ds\\sum_{i=1}^N\\Delta x_i \\frac{\\d}{\\d t}\\rho_i\\\\\n\t\t&= \\ds\\sum_{i = 1}^N \\Delta x_i \\frac{\\mathcal{F}_{\\ihalf} - \\mathcal{F}_{\\imhalf}}{\\Delta x_i} \\,=\\, \\mathcal{F}_{N+1\/2}-\\mathcal{F}_{1\/2} \\,=\\, 0,\n\\end{align*}\n\tby the no-flux condition. Analogously, the second species\n        remains nonnegative and its mass is conserved as well. As a consequence of the control of the  $L^1$-norm of $(\\rho_h,\\eta_h)$ we can extend the local solution to a global, nonnegative solution.\n\\end{proof}\n\n\n\\noindent\nNow, we are ready to study the evolution of the energy of the system on the semi-discrete level. The remaining part of this section is dedicated to proving the following lemma -- an estimate similar to \\eqref{eq:cts_inequality} for the semi-discrete scheme \\eqref{eq:scheme}. \n\n\\begin{lemma}[Energy control\\label{lem:entropy_control}]\nConsider a solution of the semi-discrete scheme\n\\eqref{eq:scheme_evol}-\\eqref{eq:numerical_fluxes}. Then we have\n\\begin{align*}\n\t\\ds \\frac{\\d }{\\d t} \\sum_{i=1}^N\\Delta x_i [\\rho_i \\log \\rho_i + \\eta_i \\log\\eta_i]\\ds+\\,\\sum_{i=1}^{N-1}\\!\\Delta x_{i+1\/2}\\left[\\nu\\, |\\d U_\\ihalf|^2 \n\t\t\\,+\\, \\frac{\\epsilon}{4}\\left(  |\\d \\rho_\\ihalf|^2\n\t\t\\,+\\,  |\\d \\eta_\\ihalf|^2\\right)\n\t\t\t\t\\right]\\,\\leq\\, C_{\\epsilon},\n\\end{align*}\nwhere the constant $C_{\\epsilon}>0$ is given by \n\\begin{equation}\n\\label{def:c}\nC_{\\epsilon} \\,=\\, \\frac{(b-a)}{\\epsilon} \\left(\\,\\left( \\|W_{11}'\\|_{L^\\infty}+\\|W_{21}'\\|_{L^\\infty} \\right)\\, m_1 \\,+\\,\n\\left(  \\|W_{12}'\\|_{L^\\infty} +  \\|W_{22}'\\|_{L^\\infty}\\right)\\, m_2\\,\\right).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nUpon using the scheme, Eq. \\eqref{eq:scheme_evol}, we get\n\\begin{align*}\n\t \\ds\\frac{\\d}{\\d t} \\sum_{i=1}^N \\Delta x_i\\,\\rho_i \\log \\rho_i =\\ds-\\sum_{i=1}^N(\\mathcal{F}_{\\ihalf} - \\mathcal{F}_{i-1\/2}) \\log\\rho_i.\n\\end{align*}\nBy discrete integration by parts and the no-flux condition, Eq. \\eqref{eq:num_noflux}, we obtain\n\\begin{align*}\n\\ds\\frac{\\d}{\\d t} \\sum_{i=1}^N \\Delta x_i\\,\\rho_i \\log \\rho_i\n                &=\\ds\\sum_{i=1}^{N-1}  \\Delta x_{i+1\/2}\\,  \\mathcal{F}_{\\ihalf}\\, \\D\\log\n             \\rho_{\\ihalf}\n\\\\\n\t&=\\ds\\nu\\sum_{i=1}^{N-1} \\Delta x_{i+1\/2} \\bigg(\n\t\t(\\d U)_{\\ihalf}^+ \\rho_i \\,+\\,(\\d\n            U)_{\\ihalf}^- \\rho_{i+1}  \\bigg)\t   \\rm{d}\\! \\log\\rho_{\\ihalf}\n\\\\\n&\\quad+\\ds\\sum_{i=1}^{N-1} \\Delta x_{i+1\/2} \\bigg( (\\d V_1)_{\\ihalf}^+ \\rho_i + (\\d\n            V_1)_{\\ihalf}^- \\rho_{i+1}  \\bigg)\t   \\rm{d}\\! \\log\\rho_{\\ihalf},\n\\\\\n&\\quad-\\ds\\frac{\\epsilon}{2}\\,\\sum_{i=1}^{N-1} \\left(\\rho_{i+1}^2 - \\rho_i^2\\right)\\, \t   \\rm{d}\\! \\log\\rho_{\\ihalf},\n\\end{align*}\nwhere, in the last equality, we substituted the definition of the numerical flux, Eq. \\eqref{eq:numerical_fluxes}.\nLet us define\n\\begin{align}\n\\label{eq:tilde_rho_ihalf}\n\t\\tilde \\rho_\\ihalf := \\left\\{\n\t\t\\begin{array}{ll}\n\t\t\t\\dfrac{\\rho_{i+1} - \\rho_i}{\\log\\rho_{i+1} - \\log \\rho_i}, & \\text{if $\\rho_i \\neq \\rho_{i+1}$},\\\\[1em]\n\t\t\t\\dfrac{\\rho_i + \\rho_{i+1}}{2}, & \\text{else},\n\t\t\\end{array}\n\t\t\\right.\n\\end{align}\nfor $i\\in\\{1,\\ldots,N-1\\}$, and note that then $\\tilde \\rho_\\ihalf \\in\n[\\rho_i, \\rho_{i+1}]$ by concavity of the $\\log$. Reordering the terms, we obtain\n\\begin{align}\n\\label{eq:someeqn}\n\\begin{split}\n\t\\frac{\\d}{\\d t} \\sum_{i=1}^N &\\Delta x_i \\rho_i \\log \\rho_i \\,-\\, \\sum_{i=1}^{N-1} \\Delta x_{i+1\/2}\\, \\left[ \\nu \\,\\d U_\\ihalf \\tilde \\rho_\\ihalf \\,-\\,\\frac{\\epsilon}{2}\\,\\d \\rho_\\ihalf^2 \\right]\\D\\log\\rho_\\ihalf\\\\\n\t=&\\,\\nu\\sum_{i=1}^{N-1}\\Delta x_{i+1\/2}\\left((\\d U)_{\\ihalf}^+ (\\rho_i - \\tilde \\rho_\\ihalf) \n\t\t  \\,+\\, (\\d U)_{\\ihalf}^-\n                  (\\rho_{i+1}-\\tilde\\rho_\\ihalf)\\right)\\, \\D\\log\\rho_{\\ihalf}\\\\\n       &+\\,\\sum_{i=1}^{N-1}\\Delta x_{i+1\/2}\\left( (\\d V_1)_{\\ihalf}^+ (\\rho_i - \\tilde \\rho_\\ihalf) \n\t\t  \\,+\\,(\\d V_1)_{\\ihalf}^- (\\rho_{i+1}-\\tilde\\rho_\\ihalf) \\right)\\,\\D\\log\\rho_{\\ihalf}\\\\\n\t&+\\, \\sum_{i=1}^{N-1}\\Delta x_{i+1\/2} \\, \\tilde\\rho_\\ihalf\\,\\d V_{1,\\ihalf} \\,\\D\\log\\rho_{\\ihalf}.\n\\end{split}\n\\end{align}\nThus, using $\\tilde \\rho_\\ihalf \\in\n[\\rho_i, \\rho_{i+1}]$ and the monotonicity of $\\log$, we note that\n\\begin{align}\n\t\\label{eq:numerical_artifacts}\n\t\\left\\{\n\t\\begin{array}{r}\n\t\\displaystyle\n\t(\\rho_i - \\tilde \\rho_\\ihalf) \\D\\log \\rho_\\ihalf \\big(\\nu(\\d U)_{\\ihalf}^+ + (\\d V_1)_{\\ihalf}^+\\big) \\leq 0,\\\\[1em]\n\t\\displaystyle\n\t(\\rho_{i+1} - \\tilde \\rho_\\ihalf) \\D\\log \\rho_\\ihalf \\big(\\nu (\\d U)_{\\ihalf}^- + (\\d V_1)_{\\ihalf}^-\\big) \\leq 0.\n\t\\end{array}\n\t\\right.\n\\end{align}\nThis is easy to see, for if $\\rho_i = \\rho_{i+1}$ we observe $ \\D\\log \\rho_\\ihalf=0$ and Eqs. \\eqref{eq:numerical_artifacts} hold with equality. In the case of $\\rho_i< \\rho_{i+1}$ we observe\n\\begin{align*}\n\t\\underbrace{(\\rho_i - \\tilde\\rho_\\ihalf)}_{\\leq  0} \\underbrace{\\D\\log\\rho_\\ihalf}_{\\geq 0} \\underbrace{\\big(\\nu(\\d U)_\\ihalf^+ + (\\d V_1)_\\ihalf^+ \\big)}_{\\geq 0} \\leq 0,\n\\end{align*}\nwhile, for $\\rho_i > \\rho_{i+1}$ there also holds\n\\begin{align*}\n\t\\underbrace{(\\rho_i - \\tilde\\rho_\\ihalf)}_{\\geq  0} \\underbrace{\\D\\log\\rho_\\ihalf}_{\\leq 0} \\underbrace{\\big(\\nu(\\d U)_\\ihalf^+ + (\\d V_1)_\\ihalf^+ \\big)}_{\\geq 0} \\leq 0,\n\\end{align*}\nwhence we infer the inequality. The same argument can be applied in order to obtain the second line of Eq. \\eqref{eq:numerical_artifacts}. Thus we may infer from Eq. \\eqref{eq:someeqn} that\n\\begin{align*}\n\\begin{split}\n\t\\frac{\\d}{\\d t} \\sum_{i=1}^N &\\Delta x_i \\rho_i \\log \\rho_i -  \\sum_{i=1}^{N-1} \\Delta x_{i+1\/2} \\left(\\nu \\tilde \\rho_\\ihalf \\, \\d U_\\ihalf \\,-\\,\\frac{\\epsilon}{2}\\,\\d \\rho_\\ihalf^2\\right)\\, \\D\\log\\rho_\\ihalf\\\\\n\t&\\leq\\sum_{i=1}^{N-1}\\Delta x_{i+1\/2} \\,\\tilde \\rho_\\ihalf\\,\\D\\log\\rho_{\\ihalf} \\,\\d V_{1,\\ihalf}.\n\\end{split}\n\\end{align*}\nNote that the definition of $\\tilde \\rho_\\ihalf$ in Eq. \\eqref{eq:tilde_rho_ihalf}, is consistent with the case $\\rho_i = \\rho_{i+1}$ and there holds\n\\begin{align*}\n\t\\tilde \\rho_\\ihalf \\,\\D\\log\\rho_\\ihalf \\,=\\, \\d \\rho_\\ihalf,\n\\end{align*}\nwhence we get \n\\begin{align*}\n\\begin{split}\n\t\\frac{\\d}{\\d t} \\sum_{i=1}^N &\\Delta x_i \\,\\rho_i \\,\\log \\rho_i\n        \\,-\\, \\nu\\,\\sum_{i=1}^{N-1} \\Delta x_{i+1\/2} \\,\\d\\rho_\\ihalf\\,\\d\n        U_\\ihalf  \\\\\n& +\\,\\frac{\\epsilon}{2}\\, \\sum_{i=1}^{N-1} \\Delta x_{i+1\/2} \\,\\d \\rho_\\ihalf^2 \\,\\D\\log\\rho_\\ihalf\t\\,\\leq\\,\\sum_{i=1}^{N-1}\\Delta x_{i+1\/2} \\,\\d\\rho_{\\ihalf} \\,\\d V_{1,\\ihalf}.\n\\end{split}\n\\end{align*}\nFurthermore, we notice that \n$$\n\t\\frac{1}{2}\\,\\d \\rho_\\ihalf^2 \\D\\log\\rho_\\ihalf  \\,=\\;  \\frac{\\rho_{i+1} +\n          \\rho_i}{2\\,\\tilde \\rho_\\ihalf} |\\d\\rho_{i+1\/2}|^2\\,\\geq\\,\\frac{1}{2}\n        |\\d\\rho_{i+1\/2}|^2,\n$$\nwhere we employed Eq. \\eqref{eq:tilde_rho_ihalf}. Hence we have \n\\begin{align}\n\\label{eq:291116_0954}\n\\begin{split}\n\t\\frac{\\d}{\\d t} \\sum_{i=1}^N &\\Delta x_i \\,\\rho_i \\,\\log \\rho_i\n        \\,-\\, \\nu\\,\\sum_{i=1}^{N-1} \\Delta x_{i+1\/2} \\,\\d\\rho_\\ihalf\\,\\d\n        U_\\ihalf  \\\\\n& +\\,\\frac{\\epsilon}{2}\\, \\sum_{i=1}^{N-1} \\Delta x_{i+1\/2}\\, |\\d \\rho_\\ihalf|^2 \t\\,\\leq\\,\\sum_{i=1}^{N-1}\\Delta x_{i+1\/2} \\,\\d\\rho_{\\ihalf} \\,\\d V_{1,\\ihalf}.\n\\end{split}\n\\end{align}\nA similar computation can be  applied to the second species, which  yields\n\\begin{align}\n\\label{eq:220916_1137}\n\\begin{split}\n\t\\frac{\\d}{\\d t} \\sum_{i=1}^N &\\Delta x_i \\,\\eta_i \\,\\log \\eta_i\n        \\,-\\,  \\nu \\sum_{i=1}^{N-1} \\Delta x_{i+1\/2}\\, \\d \\eta_\\ihalf \\,\\d\n        U_\\ihalf \\\\\n& +\\,\\frac{\\epsilon}{2}\\sum_{i=1}^{N-1}\\Delta x_{i+1\/2} \\,|\\d \\eta_\\ihalf|^2\t\\,\\leq\\,\\sum_{i=1}^{N-1}\\Delta x_{i+1\/2} \\,\\d \\eta_\\ihalf \\d V_{2,\\ihalf}.\n\\end{split}\n\\end{align}\n\\noindent\nUpon adding up equations \\eqref{eq:291116_0954} and \\eqref{eq:220916_1137}, we obtain\n\\begin{align}\n\\label{eq:220916_1212}\n\\begin{split}\n\t\\frac{\\d }{\\d t}\\sum_{i=1}^N &\\Delta x_i\\,[\\rho_i \\log \\rho_i\n        + \\eta_i \\log \\eta_i] \\,-\\, \\nu\\,\\sum_{i=1}^{N-1}\\Delta\n        x_{i+1\/2}\\,\\d U_\\ihalf \\,\\d\\left(\\rho+\\eta\\right)_\\ihalf \\\\\n\t&\\ds\\,+\\,\\frac{\\epsilon}{2}\\,\\sum_{i=1}^{N-1}\\Delta x_{i+1\/2}\n        \\,\\bigg(|\\d \\rho_\\ihalf|^2  \\,+\\, |\\d\n        \\eta_\\ihalf|^2 \\bigg)\\,\\leq\\, \\mathcal{R}_h,\n\\end{split}\n\\end{align}\t\nwhere $\\mathcal{R}_h$ is given by\n$$\n\\mathcal{R}_h \\,=\\, \\sum_{i=1}^{N-1}\\Delta x_{i+1\/2} \\,\\left[ \\d V_{1,\\ihalf}\n\\,\\d\\rho_\\ihalf \\,+\\, \\d V_{2,\\ihalf} \\,\\d\\eta_\\ihalf\\right].\n$$\nFinally we notice  that\n\\begin{equation}\n\t\\label{eq:19_11_17_0911}\n\t\\mathcal{R}_h\\,\\leq\\,\\frac{1}{2}\\sum_{i=1}^{N-1} \\Delta x_{i+1\/2} \\left[\\frac{|\\d V_{1,\\ihalf}|^2}{\\alpha} \\,+\\, \\alpha\\,|\\d \\rho_\\ihalf|^2 \\,+\\, \\frac{|\\d V_{2,\\ihalf}|^2}{\\alpha} \\,+\\, \\alpha \\,|\\d \\eta_\\ihalf|^2\\right],\n\\end{equation} \nfor any $\\alpha>0$, by Young's inequality. Observing, that for $k=1,\\,2$,\n\\begin{align*}\n|\\d V_{k,\\ihalf}| &\\leq \\left|\\sum_{j=1}^{N-1} \\rho_j\\,\\int_{C_j}\n  \\frac{W_{k1}(x_{i+1}-y)-W_{k1}(x_{i}-y)}{\\Delta x_{i+1\/2}} \\d\n  y\\right| \n\\\\\n&\\quad + \\left|\\sum_{j=1}^{N-1} \\eta_j\\,\\int_{C_j}\n  \\frac{W_{k2}(x_{i+1}-y)-W_{k2}(x_{i}-y)}{\\Delta x_{i+1\/2}} \\d\n  y\\right| \n\\\\\n &\\leq \\|W_{k1}'\\|_{L^\\infty} \\, m_1 \\,+\\, \\|W_{k2}'\\|_{L^\\infty} \\, m_2\n\\end{align*}\nand by conservation of positivity and mass, it gives for $k=1,\\,2$,\n\\begin{eqnarray*}\n\\frac{1}{2\\alpha}\\sum_{i=1}^{N-1}\\Delta x_{i+1\/2} |\\d V_{k,\\ihalf}|^2 \\,\\leq\\, \\frac{ (b-a)}{2\\,\\alpha} \\, \\left(m_1\\,\\|W_{k1}'\\|_{L^\\infty} \\, +\\,  m_2\\|W_{k2}'\\|_{L^\\infty}  \\right)^2.\n\\end{eqnarray*}\nThus Eq. \\eqref{eq:19_11_17_0911} becomes\n$$\n\\mathcal{R}_h \\,\\leq\\, \\frac{\\alpha}{2}\\sum_{i=1}^{N-1} \\Delta x_{\\ihalf}\\left(|\\d\n  \\rho_\\ihalf|^2+|\\d \\eta_\\ihalf|^2\\right)  \\,+ \\, C_{2\\alpha},\n$$\nwhere $C_{2\\alpha}$ is given in Eq. \\eqref{def:c}. Finally, substituting the latter\nestimate into  Eq. \\eqref{eq:220916_1212},  we obtain\n\\begin{align*}\n\t \\frac{\\d }{\\d t} \\sum_{i=1}^N\\Delta x_i [\\rho_i \\log \\rho_i + \\eta_i \\log\\eta_i]\n\t\t\\,&+\\, \\nu\\,\\sum_{i=1}^{N-1}\\!\\Delta x_{\\ihalf} |\\d  U_\\ihalf|^2 \\\\\n&+ \\frac{\\epsilon \\,-\\,\\alpha}{2}\\, \\sum_{i=1}^{N-1}\\!\\Delta x_{\\ihalf} \\left( |\\d \\rho_\\ihalf|^2\\,+\\, |\\d \\eta_\\ihalf|^2\\right)\n\t\t\t\t\\,\\leq\\, C_{2\\alpha},\n\\end{align*}\nfor any solution $(\\rho_i)_{i\\in I}, (\\eta_i)_{i\\in I}$ of the\nsemi-discrete scheme \\eqref{eq:scheme}. Hence choosing\n$\\alpha=\\epsilon\/2$ concludes the proof. \n\\end{proof}\n\\begin{corollary}[A priori bounds]\n\\label{cor:apriori}\n\tLet $(\\rho_i)_{i\\in I}, (\\eta_i)_{i\\in I}$ be solutions of the\n        semi-discrete scheme \\eqref{eq:scheme}. Then there exists a\n        constant $C>0$ such that \n\t\\begin{align*}\n\t\t\\int_0^T \\sum_{i=1}^{N-1} \\Delta x_{\\ihalf} \\left(\\,|\\d\\rho_\\ihalf|^2 + |\\d\\eta_\\ihalf|^2 + \\,|\\d U_\\ihalf|^2 \\right) \\,\\d\n          t&\\leq C.\n\t\\end{align*}\n\\end{corollary}\n\\begin{proof}\nUsing the fact that  $x \\log x \\geq -\\log(e)\/e$, \\emph{i.e.} $x\\log x$\nis bounded from below, yields\n\\begin{align*}\n\\sum_{i\\in I}\t\\Delta x_i \\, [\\rho_i \\log \\rho_i + \\eta_i \\log\\eta_i]\n  (t) \\geq -2\\frac{\\log(e)}{e}\\,  (b-a)=: -C_1.\n\\end{align*}\nHence the discrete version of the classical entropy functional is bounded from below. Therefore, we integrate\nthe inequality of Lemma \\ref{lem:entropy_control} in time and get \n\\begin{align*}\n\\int_0^T \\sum_{i=1}^{N-1} &\\Delta x_{i+1\/2}\\left[\\nu\\, |\\d U_\\ihalf(t)|^2 \n\t\t\\,+\\, \\frac{\\epsilon}{4}\\left(  |\\d \\rho_\\ihalf(t)|^2\n\t\t\\,+\\,  |\\d \\eta_\\ihalf(t)|^2\\right)\n\t\t\t\t\\right]\\,\\d t \\\\\n&\\leq C_\\epsilon\\, T \\,+\\,C_1\\,+\\,\\int_a^b\\rho_0 |\\log \\rho_0| + \\eta_0 |\\log\\eta_0|\\d x,\n\\end{align*}\nwhich proves the statement.\n\\end{proof}\n\n\nThanks to a classical discrete Poincar\\'e inequality \\cite[Lemma 3.7]{EGH00} and \\cite{BCF}, we get\nuniform $L^2$-estimates on the discrete approximation $(\\rho_h,\\eta_h)_{h>0}$.\n\\begin{lemma}\n\\label{lem:aprioriL2}\n\tLet $(\\rho_i)_{i\\in I}, (\\eta_i)_{i\\in I}$ be the numerical\n        solutions obtained from scheme \\eqref{eq:scheme}. Then there holds\n\t\\begin{align*}\n\t\t\\|\\rho_{h}\\|_{L^2(Q_T)} \\,+\\, \\|\\eta_{h}\\|_{L^2(Q_T)} \\leq C,\n\t\\end{align*}\n\tfor some constant $C>0$ independent of $h>0$.\n\\end{lemma}\n\n\n\n\\section{\\label{sec:convergence}Proof of Theorem \\ref{thm:convergence}}\nThis section is dedicated to proving compactness of both species, the fluxes, and the regularising porous-medium type diffusion. Upon establishing the compactness result we identify the limits as weak solutions in the sense of Definition \\ref{def:WeakSolution}.  \n\nFirst by application of Lemma \\ref{lem:nonnegative}, we get existence\nand uniqueness  of a nonnegative  approximate solution\n$(\\rho_h,\\eta_h)$ to\n(\\ref{eq:scheme_evol})-(\\ref{eq:numerical_fluxes}). Hence the first\nitem of Theorem \\ref{thm:convergence} is proven. Now let us\ninvestigate the asymptotic $h\\rightarrow 0$. \n\n\n\\subsection{Strong compactness of approximate solutions}\nWe shall now make use of the above estimates in order to obtain strong\ncompactness of both species, $(\\rho_{h},\\eta_{h})$ in $L^2(Q_T)$. \n\n\\begin{lemma}[Strong compactness in $L^2(Q_T)$]\n\\label{lem:strongconvergence}\n\tLet $(\\rho_{h}, \\eta_{h})_{h>0}$ be the approximation to system \\eqref{eq:crossdiffsystem} obtained by the semi-discrete scheme \\eqref{eq:scheme}. Then there exist functions $\\rho, \\eta \\in L^2(Q_T)$ such that\n\t\\begin{align*}\n\t\t\\rho_{h}\\rightarrow \\rho,\\qquad \\text{and} \\qquad \\eta_{h} \\rightarrow \\eta,\n\t\\end{align*}\n\tstrongly in $L^2(Q_T)$, up to a subsequence.\n\\end{lemma}\n\\begin{proof}\n\n\tWe invoke the compactness criterion by Aubin and Lions \\cite{CJL14}.\n\tAccordingly, a set $P\\subset L^2(0,T;B)$ is relatively compact if $P$ is bounded in $L^2(0,T;X)$ and the set of derivatives $\\{\\partial_t \\rho \\big| \\rho\\in P\\}$ is bounded in a third space $L^1(0,T;Y)$, whenever the involved Banach spaces satisfy $X \\hookrightarrow\\hookrightarrow B \\hookrightarrow Y$, \t\\emph{i.e.} the first embedding is compact and the second one continuous. For our purpose we choose $X:=BV(a,b)$, $B:=L^2(a,b)$, and $Y:=H^{-2}(a,b)$. \nThe first embedding is indeed compact,  \\emph{e.g.} Ref. \\cite[Theorem 10.1.4]{ABM14} and the second one is continuous. \n\t\t\n\t\t\n\t\tIn the second step we show the time derivatives are bounded in $L^1(0,T;H^{-2}(a,b))$. To this end, let $\\varphi\\in C_c^\\infty((a,b))$. Throughout, we write $\\langle \\cdot, \\cdot\\rangle$ for $\\langle \\cdot, \\cdot\\rangle_{H^{-2}, H^2}$ for the dual pairing. Making use of the scheme, there holds\n\t\\begin{align*}\n\t\\begin{split}\n\t\t\\left\\langle\\frac{\\d  \\rho_{h}}{\\d t}, \\varphi\\right\\rangle \n\t\t&=\\sum_{i=1}^N \\int_{C_i} \\frac{\\d \\rho_i }{\\d t} \\,\\varphi \\,\\d x =-\\sum_{i=1}^N  \\frac{\\mathcal{F}_\\ihalf - \\mathcal{F}_\\imhalf}{\\Delta x_i}  \\int_{C_i}\\varphi \\,\\d x,\n\t\\end{split}\n\t\\end{align*}\n\thaving used the scheme, Eq. \\eqref{eq:scheme_evol}. Next we\n        set\n$$\n\t\\varphi_i :=  \\frac{1}{\\Delta x_i}\\int_{C_i}\\varphi \\,\\d x,\n$$\nperform a discrete integration by parts and use the no-flux boundary conditions, Eq. \\eqref{eq:num_noflux}, to obtain\n$$\t\t\n\\left\\langle\\frac{\\d \\rho_{h}}{\\d t} , \\varphi\\right\\rangle \n\t\t\\,=\\, \\sum_{i=1}^{N-1}  \\mathcal{F}_\\ihalf \\, \\left( \\varphi_{i+1} -\n                  \\varphi_i\\right).\n$$\nUsing the definition of the numerical flux, Eq. \\eqref{eq:numerical_fluxes}, we  get\n\t\\begin{align*}\n\t\\begin{split}\n\t\t\\left\\langle\\frac{\\d \\rho_h}{\\d t}, \\varphi\\right\\rangle\n                \\,=\\,&\\sum_{i=1}^{N-1}\\left[\\left(\\nu(\\d U)_\\ihalf^++(\\d V_1)_\\ihalf^+\\right) \\,\\rho_i  \\,+\\,\n                  \\left(\\nu(\\d U)_\\ihalf^- \\,+\\, (\\d V_1)_\\ihalf^-\\right)\\,\\rho_{i+1} \\right]\\, \\left(\n                  \\varphi_{i+1} -\\varphi_i\\right)\n\\\\\n\t\t&- \\frac{\\epsilon}{2} \\,\\sum_{i=1}^{N-1}\\frac{\\rho_{i+1}^2 - \\rho_i^2}{\\Delta x_{i+1\/2}}\\, \\left(\n                  \\varphi_{i+1} -\\varphi_i\\right)\n\t\\end{split}\n\t\\end{align*}\n\tLet us begin with the self-diffusion part. Using the\n        Cauchy-Schwarz inequality, we estimate the discrete gradient and $\\rho$\n        itself by Corollary \\ref{cor:apriori} and Lemma \\ref{lem:aprioriL2}\n\t\\begin{align}\n\t\t\\label{eq:CS_estimate\n\t\t\\begin{split}\n\t\t\\frac{\\epsilon}{2}  &\\int_0^T \\sum_{i=1}^{N-1}\\frac{\\rho_{i+1}^2 - \\rho_i^2}{\\Delta x_{i+1\/2}}\\, \\left(\n                  \\varphi_{i+1} -\\varphi_i\\right) \\,\\d t \\\\\n\t\t&\\leq \\,\\frac{\\epsilon}{2} \\left\\|\\frac{\\partial \\varphi}{\\partial x} \\right\\|_{L^\\infty} \\int_0^T \\sum_{i=1}^{N-1} \\Delta x_\\ihalf |\\d \\rho_\\ihalf| \\, (\\rho_{i+1} + \\rho_i)\\, \\d t\\\\\n\t\n\t\t&\\leq \\,\\frac{\\epsilon}{2} \\left\\|\\frac{\\partial \\varphi}{\\partial x} \\right\\|_{L^\\infty} \\, \\left(\\int_0^T \\sum_{i=1}^{N-1} \\Delta x_\\ihalf |\\d \\rho_\\ihalf|^2 \\d t\\right)^{1\/2} \\,\\left(\\int_0^T  \\sum_{i=1}^{N} 4\\xi^{-1} \\Delta x_{i} |\\rho_i|^2 \\d t\\right)^{1\/2}\\\\\n\t\n\t\t&\\leq  C \\,\\|\\varphi\\|_{H^2(a,b)},\n\t\t\\end{split}\n\t\\end{align}\n\twhere we used that $\\varphi' \\in H^1 \\subset L^\\infty$.\n\n\tNext, we address the cross-diffusion and non-local interactions terms\n        using the same argument. For instance for the cross-diffusive\n        part, we have \n\t\\begin{align*}\n\t\t\\nu \\int_0^T &\\left|\\sum_{i=1}^{N-1}\\left[(\\d U)_\\ihalf^+ \\rho_i  \\,+\\,(\\d U)_\\ihalf^- \\rho_{i+1}\\right]\\right| \\, \\left|\n                  \\varphi_{i+1} -\\varphi_i\\right| \\,\\d t \\\\\n\t\t&\\leq \\frac{2\\,\\nu}{\\sqrt{\\xi}} \\,\\left\\|\\frac{\\partial \\varphi}{\\partial x}\\right\\|_{L^\\infty} \\left(\\int_0^T \\sum_{i=1}^{N-1} \\Delta x_\\ihalf |\\d U_\\ihalf|^2 \\d t\\right)^{1\/2}   \\left(\\int_0^T\\sum_{i=1}^{N}\\Delta x_i \\rho_i^2 \\d t\\right)^{1\/2}\\\\\n\t\t&\\leq C\\, \\|\\varphi\\|_{H^2(a,b)},\n\t\\end{align*}\n\twhere we used Corollary \\ref{cor:apriori} and Lemma\n        \\ref{lem:aprioriL2} again. The non-local interaction term is\n        estimated in the same way, thus there holds\n\t\\begin{align*}\n\t\t\\int_0^T \\left|\\left\\langle \\frac{\\d \\rho_h}{\\d t}, \\varphi \\right\\rangle\\right| \\d t \\leq C \\|\\varphi\\|_{H^2(a,b)}.\n\t\\end{align*}\n\tBy density of $C_c^\\infty((a,b))$ in $H_0^{2}(a,b)$  we may infer the boundedness of $(\\frac{\\d\\rho_h}{\\d t})_{h>0}$ in $L^1(0,T; H^{-2}(a,b))$, which concludes the proof.\n\\end{proof}\n\nFrom the latter result we can prove the convergence of the discrete advection\nfield $\\d V_{1,h}$ and $\\d V_{2,h}$ defined as in Definition \\ref{def:ApproximateSolutions}.\n\n\\begin{lemma}\n\\label{dVh}\nFor any $1\\leq p \\leq \\infty$ and $k\\in\\{1,2\\}$, the piecewise constant approximation  $\\d\nV_{k,h}$ converges strongly in $L^2(0,T;L^2(a,b))$  to $-(W_{k\\, 1}^\\prime\\star\n\\rho + W_{k\\, 2}^\\prime\\star \\eta)$, where $(\\rho,\\eta)$\ncorresponds to the limit obtained in Lemma \\ref{lem:strongconvergence}. \n\\end{lemma} \n\\begin{proof}\nLet $k\\in\\{1,2\\}$. For each $i=0,\\ldots,N-1$, and $x\\in [x_i, x_{i+1})$ we have\n\\begin{align*}\n\\d V_{k,\\, h}(x)  \\,=\\, (\\d V_{k})_\\ihalf &= -\\sum_{j=1}^{N} \\int_{C_j} \\frac{ W_{k\\,1}(x_{i+1}-y)-\n  W_{k\\,1}(x_i-y)}{\\Delta x_\\ihalf} \\,\\rho_j\\,\\d y\n\\\\\n&\\quad- \\sum_{j=1}^{N} \\int_{C_j} \\frac{W_{k\\,2}(x_{i+1}-y)- W_{k\\, 2}(x_i-y)}{\\Delta x_\\ihalf}\n\\,\\eta_j\\,\\d y.\n\\end{align*}\nWe define $V_{k, h}'$ and $V_{k}'$  as \n$$\n\\left\\{\n\\begin{array}{ll} \n\\ds {V}_{k, h}'(x) &\\ds \\,:=\\, - W_{k\\,1}^\\prime\\star \\rho_h \\,-\\,   W_{k\\,2}^\\prime\\star \\eta_h,\n\\\\ \\, \\\\\n\\ds V_{k}'(x)  & \\ds \\,:=\\, -W_{k\\,1}^\\prime\\star \\rho \\, - \\,   W_{k\\,2}^\\prime\\star \\eta.\n\\end{array}\\right.\n$$\nOn the one hand from the strong convergence of $(\\rho_h,\\eta_h)$ to $(\\rho,\\eta)$\nin $L^2(0,T;L^2(a,b))$ and the convolution product's properties, we obtain \n\\begin{equation}\n\\|  V_{k,\n  h}'- V_{k}'\\|_{L^2(0,T;L^2(a,b))}\\rightarrow 0, \\textrm{\\,when\\, }h\\rightarrow\n0.\n\\label{res:1}\n\\end{equation} \nOn the other hand, we have for any $x\\in [x_i,x_{i+1})$ \n\\begin{align*}\n|\\d V_{k,\\, h}(x)-V_{k,\\, h}^\\prime(x)|   &\\leq \\sum_{j=1}^{N} \\int_{C_j} \\left|\\frac{ W_{k\\,1}(x_{i+1}-y)-\n  W_{k\\,1}(x_i-y)}{\\Delta x_\\ihalf}-W_{k\\,1}^\\prime(x-y)\\right| \\,\\rho_j\\,\\d y,\n\\\\\n&\\quad+ \\sum_{j=1}^{N} \\int_{C_j} \\left|\\frac{W_{k\\,2}(x_{i+1}-y)- W_{k\\, 2}(x_i-y)}{\\Delta x_\\ihalf}\n-W_{k\\,2}^\\prime(x-y)\\right| \\,\\eta_j\\,\\d y,\n\\\\\n&\\leq  \\left( \\,\\|W_{k\\,1}^{\\prime\\prime}\\|_{L^\\infty} m_1\n        \\,+\\,\n        \\|W_{k\\,2}^{\\prime\\prime}\\|_{L^\\infty}\\,m_2\\;\\right) \\, h,\n\\end{align*}\nhence there exists a constant $C>0$ such that\n$$\n\t|\\d V_{k,\\, h}(x)-V_{k,\\, h}^\\prime(x)|^2 \\,\\leq \\, C^p \\, h^2.\n$$\nIntegrating over $x\\in [x_i, x_{i+1})$ and summing over $i\\in\n\\{1,\\ldots,N-1\\}$, we get that \n\\begin{equation}\n\\|  \\d V_{k,\n  h}- V_{k}'\\|_{L^2(0,T;L^2(a,b))}\\rightarrow 0, \\textrm{\\,when\\, }h\\rightarrow\n0.\n\\label{res:2}\n\\end{equation} \nNotice that $(x_1,x_{N})\\subset (a,b)$ where $x_1\\rightarrow a$ and\n$x_N\\rightarrow b$ as $h\\rightarrow 0$. From Eqs. (\\ref{res:1}) and (\\ref{res:2}) we  get that \n$\\|\\d V_{k,h} - V_{k,}^\\prime\\|_{L^2(0,T;L^2(a,b))}$ goes to zero as $h$\ntends to zero.\n\\end{proof}\n\n\\subsection{Weak compactness for the discrete gradients}\nIn the previous section we have established the strong $L^2$-convergence of both species, $(\\rho_{h})_{h>0}$ and $(\\eta_{h})_{h>0}$. However, in order to be able to pass to the limit in the cross-diffusion term $\\rho_{h} (\\d \\rho_{h} +  \\d \\eta_{h})$ we need to establish weak convergence in the discrete gradients in $L^2$. This is done in the following proposition.\n\\begin{proposition}[\\label{prop:WeakConvergenceDerivatives}Weak convergence of the derivatives]\n\tThe discrete spatial derivatives, defined in Definition\n        \\ref{def:ApproximateSolutions}, satisfy   $\\d \\beta_h$ \n        converges weakly to $\\frac{\\partial \\beta}{\\partial x}$ in $L^2(Q_T)$ and $\\beta\\in L^2(0,T;H^1(a,b))$, where $\\beta \\in \\{\\rho, \\eta, U\\}$\n\\end{proposition}\n\\begin{proof}\n\tTake $\\beta\\in \\{\\rho, \\eta, U\\}$, hence from Lemma \\ref{lem:strongconvergence}, we\n        know that $\\beta_h\\rightarrow \\beta$ strongly in\n        $L^2(Q_T)$. Furthermore, from Corollary \\ref{cor:apriori} we also\n        deduce that $\\d \\beta_h$ weakly converges to some function $r \\in\n        L^2(Q_T)$. \n\nLet us show that $\\beta\\in L^2(0,T, H^1(a,b))$ and\n        $r=\\frac{\\partial\\beta}{\\partial x}$. First, we have for any $t\\in [0,T]$ and any $\\varphi\\in\n        \\mathcal{C}^\\infty_c((0,T)\\times(a,b))$,\n\t\\begin{align*}\n\t\t\\int_{Q_T} \\beta_{h}(t)\\,\\frac{\\partial\n          \\varphi}{\\partial x} \\,\\d x &\\,=\\, \\int_0^T \\sum_{i=1}^N\\beta_i(t) \\,\\left[ \\varphi(t,x_\\ihalf) - \\varphi(t,x_\\imhalf)\\right]\\, \\d t\\\\\n\t\t&\\,=\\,-\\int_0^T \\sum_{i=1}^{N-1} \\Delta x_\\ihalf \\,\\d \\beta_\\ihalf(t)\\, \\varphi(t,x_\\ihalf)\\, \\d t,\n\t\\end{align*}\n\thaving used discrete integration by parts and the fact that\n        $\\varphi$ is  compactly supported, \\emph{i.e.} $\\varphi(t,\n        x_{N+1\/2})=\\varphi(t, x_{1\/2})=0$. Then, by\n        Definition \\ref{def:ApproximateSolutions} on the discrete\n        gradient, we may consider\n\t\\begin{align*}\n\t&\\left|\\int_0^T\\sum_{i=1}^{N-1}\\int_{x_i}^{x_{i+1}} \\d\n                        \\beta_{\\ihalf} \\varphi(t,x)\\d x \\d t +\n                        \\int_0^T\\int_a^b \\beta_{h}\\,\n                        \\frac{\\partial\\varphi}{\\partial x} \\,\\d x\\, \\d t \\right|\\\\\n\t&\\quad\\,\\leq\\, \\int_0^T\\sum_{i=1}^{N-1} \\int_{x_i}^{x_{i+1}} \\left| \\d \\beta_\\ihalf\\right|\\, \\left|\\varphi(t,x) - \\varphi(t,x_\\ihalf)\\right|\\,\\d x \\,\\d t\\\\\n\t&\\quad\\,\\leq\\,\\left\\|\\frac{\\partial \\varphi}{\\partial x}\\right\\|_\\infty\\,  \\left(\\int_0^T\\sum_{i=1}^{N-1}\\Delta x_{i+1\/2} \\left|\\d \\beta_\\ihalf\\right|^2\\d t\\right)^{1\/2} \\left(\\int_0^T\\sum_{i=1}^{N-1}\\Delta x_{i+1\/2}^3\\d t\\right)^{1\/2}\\\\\n\t&\\quad\\,\\leq\\,\\left\\|\\frac{\\partial \\varphi}{\\partial x}\\right\\|_\\infty\\, C^{1\/2} \\,T^{1\/2} \\,\\sqrt{b-a}\\,h,\n\\end{align*}\nThis yields the statement, when $h\\rightarrow 0$, for we have \n\t\\begin{align}\n\t\t\\label{eq:convergence_weak_derivative}\n\t\t\\int_0^T\\sum_{i=1}^{N-1} \\int_{x_i}^{x_{i+1}} \\d\n          \\beta_\\ihalf \\varphi(t,x)\\,\\d x\\, \\d t \\,+\\,\n          \\int_0^T\\int_a^b \\beta_{h}\\, \\frac{\\partial\\varphi}{\\partial\n          x}\\, \\d x\\, \\d t \\,\\rightarrow\\, 0,\n\t\\end{align}\n\twhich proves that $\\d \\beta_h$ converges  weakly to\n        $\\frac{\\partial \\beta}{\\partial x}$, as $h\\rightarrow 0$ and thus  $\\beta \\in L^2(0,T;H^1(a,b))$.\n\\end{proof}\n\n\n\n\\subsection{Passing to the limit}\nWe have now garnered all information necessary to  prove  Theorem \\ref{thm:convergence}. For brevity we shall\nonly show the convergence result for $\\rho$, as it follows for $\\eta$\nsimilarly, using the same arguments. Let $\\varphi\\in\nC_c^\\infty([0,T)\\times (a,b))$ be a test function. We introduce the\nfollowing notations: \n$$\n\\left\\{ \n\\begin{array}{lll}\n\\mathcal{E}_h &:=&\\ds \\int_0^T\\int_{a}^b \\rho_{h} \\frac{\\partial\n  \\varphi}{\\partial t} \\,\\d x\\, \\d t \\,+\\, \\int_a^b \\rho_{h}(0)\\,\\varphi(0)\\,\\d x, \n\\\\ \\, \\\\ \n\\mathcal{A}_h &:=&\\ds \\int_0^T\\int_{a}^b  \\d V_{1,h}\n          \\,\\rho_h\\,\\frac{\\partial\\varphi}{\\partial x}\\,\\d x\\, \\d t,\n\\\\ \\, \\\\ \n\\mathcal{C}_h &:=&\\ds\\nu\\,\\int_0^T\\int_{a}^b   \\d U_{h} \\,\\rho_h\\,\\frac{\\partial\\varphi}{\\partial x}\\,\\d x\\, \\d t,\n\\\\ \\, \\\\ \n\\mathcal{D}_h &:=&\\ds\\frac{\\epsilon}{2}\\int_0^T \\int_{a}^b\\rho_{h}^2\\,\\frac{\\partial^2\\varphi}{\\partial x^2}\\,\\d x\\, \\d t.\n\\end{array}\\right.\n$$\nand \n$$\n\\varepsilon(h) \\,:=\\, \\mathcal{E}_h \\,+\\, \\mathcal{A}_h \\,+\\, \\mathcal{C}_h \\,+\\, \\mathcal{D}_h.\n$$\n\nOn the other hand, we set \n$$\n\\varphi_i(t) = \\frac{1}{\\Delta x_i} \\int_{C_i} \\varphi(t,x)\\,\\d x,\n$$ \nand multiply the scheme, Eq. \\eqref{eq:scheme_evol}, by the test function and integrate in time and space to get\n\\begin{equation}\n \\mathcal{E}_h \\,+\\, \\mathcal{A}_{1,h} \\,+\\, \\mathcal{C}_{1,h} \\,+\\, \\mathcal{D}_{1,h} \\,=\\, 0,\n\\label{cata:0}\n\\end{equation}\nwhere \n$$\n\\left\\{ \n\\begin{array}{lll}\n\\mathcal{A}_{1,h} &\\,:=\\,&\\ds\\sum_{i=1}^{N-1} \\int_0^T \\Delta x_\\ihalf\\,\\left[(\\d V_1)_{\\ihalf}^+ \\,\\rho_i\n          + (\\d V_1)_{\\ihalf}^- \\,\\rho_{i+1}\\right]\\,\\d\\varphi_{i+1\/2}(t)\\, \\d t,\n\\\\ \\, \\\\ \n\\mathcal{C}_{1,h} &\\,:=\\,&\\ds\\nu\\,\\sum_{i=1}^{N-1} \\int_0^T \\Delta x_\\ihalf\\,\\left[(\\d U)_{\\ihalf}^+ \\,\\rho_i\n          + (\\d U)_{\\ihalf}^- \\,\\rho_{i+1}\\right]\\,\\d\\varphi_{i+1\/2}(t)\\, \\d t,\n\\\\ \\, \\\\ \n\\mathcal{D}_{1,h} &\\,:=&\\,\\ds-\\frac{\\epsilon}{2}\\sum_{i=1}^{N-1} \\int_0^T \\left[\\rho_{i+1}^2\n          - \\rho_{i}^2\\right]\\,\\d\\varphi_{i+1\/2}(t)\\, \\d t.\n\\end{array}\\right.\n$$\n\nWhen $h$ tends to zero and from the strong convergence of\n$(\\rho_h,\\eta_h)_{h>0}$ to $(\\rho,\\eta)$ in $L^2(Q_T)$, the strong convergence of $(\\d V_{k,h})_{h>0}$ to\n$V_k^\\prime$ in $L^2(Q_T)$  and the weak convergence of the discrete gradient\n$(\\d U_h)_{h>0}$ to $-\\frac{\\partial \\sigma}{\\partial x}$ in $L^2(Q_T)$, it\nis easy to see that \n\\begin{eqnarray*}\n\\varepsilon(h) &\\rightarrow& \\int_0^T\\int_{a}^b \\left\\{ \\rho \\left[\\frac{\\partial\n  \\varphi}{\\partial t}\\,+\\,\\left(\\frac{\\partial\n  V_1}{\\partial x} - \\nu\\frac{\\partial\n  }{\\partial x}\\left(\\rho+\\eta\\right)\\right)\\,\\frac{\\partial\n  \\varphi}{\\partial x} \\right] \\,+\\, \\frac{\\epsilon}{2}\\,\\rho^2 \\, \\frac{\\partial^2\n  \\varphi}{\\partial x^2} \\right\\}  \\,\\d x\\, \\d t\\\\\n&\\,& \\,+\\, \\int_a^b \\rho(0)\\,\\varphi(0)\\,\\d x,\n\\end{eqnarray*}\nwhen $h\\rightarrow 0$. Therefore it suffices to prove that\n$\\varepsilon(h)\\rightarrow 0$, as $h$ goes to zero, which will be\nachieved by proving that $\\mathcal{A}_h-\\mathcal{A}_{1,h}$, $\\mathcal{C}_h-\\mathcal{C}_{1,h}$ and\n$\\mathcal{D}_h-\\mathcal{D}_{1,h}$ vanish  in the limit $h\\rightarrow 0$. \n\n\n\t\n\\subsubsection*{The self-diffusion part $\\mathcal{D}_h-\\mathcal{D}_{1,h}$.} On the one\nhand,  after a simple integration we get\n\t\\begin{eqnarray*}\n\\mathcal{D}_h\t&=&\n            \\frac{\\epsilon}{2}\\,\\sum_{i=1}^N \\,\\int_0^T\\rho_i^2(t)\n            \\,\\left[\\frac{\\partial \\varphi}{\\partial x}(t,x_\\ihalf)\n            -\\frac{\\partial \\varphi}{\\partial x}(t,x_{i-1\/2})\\right]\n            \\,\\d t \n\\\\\n&=&\n            -\\frac{\\epsilon}{2}\\,\\sum_{i=1}^{N-1} \\,\\int_0^T\\left[\\rho_{i+1}^2(t)\\,-\\, \\rho_{i}^2(t) \\right] \n            \\,\\frac{\\partial \\varphi}{\\partial x}(t,x_\\ihalf)\\,\\d t. \n\t\\end{eqnarray*}\nHence, we have\n\\begin{eqnarray*}\n\\mathcal{D}_h-\\mathcal{D}_{1,h} &=& -\\frac{\\epsilon}{2}\\,\\sum_{i=1}^{N-1} \\,\\int_0^T\\left[\\rho_{i+1}^2(t)\\,-\\, \\rho_{i}^2(t) \\right] \n            \\,\\left[\\frac{\\partial \\varphi}{\\partial x}(t,x_\\ihalf)-\\d\n                    \\varphi_\\ihalf(t)\\right]\\,\\d t \n\\end{eqnarray*}\nand observing that \n$$\n\\left|\\frac{\\partial \\varphi}{\\partial x}(t,x_\\ihalf)-\\d\n                    \\varphi_\\ihalf(t)\\right| \\,\\leq \\;\\left\\|\n                    \\frac{\\partial^2 \\varphi}{\\partial\n                      x^2}\\right\\|_{L^\\infty} h,\n$$\nwe obtain, in conjunction with the Cauchy-Schwarz inequality and the {\\it a priori} bounds established in  Corollary \\ref{cor:apriori}, that\nand Lemma \\ref{lem:aprioriL2},\n\\begin{eqnarray}\n\\label{resu:D}\n|\\mathcal{D}_h-\\mathcal{D}_{1,h}| &\\leq& \\frac\\epsilon2 \\left\\|\n                    \\frac{\\partial^2 \\varphi}{\\partial\n                      x^2}\\right\\|_{L^\\infty}\\,\\left(\\sum_{i=1}^{N-1}\n  \\int_0^T\\Delta x_{\\ihalf}|\\d\\rho_{\\ihalf}|^2 \\d t\\right)^{1\/2}\\,\n\\frac{2\\,\\|\\rho_h\\|_{L^2(Q_T)}}{\\xi^{1\/2}}\\,h \\\\\n&\\leq&\\, C\\, h,\n\\nonumber\n\\end{eqnarray}\nin the virtue of the estimate Eq. \\eqref{eq:CS_estimate}.\n\t\n\\subsubsection*{The cross-diffusion part.} \nLet us now treat the cross-diffusion part. This term is more complicated\nsince it involves the piecewise constant functions $\\rho_h$ and $\\d\nU_h$, which are not defined on the same mesh. Thus, on the one hand we\nreformulate the discrete cross-diffusion term \n$\\mathcal{C}_{1,h}$ as $\\mathcal{C}_{1,h}=\\mathcal{C}_{10,h}+\\mathcal{C}_{11,h}$ with\n$$\n\\mathcal{C}_{10,h} \\,=\\,\\nu\\,\\sum_{i=1}^{N-1} \\int_0^T \\Delta x_\\ihalf\\, (\\d U)_{\\ihalf}^-\n          \\,\\left[\\rho_{i+1}-\\rho_{i}\\right]\\,\\d\\varphi_{i+1\/2}(t)\\, \\d t\n$$\nand\n$$\n\\mathcal{C}_{11,h} \\,=\\,\\nu\\,\\sum_{i=1}^{N-1} \\int_0^T \\Delta x_\\ihalf \\,\\rho_i\\,\\d U_{\\ihalf} \\,\\d\\varphi_{i+1\/2}(t)\\, \\d t,\n$$\nwhere a direct computation and the application of  Corollary \\ref{cor:apriori}  and Lemma \\ref{lem:aprioriL2} yield\n\\begin{eqnarray}\n\\label{resu:2}\n|\\mathcal{C}_{10,h}| &\\leq&\\nu\\, \\left\\|\n                    \\frac{\\partial \\varphi}{\\partial\n                      x}\\right\\|_{L^\\infty} \\,\\|\\d U_{h}\\|_{L^2(Q_T)}\n                    \\,\\|\\d\\rho_{h}\\|_{L^2(Q_T)} \\, h\\\\\n&\\leq&  C\\, h.\n\\nonumber \n\\end{eqnarray}\nOn the other hand, the term $\\mathcal{C}_h$ can be rewritten as\n$$\n\\mathcal{C}_h \\,=\\,\\nu\\,\\int_0^T\\sum_{i=1}^{N-1} \\d U_{\\ihalf}(t)\\,   \\int_{x_i}^{x_{i+1}} \\rho_h\\, \\frac{\\partial\\varphi}{\\partial x}\\,\\d x\\, \\d t.\n$$\nSince  \n\\begin{eqnarray*}\n\\int_{x_i}^{x_{i+1}}   \\rho_h\\, \\frac{\\partial\\varphi}{\\partial x}\\,\\d\n  x & =&\n\\rho_i \\,\\left[\\varphi(t,x_\\ihalf) - \\varphi(t,x_i)\\right] \\,+\\,\n  \\rho_{i+1}\\, \\left[\\varphi(t,x_{i+1})   - \\varphi(t,x_{\\ihalf})\\right],\n\\\\\n&=& \\left[ \\rho_i - \\rho_{i+1}\\right] \\,\\left[\\varphi(t,x_\\ihalf) - \\varphi(t,x_i)\\right] \\,+\\,\n  \\rho_{i+1}\\, \\left[\\varphi(t,x_{i+1})   - \\varphi(t,x_{i})\\right],\n\\end{eqnarray*}\nthe term $\\mathcal{C}_h$ can be decomposed as $\\mathcal{C}_h=\\mathcal{C}_{00,h}+\\mathcal{C}_{01,h}$\nwith \n$$\n\\mathcal{C}_{00,h} \\,=\\, -\\nu\\,\\int_0^T \\sum_{i=1}^{N-1}  \\d U_{\\ihalf}\\, \\left[\\rho_{i+1}\n     -\\rho_i\\right]\n          \\,\\left[\\varphi(t,x_\\ihalf)- \\varphi(t,x_i)\\right]\\,\\d t \n$$\nand\n$$\n\\mathcal{C}_{01,h} \\,=\\,\\nu\\,\\int_0^T \\sum_{i=1}^{N-1}  \\d\n    U_{\\ihalf} \\,\\rho_i\\,\\left[\\varphi(t,x_{i+1})-\\varphi(t,x_i)\\right]\\,\\d t.\n$$\nSimilarly to (\\ref{resu:2}), the first term $\\mathcal{C}_{00,h}$  can\nbe estimated as\n\\begin{equation}\n\\label{resu:3}\n|\\mathcal{C}_{00,h} |\\,\\leq  C\\, h,\n\\end{equation}\nwhereas the second term $\\mathcal{C}_{01,h}$ is compared to $\\mathcal{C}_{11, h}$ \n$$\n|\\mathcal{C}_{01, h}-\\mathcal{C}_{11, h}|\\,\\leq\\,\\nu \\int_0^T \\sum_{i=1}^{N-1} \\Delta x_\\ihalf |\\d\n    U_{\\ihalf}| \\,\\rho_i\\,\\left|\\frac{\\varphi(t,x_{i+1})-\\varphi(t,x_i)}{\\Delta x_\\ihalf}-d\\varphi_{i+1\/2}(t)\\right|\\,\\d t.\n$$\nUsing a second order Taylor expansion of $\\varphi$ at $x_i$ and\n$x_{i+1}$, it yields that \n$$\n\\left|\\frac{\\varphi(t,x_{i+1})-\\varphi(t,x_i)}{\\Delta\n    x_\\ihalf}-d\\varphi_{i+1\/2}(t)\\right|\\,\\leq\\, C\\, h,\n$$\nhence we get  from  Corollary \\ref{cor:apriori}  and Lemma \\ref{lem:aprioriL2} that\n\\begin{equation}\n\\label{resu:4}\n|\\mathcal{C}_{01, h}-\\mathcal{C}_{11, h}| \\leq C\\, h.\n\\end{equation}\nfrom  Corollary \\ref{cor:apriori}  and Lemma \\ref{lem:aprioriL2}. Gathering Eqs. (\\ref{resu:2}), (\\ref{resu:3}), and (\\ref{resu:4}), we finally\nobtain that\n\\begin{equation}\n\\label{resu:C}\n|\\mathcal{C}_{h}-\\mathcal{C}_{1, h}| \\leq C\\, h.\n\\end{equation}\n\t\n\\subsubsection*{The advective part.} The evaluation of\n$\\mathcal{A}_{h}-\\mathcal{A}_{1,h}$ is along the same lines of the\ncross-diffusion terms $\\mathcal{C}_{h}-\\mathcal{C}_{1,h}$ since the latter is treated\nas an advective term. Hence, thanks to Lemma \\ref{dVh}, we get that\n\\begin{equation}\n\\label{resu:A}\n|\\mathcal{A}_{h}-\\mathcal{A}_{1, h}| \\leq C\\, h.\n\\end{equation}\n\nFinally by definition of $\\varepsilon(h)$ and using  Eq.\n(\\ref{cata:0}) together with Eqs. (\\ref{resu:D}),  (\\ref{resu:C}), and\n(\\ref{resu:A}),  we obtain\n\\begin{eqnarray*}\n|\\varepsilon(h)| &=& | -(\\mathcal{A}_{1,h} \\,+\\, \\mathcal{C}_{1,h} \\,+\\, \\mathcal{D}_{1,h}) + \\mathcal{A}_{h} \\,+\\, \\mathcal{C}_{h} \\,+\\, \\mathcal{D}_{h} |\n\\\\\n&\\leq & | \\mathcal{A}_{h} -\\mathcal{A}_{1,h} | \\,+\\, | \\mathcal{C}_{h} -\\mathcal{C}_{1,h} | \\,+\\,  |\n        \\mathcal{D}_{h} -\\mathcal{D}_{1,h} | \n\\\\\n&\\leq&  C\\, h,\n\\end{eqnarray*}\nthat is, $\\varepsilon(h)\\rightarrow 0$, when $h\\rightarrow 0$, which\nproves that $(\\rho,\\eta)$ is a weak solution to\nEq. \\eqref{eq:crossdiffsystem}. This proves the second item of Theorem\n\\ref{thm:convergence}.\n\nFinally the last item concerning the existence of solutions to\n\\eqref{eq:crossdiffsystem} is a direct consequence of the convergence.  \n\n\n\\section{A Fully Discrete Implicit Scheme}\nIn this section we shall comment on a discrete-in-time version of the semi-discrete scheme  \\eqref{eq:scheme}. To this end we replace the time derivative in Eq. \\eqref{eq:scheme_evol} by simple forward differences and obtain the following implicit and fully-discrete scheme\n\\begin{align}\n\t\\label{eq:discrete_scheme_evol}\n\t\\left\\{\n\t\\begin{array}{l}\n\t\\displaystyle\n\t\\frac{\\rho_i^{n+1}-\\rho_i^{n}}{\\Delta t}=\\displaystyle- \\frac{\\mathcal{F}_{\\ihalf}^{n+1} - \\mathcal{F}_{i-1\/2}^{n+1}}{\\Delta x_i},\\\\[1em]\n\t\\displaystyle\n\t\\frac{\\eta_i^{n+1} - \\eta_i^{n}}{\\Delta t}=\\displaystyle - \\frac{\\mathcal{G}_{\\ihalf}^{n+1} - \\mathcal{G}_{i-1\/2}^{n+1}}{\\Delta x_i},\n\t\\end{array}\n\t\\right.\n\\end{align}\nwhere $\\Delta t >0$. System \\eqref{eq:discrete_scheme_evol} gives rise\nto two approximating sequences $(\\rho_i^n)_{1\\leq i \\leq N}$ and\n$(\\eta_i^n)_{1\\leq i \\leq N}$, for $0\\leq n\\leq M$ where $M:= \\lceil T \/ \\Delta t\\rceil$ and the discrete time instances $t^n:=n\\Delta t$; cf. Theorem \\ref{thm:existence_discrete}. Here the numerical fluxes are given by\n\\begin{align}\n\t\\label{eq:implicit_numerical_fluxes}\n\t\\left\\{\n\t\\begin{array}{l}\n\t\\displaystyle\n\t\\mathcal{F}_{\\ihalf}^{n+1} =\t\\displaystyle \\left[\\nu\\,(\\d U^{n+1})_{\\ihalf}^+ +\n                       (\\d V_1^{n})_{\\ihalf}^+\\right] \\,\\rho_i^{n+1} \\,+\\,\n                       \\left[\\nu\\,(\\d U^{n+1})_{\\ihalf}^- + (\\d\n                       V_1^n)_{\\ihalf}^-\\right]\\,\\rho_{i+1}^{n+1}\\\\ [1.5em]\n\t\\phantom{\\mathcal{F}_{\\ihalf}=\t}- \\ds\\frac{\\epsilon}{2} \\,\\frac{(\\rho_{i+1}^{n+1})^2 - (\\rho_i^{n+1})^2}{\\Delta x_{i+1\/2}},\\\\[1.5em]\n\t\\displaystyle\n\t\\mathcal{G}_{\\ihalf}^{n+1} =\t\\displaystyle \\left[ \\nu\\,(\\d U^{n+1})_{\\ihalf}^+\n                       +(\\d V_2^n)_{\\ihalf}^+\\right]\\,   \\eta_i^{n+1} \\,+\\,\n                       \\left[\\nu\\, (\\d U^{n+1})_{\\ihalf}^- + (\\d V_2^{n})_{\\ihalf}^-\\right]\\,\\eta_{i+1}^{n+1}\\\\[1.5em]\n\t\\phantom{\\mathcal{G}_{\\ihalf} =\t}- \\ds\\,\\frac{\\epsilon}{2}\\, \\frac{(\\eta_{i+1}^{n+1})^2 - (\\eta_i^{n+1})^2}{\\Delta x_{i+1\/2}},\n\t\\end{array}\n\t\\right. \\tag{\\ref{eq:discrete_scheme_evol}a}\n\\end{align}\nfor $i = 1,\\ldots, N-1$, with the numerical no-flux boundary condition\n\\begin{align}\n\\label{eq:implicit_num_noflux}\n\t\\mathcal{F}_{1\/2}^{n+1} = \\mathcal{F}_{N+1\/2}^{n+1} &= 0, \\quad \\text{and} \\quad \\mathcal{G}_{1\/2}^{n+1} = \\mathcal{G}_{N+1\/2}^{n+1} = 0,\\tag{\\ref{eq:discrete_scheme_evol}b}\n\\end{align}\nfor $n=0,\\ldots,M$. Recall that\n\\begin{align*}\n\t(\\d U^{n+1})_\\ihalf^{\\pm} = \\left(\\frac{(\\rho_{i+1}^{n+1} + \\eta_{i+1}^{n+1}) - (\\rho_{i}^{n+1} + \\eta_{i}^{n+1})}{\\Delta x_\\ihalf}\\right)^\\pm,\n\\end{align*}\nand\n\n\\begin{align*}\n(\\d V_{k}^n)_\\ihalf^\\pm &= \\bigg(-\\sum_{j=1}^{N} \\int_{C_j} \\frac{ W_{k\\,1}(x_{i+1}-y)-\n  W_{k\\,1}(x_i-y)}{\\Delta x_\\ihalf} \\,\\rho_j^n\\,\\d y\n\\\\\n&\\qquad - \\sum_{j=1}^{N} \\int_{C_j} \\frac{W_{k\\,2}(x_{i+1}-y)- W_{k\\, 2}(x_i-y)}{\\Delta x_\\ihalf}\n\\,\\eta_j^n\\,\\d y\\bigg)^\\pm,\n\\end{align*}\nfor $k=1,2$.\n\n\n\n\nSimilarly to Definition \\ref{def:ApproximateSolutions} we define the piecewise constant interpolation by\n\t\\begin{align*}\n\t\t\\rho_{h}(t,x):= \\rho_i^n,\\qquad \\text{and} \\qquad \\eta_{h}(t,x):=\\eta_i^n,\n\t\\end{align*}\n\tfor all $(t,x) \\in [t^n, t^{n+1})\\times C_i$, with $i=1,\\ldots,N$, and $n=0,\\ldots, M$. Moreover, we define the discrete approximation of the spatial gradients as \n\t\\begin{align*}\n\t\t\\d_x \\rho_{h}(t,x) = \\frac{\\rho_{i+1}^n-\\rho_i^n}{\\Delta x_{i+1\/2}}, \\qquad \\mbox{and}\\qquad \\d_x \\eta_{h}(t,x) = \\frac{\\eta_{i+1}^n - \\eta_i^n}{\\Delta x_{i+1\/2}},\n\t\\end{align*}\n\tfor $(t,x)\\in[t^n,t^{n+1})\\times [x_i,x_{i+1})$, for\n        $i=1,\\ldots,N-1$ and $n=0,\\ldots,M$. As above, we set the discrete gradients to zero\n         on $(a,x_1)$ and  $(x_N,b$). Furthermore, we define the discrete time derivative as  \n\t\\begin{align*}\n\t\t\\d_t \\rho_{h}(t,x) = \\frac{\\rho_{i}^{n+1}-\\rho_i^n}{\\Delta t}, \\qquad \\mbox{and}\\qquad \\d_t \\eta_{h}(t,x) = \\frac{\\eta_{i}^n - \\eta_i^n}{\\Delta t},\n\t\\end{align*}\n\tfor $(t,x)\\in[t^n,t^{n+1})\\times C_i$, for\n        $i=1,\\ldots,N$ and $n=0,\\ldots,M-1$.\n        \n        \n\n\\begin{theorem}[\\label{thm:existence_discrete}Existence and uniqueness\n  result]\nLet $\\rho_i^0, \\eta_i^0$ be nonnegative initial data with mass $m_1$ and $m_2$, respectively, and \n\tassume the following CFL condition\n\t\\begin{align}\n\t\t\\label{eq:CFL}\n16\\, (m_1+m_2)\\,\\frac{\\Delta  t}{(\\xi\\,h)^3}\\,<\\, 1.\n\t\\end{align}\nThen there exists a unique nonnegative solution $(\\rho_i^n, \\eta_i^n)$\nto scheme \\eqref{eq:discrete_scheme_evol},\n        \\eqref{eq:implicit_numerical_fluxes} and\n        \\eqref{eq:implicit_num_noflux}. \n\\end{theorem}\n\\begin{proof}\nWe show existence first and prove uniqueness later. Suppose we are given $(\\rho_i^n)_{1\\leq i\\leq N}$ and $(\\eta_i^n)_{1\\leq i\\leq N}$ from some previous iteration. In order to construct the next iteration we shall employ Brouwer's fixed point theorem. It is easy to verify that the set\n\\begin{align*}\n\t\\mathcal{X}:=\\left\\{ (\\rho, \\eta) \\in \\mathbb{R}^{2N} \\,|\\, \\forall 1\\leq i\\leq N: \\rho_i, \\eta_i \\geq 0,  \\sum_{i=1}^N \\Delta x_i \\rho_i \\leq m_1, \\text{ and } \\sum_{i=1}^N \\Delta x_i \\eta_i \\leq  m_2\\right\\},\n\\end{align*}\nis a convex and compact subset of $\\mathbb{R}^N\\times\\mathbb{R}^N$. Hence, we define the\nfixed point operator $\\mathfrak{S}:\\mathcal{X} \\rightarrow \\mathbb{R}^N \\times \\mathbb{R}^N$ by setting $(\\rho^\\star, \\eta^\\star) = \\mathfrak{S}(\\rho, \\eta)$\nwhere $(\\rho^\\star,\\eta^\\star)$ are implicitly given as\n\\begin{align*}\n\\rho^\\star_i \\,=\\, \\rho_i^n - \\frac{\\Delta t}{\\Delta x_i} \\left(\\mathcal{F}_\\ihalf^\\star - \\mathcal{F}_\\imhalf^\\star\\right),\\;\\,\\, \\eta^\\star_i = \\eta_i^n - \\frac{\\Delta t}{\\Delta x_i} \\left(\\mathcal{G}_\\ihalf^\\star - \\mathcal{G}_\\imhalf^\\star\\right)\n\\end{align*}\nfor any $(\\rho, \\eta) \\in \\mathcal{X}$, where $\\mathcal{F}^\\star$ and $\\mathcal{G}^\\star$ denote the numerical fluxes\n\\begin{align}\n\\label{eq:LIN_implicit_numerical_fluxes}\n\t\\left\\{\n\t\\begin{array}{l}\n\t\\displaystyle\n\t\\mathcal{F}_{\\ihalf}^\\star =\t\\displaystyle \\left[\\nu\\,(\\d U^{})_{\\ihalf}^+ +\n                       (\\d V_1^n)_{\\ihalf}^+\\right] \\,\\rho_i^{\\star} \\,+\\,\n                       \\left[\\nu\\,(\\d U)_{\\ihalf}^- + (\\d\n                       V_1^n)_{\\ihalf}^-\\right]\\,\\rho_{i+1}^{\\star}\n\\\\ [1.5em]\n\t\\phantom{\\mathcal{F}_{\\ihalf}=\t}- \\ds\\frac{\\epsilon\\,\n          (\\rho_{i+1}^{}+ \\rho_i^{})}{2} \\,\\frac{\\rho_{i+1}^{\\star} -\n          \\rho_i^{\\star}}{\\Delta x_{i+1\/2}},\n\\\\[1.5em]\n\t\\displaystyle\n\t\\mathcal{G}_{\\ihalf}^{\\star} =\t\\displaystyle \\left[ \\nu\\,(\\d U^{})_{\\ihalf}^+\n                       +(\\d V_2^n)_{\\ihalf}^+\\right]\\,   \\eta_i^{\\star} \\,+\\,\n                       \\left[\\nu\\, (\\d U^{})_{\\ihalf}^- + (\\d\n          V_2^n)_{\\ihalf}^-\\right]\\,\\eta_{i+1}^{\\star}\n\\\\[1.5em]\n\t\\phantom{\\mathcal{G}_{\\ihalf} =\t}- \\ds\\,\\frac{\\epsilon \\,(\\eta_{i+1} + \\eta_i)}{2}\\, \\frac{\\eta_{i+1}^{\\star} - \\eta_i^{\\star}}{\\Delta x_{i+1\/2}},\n\t\\end{array}\n\t\\right.\n\\end{align}\nfor $i = 1,\\ldots, N-1$ where  $\\d U$ is computed from $(\\rho,\\eta)$,  with the numerical no-flux boundary condition\n\\begin{align}\n\\label{eq:LIN_implicit_num_noflux}\n\t\\mathcal{F}_{1\/2}^{\\star} = \\mathcal{F}_{N+1\/2}^{\\star} &= 0, \\quad \\text{and} \\quad \\mathcal{G}_{1\/2}^{\\star} = \\mathcal{G}_{N+1\/2}^{\\star} = 0.\n\\end{align}\nNotice that for any given $(\\rho,\\eta)\\in \\mathcal{X}$, the viscosity\nterms in front of the discrete gradients involved in the definition of\nthe fluxes $\\mathcal{F}_{\\ihalf}^\\star$ and\n$\\mathcal{G}_{\\ihalf}^\\star$ are indeed nonnegative, hence  the couple $(\\rho^\\star,\\eta^\\star)$ is well\ndefined since it corresponds to the unique solution of a classical\nfully implicit scheme in time with an upwind discretisation for the convective\nterms and a centred approximation for diffusive terms \\cite{EGT00}.  Moreover, since $\\rho^n$ and $\\eta^n$ are\nnonnegative and using  the monotonicity of the numerical flux with respect to\n$(\\rho^\\star,\\eta^\\star)$, we prove that both densities $\\rho^\\star$\nand $\\eta^\\star$ are also nonnegative. Furthermore, using the nonnegativity and the no-flux conditions, we get\n\\begin{align*}\n\t\\|\\rho^\\star\\|_{L^1} = \\sum_{i=1}^N\\Delta x_i \\rho_i^\\star = m_1, \\qqand \\|\\eta^\\star\\|_{L^1} = \\sum_{i=1}^N\\Delta x_i \\eta_i^\\star = m_2, \n\\end{align*}\nwhich yields that $\\mathfrak{S}(\\mathcal{X})\\subset \\mathcal{X}$.   Finally,\n$\\mathfrak{S}$ is continuous as the composition of continuous\nfunctions. Thus, we may apply Brouwer's fixed point theorem to infer\nthe existence of a fixed point, $(\\rho^{n+1}, \\eta^{n+1})$. It now remains to show uniqueness of the fixed point. \n\nTo treat in a systematic way the boundary conditions and simplify the presentation, we define ghost values\nfor $(\\rho,\\eta)$ by setting for $\\alpha\\in\\{\\rho,\\,\\eta\\}$, and $k\\in\\{1,\\,2\\}$,\n$$\n\\alpha_{N+1}\\,=\\,\\alpha_N \\quad  \\alpha_{0}\\,=\\,\\alpha_1 \\quad{\\rm and}\\quad\n(\\d V_k^n)_{N+1\/2} \\,=\\,(\\d V_k^n)_{1\/2} = 0.\n$$\nThen we consider two solutions $(\\tilde \\rho,\n\\tilde \\eta)$ and $(\\rho, \\eta)$ to  (\\ref{eq:discrete_scheme_evol}).\nSetting $h(x):=x^2\/2$,  $s := \\rho-\\tilde\\rho$ and $r :=\n\\eta - \\tilde \\eta$, we  get after substituting the two solutions to\n(\\ref{eq:discrete_scheme_evol}), for $i=1,\\ldots, N$,\n\\begin{align*}\n\ts_i = &- \\frac{\\Delta t}{\\Delta x_i} \\left((\\d V_1^n)_\\ihalf^+\n                \\,s_i + (\\d V^n_1)_\\ihalf^- \\, s_{i+1}-(\\d\n                V_1^n)_\\imhalf^+ \\, s_{i-1} - (\\d V^n_1)_\\imhalf^- \\, s_i\\right)\\\\\n\t&+ \\frac{\\Delta t}{\\Delta x_i} \\left(\\frac{[h(\\rho_{i+1})\n          -h(\\tilde \\rho_{i+1})] \\,-\\,  \n          [h(\\rho_i) -\n          h(\\tilde \\rho_i)]}{\\Delta x_\\ihalf } \\,-\\, \\frac{[h(\\rho_{i})- h(\\tilde\n          \\rho_i)] \\,-\\, [h(\\rho_{i-1})-h(\\tilde \\rho_{i-1})]}{\\Delta x_\\imhalf}\\right)\\\\\n\t&- \\frac{\\Delta t}{\\Delta x_i} \\left( (\\d U)_\\ihalf^+ \\, s_i\n          + (\\d U)_\\ihalf^- \\, s_{i+1} -(\\d U)_\\imhalf^+ \\, s_{i-1}\n          -(\\d U)_\\imhalf^- \\, s_{i} \\right) \\\\\n\t&- \\frac{\\Delta t}{\\Delta x_i} \\left([(\\d U)_\\ihalf^+  - (\\d\n          \\tilde U)_\\ihalf^+]\\,\\tilde \\rho_i \\,+\\, [(\\d U)_\\ihalf^-  - (\\d\n          \\tilde U)_\\ihalf^-]\\, \\tilde \\rho_{i+1} \\right) \\\\\n\t&+\\frac{\\Delta t}{\\Delta x_i} \\left( [(\\d U)_\\imhalf^+  - (\\d\n          \\tilde U)_\\imhalf^+] \\,\\tilde \\rho_{i-1} \\,+\\, [(\\d U)_\\imhalf^- - (\\d \\tilde U)_\\imhalf^-] \\,\\tilde \\rho_{i} \\right)\n\\end{align*}\nand a similar relation for $(r_i)_{1\\leq i\\leq N}$. Applying a Taylor\nexpansion on $h(\\rho)=h(\\tilde\\rho) + h'(\\hat\\rho)\\,s$,\nwith $\\hat\\rho$ a convex combination of $\\rho$ and $\\tilde\\rho$, we\nmay write  \n\\begin{equation}\n\\label{tmp:0}\n\\left\\{\n\\begin{array}{ll}\n(\\Delta x_i + \\Delta t \\,A_i )\\,s_i = &  \\ds\\Delta t \\left( B_{i-1}\\,s_{i-1} \\,+\\, C_{i+1} \\, s_{i+1}  \\right) \\\\[0.75em]\n\t&-\\ds\\Delta t \\,\\left([(\\d U)_\\ihalf^+  - (\\d\n          \\tilde U)_\\ihalf^+]\\,\\tilde \\rho_i \\,+\\, [(\\d U)_\\ihalf^-  - (\\d\n          \\tilde U)_\\ihalf^-]\\, \\tilde \\rho_{i+1} \\right) \\\\[0.75em]\n\t&+\\ds\\Delta t \\,\\left( [(\\d U)_\\imhalf^+  - (\\d\n          \\tilde U)_\\imhalf^+] \\,\\tilde \\rho_{i-1} \\,+\\, [(\\d U)_\\imhalf^- - (\\d \\tilde U)_\\imhalf^-] \\,\\tilde \\rho_{i} \\right),\n\\end{array}\\right.\n\\end{equation}\nwhere $A_i$, $B_{i-1}$, $C_{i+1}$ are nonnegative coefficients given by\n$$\n\\left\\{\n\\begin{array}{rl}\n\\ds A_i \\!\\!\\!&=\\,\\ds +(\\d V_1^n)_\\ihalf^+- (\\d V^n_1)_\\imhalf^-  +  (\\d\n  U)_\\ihalf^+ -  (\\d U)_\\imhalf^- + \\frac{h'(\\hat\\rho_i)}{\\Delta x_\\ihalf}+ \\frac{h'(\\hat\\rho_i)}{\\Delta x_\\imhalf},\n\\\\[1.1em]\n\\ds B_{i-1} \\!\\!\\!&=\\,\\ds  +(\\d V_1^n)_\\imhalf^+ +   (\\d U)_\\imhalf^+ + \\frac{h'(\\hat\\rho_{i-1})}{\\Delta x_\\imhalf},\n \\\\[1.1em]\n\\ds C_{i+1} \\!\\!\\!&=\\, \\ds -(\\d V_1^n)_\\ihalf^-- (\\d U)_\\ihalf^- + \\frac{h'(\\hat\\rho_{i+1})}{\\Delta x_\\ihalf}.\n\\end{array}\n\\right.\n$$\nNow, we multiply  equation (\\ref{tmp:0})  by ${\\rm sign}(s_i)$ and sum\nover $i=1,\\ldots,N$, hence using that  $x\\rightarrow x^\\pm$ is\nLipschitz continuous and observing that $A_i=B_i+C_i$, with $A_i,\n\\,B_i, \\,C_i\\,\\geq\\,\n0$, it yields \n\\begin{align*}\n\\sum_{i=1}^N \\Delta x_i |s_i| \\,\\,\\leq\\,\\,&  2\\Delta t\n                                            \\,\\|\\tilde\\rho\\|_\\infty\\,\\sum_{i=1}^N\\left(\n                                            \\frac{|s_{i+1}|+|r_{i+1}|\n                                          +|s_{i}|+|r_{i}|\n                                            }{\\Delta\n                                            x_\\ihalf} + \\frac{|s_{i-1}|+|r_{i-1}|\n                                          +|s_{i}|+|r_{i}|\n                                            }{\\Delta\n                                            x_\\imhalf} \\right) \n\\end{align*}\nand in a similar way,\n\\begin{align*}\n\\sum_{i=1}^N \\Delta x_i |r_i| \\,\\,\\leq\\,\\,&  2\\Delta t\n                                            \\,\\|\\tilde\\eta\\|_\\infty \\,\\sum_{i=1}^N\\left(\n                                            \\frac{|r_{i+1}|+|s_{i+1}|\n                                          +|r_{i}|+|s_{i}|\n                                            }{\\Delta\n                                            x_\\ihalf} + \\frac{|r_{i-1}|+|s_{i-1}|\n                                          +|r_{i}|+|s_{i}|\n                                            }{\\Delta\n                                            x_\\imhalf} \\right).\n\\end{align*}\nGathering these latter inequalities and from \\eqref{admissible}, it\ngives that \n\\begin{align*}\n\\sum_{i=1}^N \\Delta x_i \\left( |s_i| + |r_i|\\right) \\,\\,\\leq\\,\\,16\\,\n  \\|(\\tilde\\rho,\\tilde\\eta)\\|_\\infty \\,\\frac{\\Delta t}{(\\xi\\,h)^2}\\, \\sum_{i=1}^N \\Delta x_i \\left( |s_i| + |r_i|\\right). \n\\end{align*}\nFinally,  from the nonnegativity and the preservation of mass,  we\nhave\n$$\n\\|(\\tilde\\rho,\\tilde\\eta)\\|_\\infty  \\,\\leq\\,\n\\frac{m_1+m_2}{\\xi h},\n$$\nhence under the condition \\eqref{eq:CFL}, we conclude that $s=r=0$ and the uniqueness follows.\n\\end{proof}\n\nIt is worth to mention here that  the condition \\eqref{eq:CFL} is not\noptimal since we only use the discrete $L^1$-estimate on $\\rho$ and $\\eta$ to\ncontrol the discrete gradient and the $L^\\infty$-norm.\n\n         \nWe are now in position to state for  \\eqref{eq:discrete_scheme_evol},\n        \\eqref{eq:implicit_numerical_fluxes} and\n        \\eqref{eq:implicit_num_noflux} an analogous results to the\nsemi-discrete case\n\\begin{theorem}[\\label{thm:implicit_thm}Convergence to a weak solution of the implicit Euler discretisation]\nUnder the assumptions of Theorem \\ref{thm:existence_discrete}, let $\\rho_0, \\eta_0 \\in L_+^1(a,b) \\cap L_+^\\infty(a,b)$ be\n        some initial data  and $Q_T := (0,T)\\times (a,b)$ as\n        above. Then, given two nonnegative sequences\n        $(\\rho_i^n)_{1\\leq i \\leq  N}$ and $(\\eta_i^n)_{1\\leq i\\leq\n          N}$ satisfying  \\eqref{eq:discrete_scheme_evol},\n        \\eqref{eq:implicit_numerical_fluxes} and\n        \\eqref{eq:implicit_num_noflux}, for any $n\\in\\{0,\\ldots,M\\}$, then\n\\begin{itemize}\n\\item[$(i)$]  up to a subsequence, the piecewise constant \n        approximations converge strongly in $L^2(Q_T)$ to\n        $(\\rho,\\eta) \\in L^2(Q_T)$, where $(\\rho,\\eta)$ is  a weak solution as in\n        Definition \\ref{def:WeakSolution}. Furthermore we have $\\rho$, $\\eta\n        \\in L^2(0,T; H^1(a,b))$;\n\\item[$(ii)$] in particular, system \\eqref{eq:crossdiffsystem} has a weak solution.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}[Sketch of the proof of Theorem \\ref{thm:implicit_thm}]\n\tIt is easily observed that the total mass is conserved due to the discrete no-flux boundary conditions, cf. \\eqref{eq:implicit_num_noflux}. Together with the nonnegativity we were able to prove a semi-discrete version of the energy estimate Eq. \\eqref{eq:cts_inequality} which is at the heart of the convergence result. Similarly as above, we are able to prove a fully discrete version of the energy estimate which then reads\n\t\\begin{align*}\n\t\t\\sum_{n=0}^M &\\Delta t\\sum_{i=1}^N \\Delta x_i \\frac{\\rho_i^{n+1} \\log \\rho_i^{n+1} - \\rho_i^n\\log \\rho_i^n}{\\Delta t} +\\frac{\\eta_i^{n+1} \\log \\eta_i^{n+1} - \\eta_i^n\\log \\eta_i^n}{\\Delta t}\\\\\n\t\t &+\\,\\sum_{n=0}^M\\Delta t\\sum_{i=1}^{N-1}\\!\\Delta x_{i+1\/2}\\left[\\nu\\, |\\d U_\\ihalf|^2 \n\t\t\\,+\\, \\frac{\\epsilon}{4}\\left(  |\\d \\rho_\\ihalf^{n+1}|^2\n\t\t\\,+\\,  |\\d \\eta_\\ihalf^{n+1}|^2\\right)\n\t\t\t\t\\right]\\,\\leq\\, C_{\\epsilon},\n\t\\end{align*}\n\twith $C_\\epsilon>0$ as before. The inequality follows from the convexity of $x (\\log x - 1)$ since\n\t\\begin{align*}\n\t\t\\rho_i^{n+1} (\\log \\rho_i^{n+1} - 1) - \\rho_i^n (\\log \\rho_i^n -  1)\\leq  \\log(\\rho_i^{n+1}) (\\rho_i^{n+1} - \\rho_i^n).\n\t\\end{align*}\n\tMultiplying this expression by $\\Delta x_i \/ \\Delta t$ and summing over $i=1,\\ldots N$ and $n=0,\\ldots, M$ we obtain \n\t\\begin{align*}\n\t\t\\sum_{n=0}^M\\Delta t \\sum_{i=1}^N\\Delta x_i \\frac{\\rho_i^{n+1} (\\log \\rho_i^{n+1} - 1) - \\rho_i^n (\\log \\rho_i^n -  1)}{\\Delta t} \\leq  \\sum_{n=0}^M\\Delta t \\sum_{i=1}^N\\Delta x_i\\log(\\rho_i^{n+1}) \\frac{\\rho_i^{n+1} - \\rho_i^n}{\\Delta t}.\n\t\\end{align*}\n\tThe right-hand side is then substituted by the scheme and simplified along the lines of the proof of Lemma \\ref{lem:entropy_control}. Since the computations are exactly the same we omit them here for brevity and only note that it is important to set\n\t\\begin{align}\n\t\\tilde \\rho_\\ihalf^{n+1} := \\left\\{\n\t\t\\begin{array}{ll}\n\t\t\t\\ds\\dfrac{\\rho_{i+1}^{n+1} -\n                  \\rho_i^{n+1}}{\\log\\rho_{i+1}^{n+1} - \\log\n                  \\rho_i^{n+1}}, & \\text{if $\\rho_i^{n+1} \\neq\n                                   \\rho_{i+1}^{n+1}$},\n\\\\ \\, \\\\%[1.1em]\n\t\t\t\\ds\\dfrac{\\rho_i^{n+1} + \\rho_{i+1}^{n+1}}{2}, & \\text{else},\n\t\t\\end{array}\n\t\t\\right.\n\\end{align}\n\tto obtain the right sign in the numerical artefacts in Eq. \\eqref{eq:numerical_artifacts}, which now read\n$$\n\t\\left\\{\n\t\\begin{array}{l}\n\t\\displaystyle\n\t(\\rho_i^{n+1} - \\tilde \\rho_\\ihalf^{n+1}) \\D\\log\n          \\rho_\\ihalf^{n+1} \\big(\\nu(\\d U)_{\\ihalf}^+ + (\\d\n          V_1)_{\\ihalf}^+\\big) \\leq 0,\n\\\\ \\, \\\\%[1.4em]\n\t\\displaystyle\n\t(\\rho_{i+1}^{n+1} - \\tilde \\rho_\\ihalf^{n+1}) \\D\\log \\rho_\\ihalf^{n+1} \\big(\\nu (\\d U)_{\\ihalf}^- + (\\d V_1)_{\\ihalf}^-\\big) \\leq 0.\n\t\\end{array}\n\t\\right.\n$$\nFollowing the lines of the proof above we obtain the fully discrete a priori bounds\n\t\\begin{align*}\n\t\t\\sum_{n=0}^N \\Delta t \\sum_{i=1}^{N-1} \\Delta x_{\\ihalf} \\left(\\,|\\d_x\\rho_\\ihalf^{n+1}|^2 + |\\d_x\\eta_\\ihalf^{n+1}|^2 + \\,|\\d_x U_\\ihalf|^2 \\right) \\leq C,\n\t\\end{align*}\n\tfor some constant $C>0$, where we used `$\\d_x$' to denote the discrete spatial gradient as before.\n\tAgain, an application of Aubin-Lions Lemma provides relative compactness in the space $L^2((0,T)\\times (a,b))$ of the piecewise constant interpolations $\\rho_h, \\eta_h$. As above, the discrete gradients are uniformly bounded in $L^2((0,T)\\times (a,b))$ and their weak convergence is a consequence of the Banach-Alaoglu Theorem. Identifying the limits as a weak solution to system \\eqref{eq:crossdiffsystem} is shown in the same way as above and we leave it as an exercise for the reader.\n\\end{proof}\n\n\\begin{rem}[Explicit Scheme]\n\tWe do not consider an explicit scheme here since its analysis\n        is much more complicated due to the lack of uniform\n        estimates. Indeed, an explicit scheme  requires a CFL\n        condition on the time step which appears in the stability\n        analysis or the energy estimate provided in Lemma\n        \\ref{lem:entropy_control}. In our case, it leads to\n$$\n\\frac{\\rho^{n+1} \\log \\rho^{n+1} - \\rho^n\\log \\rho^n}{\\Delta\n  t} = \\frac{\\rho^{n+1}  - \\rho^n}{\\Delta\n  t} \\left( \\log \\rho^n - 1\\right) \\,+\\,   \\frac{(\\rho^{n+1}  - \\rho^n)^2}{\\left|\\rho^{n+1\/2}\\right|^2\\,\\Delta\n  t}, \n$$\nwhere $\\rho^{n+1\/2}$ belongs to the interval $[\\rho^n,\n\\rho^{n+1}]$ or  $[\\rho^{n+1},\n\\rho^{n}]$. The control of this reminder term would require some\nlower bounds estimates on the density $\\rho$ (see for instance \\cite{filbet-herda}). \n\n\\end{rem}\n\n\n\n\\begin{comment}\nIn this case the numerical fluxes are given by\n\\begin{align}\n\t\\label{eq:explicit_numerical_fluxes}\n\t\\left\\{\n\t\\begin{array}{l}\n\t\\displaystyle\n\t\\mathcal{F}_{\\ihalf}^{n} =\t\\displaystyle \\left[\\nu\\,(\\d U)_{\\ihalf}^+ +\n                       (\\d V_1)_{\\ihalf}^+\\right] \\,\\rho_i^{n} \\,+\\,\n                       \\left[\\nu\\,(\\d U)_{\\ihalf}^- + (\\d\n                       V_1)_{\\ihalf}^-\\right]\\,\\rho_{i+1}^{n}\\\\ [1.5em]\n\t\\phantom{\\mathcal{F}_{\\ihalf}=\t}- \\ds\\frac{\\epsilon}{2} \\,\\frac{(\\rho_{i+1}^{n})^2 - (\\rho_i^{n})^2}{\\Delta x_{i+1\/2}},\\\\[1.5em]\n\t\\displaystyle\n\t\\mathcal{G}_{\\ihalf}^{n} =\t\\displaystyle \\left[ \\nu\\,(\\d U)_{\\ihalf}^+\n                       +(\\d V_2)_{\\ihalf}^+\\right]\\,   \\eta_i^{n} \\,+\\,\n                       \\left[\\nu\\, (\\d U)_{\\ihalf}^- + (\\d V_2)_{\\ihalf}^-\\right]\\,\\eta_{i+1}^{n}\\\\[1.5em]\n\t\\phantom{\\mathcal{G}_{\\ihalf} =\t}- \\ds\\,\\frac{\\epsilon}{2}\\, \\frac{(\\eta_{i+1}^{n})^2 - (\\eta_i^{n})^2}{\\Delta x_{i+1\/2}},\n\t\\end{array}\n\t\\right. \\tag{\\ref{eq:discrete_scheme_evol}a'}\n\\end{align}\nfor $i = 1,\\ldots, N-1$, with the numerical no-flux boundary condition\n\\begin{align}\n\\label{eq:explicit_num_noflux}\n\t\\mathcal{F}_{1\/2}^{n} = \\mathcal{F}_{N+1\/2}^{n} &= 0, \\quad \\text{and} \\quad \\mathcal{G}_{1\/2}^{n} = \\mathcal{G}_{N+1\/2}^{n} = 0,\\tag{\\ref{eq:discrete_scheme_evol}b'}\n\\end{align}\n\\end{comment}\n\n\n\n\n\\section{Numerical examples and validation\\label{sec:numerical_results}}\nIn this section we perform some numerical simulations of system \\eqref{eq:crossdiffsystem} using our scheme, Eqs. \\eqref{eq:scheme}. In Section \\ref{sec:error_numerical_convergence_order} we test our scheme by computing the error between the numerical simulation and a benchmark solution on a finer grid. Furthermore we determine the numerical convergence order. In Section \\ref{sec:stationary_states} we compute the numerical stationary states of system \\eqref{eq:crossdiffsystem} and discuss the implication of different cross-diffusivities and  the self-diffusivities, respectively.\n\n\nLet us note here, that in the case of no regularising porous-medium\ndiffusion, \\emph{i.e.} $\\epsilon=0$, and certain singular potentials\nthe stationary states of system \\eqref{eq:crossdiffsystem} are even\nknown  explicitly \\cite{CHS17}. This allows us to compare the\nnumerical solution directly to the analytical stationary state in  Section \\ref{sec:CHS17}. \n\n\n\n\n\nThroughout the remainder of this section we apply the scheme \\eqref{eq:scheme} to system \\eqref{eq:crossdiffsystem}\nusing different self-diffusions, $\\epsilon$,  and cross-diffusions, $\\nu$.\n\n\n\\subsection{Error and numerical order of convergence\\label{sec:error_numerical_convergence_order}}\nThis section is dedicated to validating our main convergence result, Theorem \\ref{thm:convergence}. Due to the lack of  explicit solutions we compute the numerical solution on a fine grid and consider it  a benchmark solution. We then compute numerical approximations on coarser grids and study the error in order to obtain the numerical convergence order.\n\nIn all our simulations we use a fourth order Runge-Kutta scheme to solve the ordinary differential equations Eqs. \\eqref{eq:scheme} with initial data Eqs. \\eqref{eq:initial_data}. The discrepancy between the benchmark solution and numerical solutions on coarser grids is measured by the error\n\\begin{align*}\n\te:=\\left(\\Delta t \\sum_{k=1}^M \\left(\\Delta x \\sum_{i=1}^N |\\rho_{\\mathrm{ex}}(t_k,x_i) - \\rho(t_k, x_i)|^2 + |\\eta_{\\mathrm{ex}}(t_k,x_i) - \\eta(t_k, x_i)|^2\\right)\\right)^{1\/2}.\n\\end{align*}\nHere $\\rho_{\\mathrm{ex}}$, $\\eta_{\\mathrm{ex}}$ denote the benchmark solutions. We use this quantity to study the convergence of our scheme as the grid size decreases. \n\\subsubsection{No non-local interactions}\nLet us begin with the purely diffusive system. We consider system \\eqref{eq:crossdiffsystem}  without any interactions, \\emph{i.e.}  $W_{ij}\\equiv 0$, for $i,j=1,2$, and we choose $\\nu=0.5$ and $\\epsilon=0.1$. \n\nIn Figure \\ref{fig:purelydiffusive} we present the convergence result as we decrease the grid size. We computed a benchmark solution on a grid of $\\Delta x = 2^{-10}$ on the time interval $[0,10]$ with $\\Delta t = 0.05$. Figure \\ref{fig:symmetric_initial_data_only_diffusion} shows the convergence for symmetric initial data whereas Figure \\ref{fig:asymmetric_initial_data_only_diffusion} shows the convergence of the same system in case of asymmetric initial data. In both cases we overlay a line of slope one and we conclude that the numerical convergence is of order one.\n\n\\begin{figure}[ht!]\n\t\\centering\n\t\\subfigure[Symmetric initial data. $\\rho_0(x)=\\eta_0(x)= \\mathds{1}_{[7,10]}$.]{\n\t\t\\includegraphics[width=0.45\\textwidth]{error_only_diffusion_symmetric_init}\n\t\t\\label{fig:symmetric_initial_data_only_diffusion}\n\t}\n\t\\hspace{1em}\n\t\\subfigure[Asymmetric initial data. $\\rho_0(x)=\\mathds{1}_{[5,7]}$ and $\\eta_0(x)=\\mathds{1}_{[10,12]}$.]{\n\t\t\\includegraphics[width=0.45\\textwidth]{error_only_diffusion_asymmetric_init}\n\t\\label{fig:asymmetric_initial_data_only_diffusion}\n\t}\n\t\\caption{In the purely diffusive system all interaction kernels are set to zero. Both graphs show the convergence to the benchmark solution. The triangular markers  denote the discrepancy between the numerical solution and the benchmark solution. A line of slope one is superimposed for the ease of comparison. On the left we start the system with symmetric initial conditions, on the right we start with asymmetric initial data. In both cases the scheme has numerical convergence order 1.}\n\t\\label{fig:purelydiffusive}\n\\end{figure}\n\n\n\\subsubsection{Gaussian cross-interactions\\label{sec:numeric_gaussians}}\nNext, we add non-local self-interaction and cross-interactions. We choose smooth Gaussians with different variances. These potentials, like the related, more singular Morse potentials, are classical in mathematical biology since oftentimes the availability of sensory information such as sight, smell or hearing is spatially limited \\cite{OCBC06, CMP13, CHM14}. For the intraspecific interaction we use\n\\begin{align*}\n\tW_{11}(x) = W_{22}(x) = 1 - \\exp\\left(-\\frac{|x|^4}{4\\times 0.1}\\right),\n\\end{align*}\nwhile we choose\n\\begin{align*}\n\tW_{12}(x) = -W_{21}(x) = 1 - \\exp\\left(-\\frac{|x|^2}{2\\times 0.1}\\right),\n\\end{align*}\nfor the interspecific interactions. \n\n\nWe consider system \\eqref{eq:crossdiffsystem} with the diffusive coefficients $\\nu = 0.4$ and $\\epsilon\\in\\{0.1, 0.5\\}$, and we initialise the system with\n\\begin{align*}\n\t\\rho(x) = \\eta(x) = c \\big((s-6.5)(9.5-s)\\big)^+,\n\\end{align*}\non the domain $[0,9]$. Here the constant $c$ is such that $\\rho$ and $\\eta$ have unit mass. Figure \\ref{fig:gaussian_interactions_error} depicts the simulation with Gaussian kernels as interaction potentials. In Figures \\ref{fig:gaussian_interactions_low_self_diffusion} \\& \\ref{fig:gaussian_interactions_high_self_diffusion} we present the error plots  corresponding to $\\epsilon\\in\\{0.1,0.5\\}$. Again we observe convergence to the reference solution with a first order accuracy in space. \n\\begin{figure}[ht!]\n\t\\centering\n\t\\subfigure[Convergence to benchmark solution for individual diffusion constant $\\epsilon=0.1$.]{\n\t\t\\includegraphics[width=0.45\\textwidth]{error_Gaussians}\n\t\t\\label{fig:gaussian_interactions_low_self_diffusion}\n\t}\n\t\\hspace{1em}\n\t\\subfigure[Convergence to benchmark solution for individual diffusion constant $\\epsilon=0.5$.]{\n\t\t\\includegraphics[width=0.45\\textwidth]{error_Gaussian_LargeDiffusion}\n\t\t\\label{fig:gaussian_interactions_high_self_diffusion}\n\t}\n\t\\caption{We choose Gaussian interaction kernels of different strengths and ranges for the self-interaction and the cross-interaction, respectively. The graphs show the numerical convergence order in the cases of $\\epsilon=0.1$ (left),  and  $\\epsilon=0.5$ (right), respectively.}\n\t\\label{fig:gaussian_interactions_error}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{General behaviour of solutions and stationary states\\label{sec:stationary_states}}\nIn this section we aim to study the asymptotic behaviour of system \\eqref{eq:crossdiffsystem} numerically.  Let us begin by going back to the set up of Section \\ref{sec:numeric_gaussians}. We note that the potentials were chosen in such a way that there is an attractive intraspecific force, and the cross-interactions are chosen as attractive-repulsive explaining the segregation observed in Figure \\ref{fig:gaussian_interactions_left}. For larger self-diffusivity, $\\epsilon=0.5$, we see that some mixing occurs. In the absence of any individual diffusion we would have expected adjacent species with a jump discontinuity at their shared boundary \\cite{CHS17}. However, this phenomenon is no longer possible as we have a control on the gradients of each individual species, by Lemma \\ref{cor:apriori}, rendering jumps impossible thus explaining the continuous transition. \n\\begin{figure}[ht!]\n\t\\centering\n\t\\subfigure[Stationary state ($\\epsilon=0.1$).]{\n\t\t\\includegraphics[width=0.47\\textwidth]{exact_Gaussian}\n\t\t\\label{fig:gaussian_interactions_left}\n\t}\n\t\\subfigure[Stationary state ($\\epsilon=0.5$).]{\n\t\t\\includegraphics[width=0.47\\textwidth]{exact_Gaussian_LargeDiffusion}\n\t\t\\label{fig:gaussian_interactions_right}\n\t}\n\t\\caption{We choose Gaussian interaction kernels of different strengths and ranges for the self-interaction and the cross-interaction, respectively. The graphs correspond to the simulationed stationary states for self-diffusivities $\\epsilon=0.1$, and $\\epsilon=0.5$, respectively.}\n\t\\label{fig:gaussian_interactions_steady_states}\n\\end{figure}\n\n\n\n\nIn the subsequent section we shall push our scheme even further by dropping the smoothness assumption on our potentials.\n\n\n\n\n\\subsubsection{\\label{sec:CHS17_regularised}\\label{sec:CHS17}Case of singular potentials}\nIn this section we go beyond the limit of what we could prove in this paper. On the one hand we consider more singular potentials and on the other hand we consider vanishing individual diffusion. We study system \\eqref{eq:crossdiffsystem} for $\\epsilon\\in\\{0,\\,0.02,\\,0.04,\\, 0.06,\\, 0.09\\}$, and $\\nu \\in\\{0.05, 0.5\\}$. \nHere the potentials are given by\n\\begin{align*}\n\tW_{11}(x) = W_{22}(x) = x^2\/2.\n\\end{align*}\nfor the self-interaction terms and  \n\\begin{align*}\n\tW_{12}(x) = |x| = \\pm W_{21}(x).\n\\end{align*}\nfor the cross-interactions. The system is posed on the domain $[0,5]$\nwith a grid size of $\\Delta x = 2^{-8}$. Note that the case of\n$\\epsilon=0$ corresponds to the absence of individual diffusion,  see\nFigures \\ref{fig:CHS_agreement_attrrep} \\&\n\\ref{fig:CHS_agreement_attrrep_adjacent}. By virtue of Corollary\n\\ref{cor:apriori}, it is the individual diffusion that regularises the\nstationary states, in the sense that we will not observe any\ndiscontinuities in either $\\rho$ or $\\eta$. As we add individual\ndiffusion we can see the immediate regularisation. While stationary\nstates may still remain segregated, as it is shown in Figure\n\\ref{fig:CHS_agreement_attrrep_increasing_eps}, adjacent solutions are\nnot possible anymore (see Figure \\ref{fig:CHS_agreement_attrrep_increasing_eps_adjacent}).\n\\begin{figure}[ht!]\n\t\\centering\n\t\\subfigure[In the case $\\nu = 0.05$, $\\epsilon=0$, we obtain a great agreement of the numerically computed stationary states and the analytical stationary states described in \\cite{CHS17}.]{\n\t\\includegraphics[width=0.47\\textwidth]{agreement_CHS17}\n\t\\label{fig:CHS_agreement_attrrep}\n\t}\\hspace{0.2cm}\n\t\\subfigure[Adding individual diffusion may still lead to segregated stationary states. However both species remain continuous as they mix.]{\n\t\\includegraphics[width=0.47\\textwidth]{increasing_eps_strict_segregation}\n\t\\label{fig:CHS_agreement_attrrep_increasing_eps}\n\t}\n\t\\subfigure[The case $\\nu = 0.5$, $\\epsilon=0$ leads to adjacent stationary states. Again we see an excellent agreement of the numerical stationary states and the analytical ones \\cite{CHS17}.]{\n\t\\includegraphics[width=0.47\\textwidth]{CHS_adjacent}\n\t\\label{fig:CHS_agreement_attrrep_adjacent}\n\t}\\hspace{0.2cm}\n\t\\subfigure[The regularising effect of the individual\n        diffusion, by Corollary \\ref{cor:apriori}, becomes apparent immediately. Instantaneously both species become continuous as they start to intermingle in a small region. This region grows as we keep increasing the individual diffusion.]{\n\t\\includegraphics[width=0.47\\textwidth]{increasing_eps_adjacent}\n\t\\label{fig:CHS_agreement_attrrep_increasing_eps_adjacent}\n\t}\t\n\t\\caption{We pushed our numerical scheme to see how it performs in regimes in which we are unable to prove convergence. We chose Newtonian cross-interactions in the attractive-repulsive case. The red curves denote the symmetric stationary states of $\\eta$ while the blue curves are the stationary distributions of $\\rho$. The different line widths and styles correspond to varying values of $\\epsilon$.}\n\\end{figure}\nIn the case of attractive-repulsive interspecific interactions, \\emph{i.e.}\n\\begin{align*}\n\tW_{12}(x) = |x| = -W_{21}(x).\n\\end{align*}\n we expect both species to segregate \\cite{CHS17}. We initialise the system with the following symmetric initial data\n\\begin{align*}\n\t\\rho(x) = \\eta(x) = c \\big((x-3)(5-x)\\big)^+,\t\n\\end{align*}\nas symmetric initial data are known to approach stationary states \\cite{CHS17}. Here the constant $c$ normalises the mass of $\\rho$ and $\\eta$ to one.\n\nFigures \\ref{fig:CHS_agreement_attrrep} \\& \\ref{fig:CHS_agreement_attrrep_adjacent} show our scheme performs well even in regimes we are unable to show convergence due to the lack of regularity in the potentials as well as the lack of regularity due to the absence of the porous medium type self-diffusion. While the schemes developed in \\cite{CCH15, CHS17} are asymptotic preserving, their convergence to weak solutions of the respective equations could not be established. We reproduce the steady states of \\cite{CHS17} that exhibit phase separation phenomena. Figure \\ref{fig:additional_numerics} displays the stationary state in the case $\\nu=0.09$ and $\\epsilon=0$. Even in the case of no regularising individual diffusion and with Newtonian cross-interactions we observe a numerical convergence order of one.\n\n\n\\begin{figure}[ht!]\n\t\\centering\n\t\\subfigure[Strictly segregated stationary state in the absence of individual diffusion, $\\epsilon=0$, and Newtonian potentials.]{\n\t\\includegraphics[width=0.47\\textwidth]{additional_exact_solution}\n\t}\n\t\\subfigure[Convergence to benchmark solution in the case of $\\epsilon=0$ and Newtonian attractive-repulsive potentials.]{\n\t\\includegraphics[width=0.47\\textwidth]{additional_error_Singular}\n\t}\n\t\\caption{Stationary state and numerical convergence order in the case of Newtonian attractive-repulsive cross-interactions. Even though our estimates fail in the analysis above we observe a numerical convergence order of one.}\n\t\\label{fig:additional_numerics}\n\\end{figure}\n\n\n\nIn the case of attractive-attractive self-interactions, \\emph{i.e.} $W_{12}(x) = |x| = W_{21}(x)$, we observe an interesting phenomenon. Even in the absence of the individual diffusion, \\emph{i.e.} $\\epsilon=0$, some additional mixing occurs even though we expect sharp boundaries, see Figure \\ref{fig:CHS_agreement_attrattr}, due to numerical diffusion. This is in contrast to the finite volume schemes proposed in \\cite{CCH15, CHS17}.\n\n\n\\begin{figure}[ht!]\n\t\\centering\n\\includegraphics[width=0.47\\textwidth]{agreement_CHS17_attrattr}\n\t\\caption{We choose $m_1 = 0.6$ and $m_2=0.1$ in order to be able to compare the stationary state with the explicit one given in \\cite{CHS17}. We can see a strong resemblance between the numerical stationary state and the one obtained analytically. However there are some regimes of mixing due to numerical diffusion.}\n\\label{fig:CHS_agreement_attrattr}\n\\end{figure}\n\n\n\n\\begin{comment}\n\\section{Discrete energy}\nIn order to understand the two-species case better we start with a simpler model for one species.\n\\begin{align}\n\t\\partial_t \\rho = \\div (\\rho \\nabla(h(\\rho) + V)).\n\\end{align}\nConsider\n\\begin{align*}\n\t\\sum \\frac{\\mathcal{F}_\\ihalf-\\mathcal{F}_\\imhalf}{\\Delta x} \\, h_i\n\t&= \\sum  \\mathcal{F}_\\ihalf \\d h_\\ihalf\\\\\n\t&= -\\sum \\big[(-\\d h)_\\ihalf^+ \\rho_i + (-\\d h)_\\ihalf^- \\rho_{i+1}\\\\\n\t&\\qquad \\qquad +(-\\d V)_\\ihalf^+ \\rho_i + (-\\d V)_\\ihalf^- \\rho_{i+1}\\big] (-\\d h_\\ihalf)\\\\[0.8em]\n\t&= -\\sum \\big[(-\\d h)_\\ihalf^+ (\\rho_i-\\hat \\rho_\\ihalf) + (-\\d h)_\\ihalf^- (\\rho_{i+1}- \\hat \\rho_\\ihalf)\\\\\n\t&\\qquad \\qquad +(-\\d V)_\\ihalf^+ (\\rho_i-\\hat \\rho_\\ihalf) + (-\\d V)_\\ihalf^- (\\rho_{i+1}-\\hat \\rho_\\ihalf)\\big] (-\\d h_\\ihalf)\\\\\n\t&\\phantom{=} - \\sum_i \\hat \\rho_\\ihalf (\\d h_\\ihalf)^2 - \\sum \\hat \\rho_\\ihalf (-\\d V)_\\ihalf (-\\d h_\\ihalf)\\\\[0.8em]\n\t&= -\\sum \\big[\\,|(-\\d h)_\\ihalf^+|^2\\, (\\rho_i-\\hat \\rho_\\ihalf) + |(-\\d h)_\\ihalf^-|^2\\, (\\rho_{i+1}- \\hat \\rho_\\ihalf)\\big]\\\\\n\t&\\phantom{=\\;} - \\sum_i \\hat \\rho_\\ihalf (\\d h_\\ihalf)^2 - \\sum \\hat \\rho_\\ihalf (-\\d V)_\\ihalf (-\\d h_\\ihalf)\\\\\n\t&\\phantom{=\\;} -\\sum \\big[ (-\\d V)_\\ihalf^+ (\\rho_i-\\hat \\rho_\\ihalf) + (-\\d V)_\\ihalf^- (\\rho_{i+1}-\\hat \\rho_\\ihalf)\\big] (-\\d h_\\ihalf)\\big],\n\\end{align*}\nand similarly\n\\begin{align*}\n\t\\sum &\\frac{\\mathcal{F}_\\ihalf-\\mathcal{F}_\\imhalf}{\\Delta x} \\, V_i\\\\\n\t&= \\sum  \\mathcal{F}_\\ihalf \\d V_\\ihalf\\\\\n\t&= -\\sum \\big[(-\\d h)_\\ihalf^+ \\rho_i + (-\\d h)_\\ihalf^- \\rho_{i+1}\\\\\n\t&\\qquad \\qquad +(-\\d V)_\\ihalf^+ \\rho_i + (-\\d V)_\\ihalf^- \\rho_{i+1}\\big] (-\\d V_\\ihalf)\\\\[0.8em]\n\t&= -\\sum \\big[(-\\d h)_\\ihalf^+ (\\rho_i-\\hat \\rho_\\ihalf) + (-\\d h)_\\ihalf^- (\\rho_{i+1}- \\hat \\rho_\\ihalf)\\\\\n\t&\\qquad \\qquad +(-\\d V)_\\ihalf^+ (\\rho_i-\\hat \\rho_\\ihalf) + (-\\d V)_\\ihalf^- (\\rho_{i+1}-\\hat \\rho_\\ihalf)\\big] (-\\d V_\\ihalf)\\big]\\\\\n\t&\\qquad \\qquad -\\sum_i  \\hat \\rho_\\ihalf (-\\d V)_\\ihalf^2 - \\sum_i \\hat \\rho_\\ihalf (-\\d h_\\ihalf)(-\\d V_\\ihalf)\\\\[0.8em]\n\t&= -\\sum \\big[\\,|(-\\d V)_\\ihalf^+|^2\\, (\\rho_i-\\hat \\rho_\\ihalf) + |(-\\d V)_\\ihalf^-|^2\\, (\\rho_{i+1}- \\hat \\rho_\\ihalf)\\big]\\\\\n\t&\\phantom{=\\;} - \\sum_i \\hat \\rho_\\ihalf (\\d V_\\ihalf)^2 - \\sum \\hat \\rho_\\ihalf (-\\d V)_\\ihalf (-\\d h_\\ihalf)\\\\\n\t&\\phantom{=\\;} -\\sum \\big[ (-\\d h)_\\ihalf^+ (\\rho_i-\\hat \\rho_\\ihalf) + (-\\d h)_\\ihalf^- (\\rho_{i+1}-\\hat \\rho_\\ihalf)\\big] (-\\d V_\\ihalf)\\big].\n\\end{align*}\nWe combine both computations and obtain\n\\begin{align}\n\t&-\\sum \\big[\\,|(-\\d h)_\\ihalf^+|^2\\, (\\rho_i-\\hat \\rho_\\ihalf) + |(-\\d h)_\\ihalf^-|^2\\, (\\rho_{i+1}- \\hat \\rho_\\ihalf)\\big]\\\\\n\t&\\phantom{=\\;} - \\sum_i \\hat \\rho_\\ihalf (\\d h_\\ihalf)^2 - \\sum \\hat \\rho_\\ihalf (-\\d V)_\\ihalf (-\\d h_\\ihalf)\\\\\n\t&\\phantom{=\\;} -\\sum \\big[ (-\\d V)_\\ihalf^+ (\\rho_i-\\hat \\rho_\\ihalf) + (-\\d V)_\\ihalf^- (\\rho_{i+1}-\\hat \\rho_\\ihalf)\\big] (-\\d h_\\ihalf)\\big]\\\\\n\t&-\\sum \\big[\\,|(-\\d V)_\\ihalf^+|^2\\, (\\rho_i-\\hat \\rho_\\ihalf) + |(-\\d V)_\\ihalf^-|^2\\, (\\rho_{i+1}- \\hat \\rho_\\ihalf)\\big]\\\\\n\t&\\phantom{=\\;} - \\sum_i \\hat \\rho_\\ihalf (\\d V_\\ihalf)^2 - \\sum \\hat \\rho_\\ihalf (-\\d V)_\\ihalf (-\\d h_\\ihalf)\\\\\n\t&\\phantom{=\\;} -\\sum \\big[ (-\\d h)_\\ihalf^+ (\\rho_i-\\hat \\rho_\\ihalf) + (-\\d h)_\\ihalf^- (\\rho_{i+1}-\\hat \\rho_\\ihalf)\\big] (-\\d V_\\ihalf)\\big]\n\\end{align}\n\n\n\n\n\n\n\\MS{\n\\noindent\nNow let us consider our attention to the second part of Eq. \\eqref{eq:220916_1212}, coming from the diffusion.\nFor the moment, we turn our attention to the following  computation.\n\\begin{align}\n\t\\D\\log \\rho_\\ihalf &=\\log \\rho_{i+1} -  \\log \\rho_i\\\\\n\t&= 2(\\log \\sqrt{\\rho_{i+1}} -  \\log \\sqrt{\\rho_i})\\\\\n\t&= \\frac{2}{\\sqrt{r}} (\\sqrt{\\rho_{i+1}} - \\sqrt{\\rho_i}).\n\\end{align}\nLet us consider\n\\begin{align}\n\t-(&\\rho_{i+1} - \\rho_i) \\D\\log \\rho_\\ihalf\\\\\n\t&= -(\\rho_{i+1} - \\rho_i) \\D\\log \\rho_\\ihalf\n\t +2 (\\sqrt{\\rho_{i+1}} - \\sqrt{\\rho_i})^2 \n\t -2 (\\sqrt{\\rho_{i+1}} - \\sqrt{\\rho_i})^2\\\\\n\t&= -\\frac{2(\\sqrt{\\rho_{i+1}} + \\sqrt{\\rho_i})}{\\sqrt{r}} (\\sqrt{\\rho_{i+1}} - \\sqrt{\\rho_i})^2\n\t +2 (\\sqrt{\\rho_{i+1}} - \\sqrt{\\rho_i})^2 \n\t -2 (\\sqrt{\\rho_{i+1}} - \\sqrt{\\rho_i})^2\\\\\n\t&=(\\sqrt{\\rho_{i+1}} - \\sqrt{\\rho_i})^2 \\bigg(2 - \\frac{2(\\sqrt{\\rho_{i+1}} + \\sqrt{\\rho_i})}{\\sqrt{r}}\\bigg) -2 (\\sqrt{\\rho_{i+1}} - \\sqrt{\\rho_i})^2.\n\\end{align}\nSince $\\sqrt{\\cdot}$ is monotonous and $\\tilde \\rho_\\ihalf \\in [\\rho_i, \\rho_{i+1}]$ we can estimate the first term as follows.\n\\begin{align}\n\\label{eq:220916_1401}\n\\begin{split}\n\t\t-(&\\rho_{i+1} - \\rho_i) \\D\\log \\rho_\\ihalf\\\\\n\t\t&\\leq(\\sqrt{\\rho_{i+1}} - \\sqrt{\\rho_i})^2 \\bigg(2 - \\frac{2(\\sqrt{\\rho_{i+1}} + \\sqrt{\\rho_i})}{\\sqrt{r}}\\bigg) -2 (\\sqrt{\\rho_{i+1}} - \\sqrt{\\rho_i})^2\\\\\n\t\t&\\leq -2(\\sqrt{\\rho_{i+1}} - \\sqrt{\\rho_i})^2.\n\\end{split}\n\\end{align}\n\n\n\\noindent\nNote that from Eq. \\eqref{eq:220916_1401} we also get the bounds\n\\begin{align}\n\t|\\rho_i - \\tilde \\rho_\\ihalf| |\\D\\log\\rho_\\ihalf|&\\leq 2 |\\sqrt{\\rho_{i+1}} - \\sqrt{\\rho_i}|^2,\\\\\n\t|\\rho_{i+1} - \\tilde \\rho_\\ihalf| |\\D\\log\\eta_\\ihalf|&\\leq 2 |\\sqrt{\\rho_{i+1}} - \\sqrt{\\rho_i}|^2,\n\\end{align}\nwhence we can see that the terms in Eq. \\eqref{eq:220916_1405} do not blow up as either $\\rho_{i+1}$ or $\\rho_i$ tend to zero. \n\n}\n\n\\section{Snippets}\t\n\\wip{From the a priori bounds, Corollary \\ref{cor:apriori}, we obtain\n\t\\begin{align}\n\t\t\\label{eq:spaceshifts}\n\t\t\\|\\rho_{\\Delta x}(\\cdot, x+\\xi) - \\rho_{\\Delta x}(\\cdot, x)\\|_{L^2(0, 1- \\xi)}^2 \\leq C \\xi (\\xi +2\\Delta x),\n\t\\end{align}\n\tfor $\\xi\\in[0,1\/2]$, by Lemma 2.4 in Ref. \\cite{EGT00}. \\\\\n\tWe set\n\t\\begin{align*}\n\t\tX:=\\overline{\\operatorname{span}\\{\\rho_{\\Delta x}\\,|\\,\\Delta x > 0\\}}^{L^2(0,1)}.\n\t\\end{align*}\n\tThen, $X$ is a Banach space closed in $L^2(0,1)$. Furthermore, any bounded set $S\\subset X$ is relatively compact since there exists $C>0$ such that for any $f \\in S$\n\t\\begin{align}\n\t\t\\|f(\\cdot + \\xi) - f(\\cdot)\\|_{L^2(0,T;L^2(0,1))}^2 \\leq C \\xi (\\xi + 2 \\Delta x) T,\n\t\\end{align}\n\tby Eq. \\eqref{eq:spaceshifts}. Thus $S$ is relatively compact by Kolmogorov's compactness theorem.}\n\\subsection{2}\n\n\tThe idea is to let $X$ be  the space of piecewise constant $L^2$ functions on the mesh $\\mathcal{T}_{\\Delta x}$ whose discrete gradient is bounded. Since we are interested in the set of approximations $(\\rho_{\\Delta x})$ we need to construct a vector space containing all those functions, i.e. for all $\\Delta x > 0$.  Thus we set \n\t\\begin{align*}\n\t\tX_\\mathcal{T}(0,1):=\\left\\{f\\in L^2(0,1) \\,|\\, f\\;\\text{is piecewise constant and} \\sum_{i\\in I} \\frac{|\\d f_\\ihalf|^2}{\\Delta x} < \\infty  \\right\\}.\n\t\\end{align*}\n\tHowever this is not a Banach space, so we need to make it one by closing it in the $L^2$ topology, i.e. we let\n\t\\begin{align}\n\t\tX:= \\overline{\\bigcup_{\\Delta x >0} X_\\mathcal{T}(0,1)}^{_{L^2(0,1)}}.\n\t\\end{align}\n\tClearly any bounded set in $X$ is bounded in $L^2$ as $X \\subset L^2(0,1)$. Now let $x\\in X$ and $h>0$. Then by the $L^2$ completion, there exists a sequence $(x_n)\\subset X$ such that $x_n\\rightarrow x$ in $X$. Thus for any arbitrary $\\epsilon> 0$ we find $n\\in \\N $ such that\n\t\\begin{align}\n\t\t\\|x_n - x\\|_{L^2(0,1)} < \\epsilon\/3.\n\t\\end{align}\n\tSince $x_n \\in X_{\\mathcal{T}}$ for some grid $\\mathcal{T}$ there exists an $\\eta>0$ such that for any $0<h<\\eta$ there holds\n\t\\begin{align}\n\t\t\\|\\tau_h x_n - x_n\\| < \\epsilon\/3.\n\t\\end{align}\n\t\n\tWe start by verifying the boundedness of the sequence of numerical approximations in the space of functions of bounded variation. Subsequently, we prove that the time derivatives are bounded in the space $(W^{1,\\infty})^\\star$. This suffices to apply the theorem and induce the statement.\n\t\n\\end{comment}\n\n\\section{Conclusion}\nIn this paper we presented a finite volume scheme for a system of non-local partial differential equations with cross-diffusion. We were able to reproduce a continuous energy estimate on the discrete level for our scheme. These discrete estimates for the approximate solution are enough to get compactness results and we are able to identify the limit of the approximate solutions as a weak solution of the equation. We complement the analytical part with numerical simulations. These back up our convergence result and we are also able to apply the scheme in cases in which we cannot show convergence. To this end we pushed the scheme to regimes of singular potentials also lacking the regularising porous medium type self-diffusion terms. Comparing them with the explicit stationary states from \\cite{CHS17} we conclude the scheme performs well even in  regimes it was not designed for.\n\n\n\\section*{Acknowledgements}\nJAC was partially supported by EPSRC grant number EP\/P031587\/1.\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nAccording to the bottom-up hierarchical scenario, Galaxy clusters (GCs) are the last bounded structures to form in our Universe (starting from about $z \\sim 2$). They reside in the deepest potential wells, which are generated by dark matter (DM) overdensities \\citep[e.g.][]{springel05}. \nGalaxy clusters are multi-component structures. In addition to DM, haloes include baryonic matter in different forms and phases \\citep[see e.g.][and references therein]{allen11}. In addition to galaxies, cold and hot gas and non-thermal plasma constitute the intra-cluster medium (ICM). This multi-component nature allows us to observe GCs at various wavelengths, taking advantage of the different physical processes involved in each case. For instance, we use X-rays and radio observations to probe the ICM or the visible\/IR wavelengths to study the galactic component. Dark matter is usually studied through the deformation of images from the background galaxies. \nThese multi-wavelength observations provide complementary information about the cluster physics. \n\nGalaxy clusters are excellent tracers of the evolution of structures throughout the history of the Universe. Their abundance as a function of redshift and mass, the so-called cluster number counts, is very sensitive to cosmological parameters, and in particular, to the matter density parameter $\\Omega_m$ and to the amplitude of the density fluctuations $\\sigma_8$ \\citep{vikhlinin09, mantz10, planck13_count, planck15_count}.\nThe accuracy in the cluster mass determination is crucial for the determination of cosmological parameters. To this end, several mass proxies can be defined through scaling relations \\citep[see e.g.][and references therein]{Pratt19}. However, each mass proxy has its own biases that are linked to the methods of observation or the assumptions made to calculate it. To understand these biases, the analysis of cosmological simulations is a valuable tool, and the comparison of the masses obtained from different proxies becomes essential.\n\n\nThe Sunyaev-Zeldovich (SZ) effect \\citep{SZ72} is a spectral distortion of the cosmic microwave background (CMB), generated by the inverse Compton scattering of the CMB photons off the hot electrons in the ICM. As the brightness temperature of the resulting distortion is redshift independent, the SZ effect has recently become a powerful tool for detecting GCs. \n\nThe ESA\\textit{Planck}{} Planck mission scanned the entire sky in microwaves with the aim of studying primary and secondary anisotropies of the CMB. The observations, which lasted four years, spanned nine bands between 30\\,GHz and 857\\,GHz. The results of these observations include two catalogues of objects detected by means of their SZ signature: PSZ1 \\citep{planck13_cat, planck13_cat_2} and PSZ2 \\citep{planck15_cat}. \nThese products were the first to provide the possibility of detecting GCs through the SZ effect in a full-sky survey. The catalogues contain 1227 and 1653 objects, respectively, with 937 objects in common. %\nThe total integrated SZ signal within a circle of radius (the so-called integrated Compton\nparameter, $Y_{\\rm SZ}$) is closely related to the cluster mass \\citep[e.g.][]{daSilva2004}. This $Y_{500}$ observable has been used as the mass proxy by the Planck Collaboration, after calibrating it to X-ray measurements \\citep{planck13_count, planck15_count}. \n\nThis paper is the third (and last) in the series of publications associated with the International Time Programme ITP13B-15A, a two-year (August 2013 to July 2014) long-term program in the Canary Islands Observatories dedicated to characterising PSZ1 sources in the northern sky without known optical counterparts at the time of publication of the PSZ1 catalogue. \nThe publications were a continuation of the validation efforts carried out in \\cite{nostro16}, within the context of the ITP 12-2 program.\nPaper I \\citep{rafa18} and paper II \\citep{rafa20} described the ITP13B-15A program in detail, which included observations of 256 SZ sources with a declination above $ -15^\\circ$ (212 of them were previously unknown), finding optical counterparts for 152 SZ sources. This paper (number III in the series) presents all the spectroscopic observations of the program, including velocity dispersion and dynamical mass estimates. \n\nIn section \\ref{sec:ITP_SDSS_sample}, we describe the PSZ1-North reference sample summarising the results of the ITP program.\nSections \\ref{subsec:POFU_spec} and \\ref{sec:dyn_mass} depict how we estimate GC velocity dispersion and dynamical masses, respectively.\nIn section \\ref{sec:ITP_SDSS_sample} we describe the ITP and the SDSS GC samples we used for the cosmological analyses of this work. In sections \\ref{subsec:relation2}, we constrain the mass bias parameter, $\\left (1- B\\right)$, explaining how much the Eddington bias affects our sample and detailing the importance of correcting for it. Finally, in section \\ref{sec:comparison}, we compare our results with those from \\textit{Planck}{} Collaboration and those from the literature, using the velocity dispersion and the weak-lensing analysis as mass proxies. We also discuss the implications on the $\\sigma_8$ tension with CMB measurements.\n\nThroughout this paper, we define $R_{200}$ and $R_{500}$ as the radius within which the mean cluster density is 200 and 500 times the critical density of the Universe at redshift $z$, respectively. The quantities with the subscripts 200 or 500 have to be considered as evaluated at or within $R_{200}$ and $R_{500}$, respectively.\nWe assume a flat $\\Lambda$CDM cosmology, with $H_0 =100 h$\\,km\\,s$^{-1}$\\,Mpc$^{-1}$, $h = 0.678$,  and $\\Omega_{\\rm M}= 0.307$ \\citep{planck13_par}.\n\n\n\\section{PSZ1-North sample}\n\\label{sec:ITP_SDSS_sample}\n\nThe PSZ1 catalogue \\citep{planck13_cat,planck13_cat_2} contains 1227 sources detected by means of their SZ signature in the all-sky maps obtained during the first 15.5 months of \\textit{Planck}\\ observations. \nThese sources are detected by at least one of the three\n\\textit{Planck}\\ cluster detection algorithms (MMF1, MMF3, and PwS) with a signal-to-noise ratio (S\/N) of $4.5$ or higher. \\cite{planck13_cat} describes these three algorithms, the selection, and the validation methods adopted in the construction of the catalogue in detail. \n\nFollowing papers I and II, we define the PSZ1-North sample as the 753 PSZ1 sources in the northern hemisphere with a declination $Dec \\geq -15\\mbox{$^{\\circ}$}$. Of these, 541 were validated by the \\textit{Planck}\\ collaboration at the time of publication of the PSZ1 catalogue. The remaining 212 sources were studied within the ITP program. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[trim=0.4cm 1.5cm 0.4cm 1.5cm, clip, width=\\textwidth]{figure\/ITP_SDSS_galactic_map_no_z_paper.png}\n\\caption{Sky distribution of all the PSZ1 clusters studied in this paper. The figure uses a Mollweide projection in Galactic coordinates. The Galactic plane is horizontal and centred at longitude zero. Red circles and blue triangles represent ITP (Sec.~\\ref{sub:itp}) and SDSS  (Sec.~\\ref{sub:sdss}) clusters, respectively, and the symbol size is proportional to the cluster mass. The shaded grey area represents the union mask used to produce the CMB map in the first \\textit{Planck}\\ data release (downloaded from the Planck Legacy Archive). This mask excludes $27\\,\\%$ of the sky, mostly in the region of the Galactic plane, the Magellanic Clouds, and point sources. }\n\\label{fig:fullsample}\n\\end{figure*}\n\n\n\\begin{table}\n\\centering\n\\caption{Summary of the data sets.}\n\\begin{tabular}{c c c|c}\n\\hline \\hline\n\\noalign{\\smallskip}\nData set    & PSZ1-North  & Beyond PSZ1 & Scaling relation\\\\\n & & (see Sect.~\\ref{sub:itp}) &  (see Sect.~\\ref{sec:subsample}) \\\\\n\\hline\n\\noalign{\\smallskip}\nITP   &  58 & 35 &  58 \\\\\nSDSS  & 212 &  0 & 149 \\\\\n\\hline\nTotal & 270 & 35 & 207 \\\\\n\\hline \\hline\n\\end{tabular}\n\\label{tab:sample}\n\\end{table}\n\n\\subsection{ITP program}\n\\label{sub:itp}\n\\cite{nostro16}, \\cite{rafa18}, and \\cite{rafa20} have described our optical follow-up of the unknown PSZ1 sources in the northern sky that we carried out between the second semester of 2012 and first semester of 2015 by means of two International Time Projects (ITP 12-2 and ITP13-08, hereafter ITP).\nTable~3 in \\cite{rafa20} presented the summary information of the full program, where we observed all the 212 PSZ1-North sources for which no counterparts were known, plus 44 of the already validated sources. \n\nPapers I and II have presented the complete imaging results of the program and partial spectroscopic results, including the mean spectroscopic redshifts to the clusters, as well as the redshift of the BCG, when available. This paper III presents for the first time the velocity dispersion and dynamical mass estimates for s with at least ${N_{\\rm gal}} \\text{seven}$ spectroscopic members. In total, 58 PSZ1-North clusters were characterised (see Table~\\ref{tab:sample})\n\nThe majority of these results were obtained using multi-object spectroscopy (MOS), although for a few cases and when the MOS mode was not available, we used a combination of long-slit measurements. \nAs described in previous papers, all the spectroscopic data for the ITP program have been acquired using three instruments at the Observatorio del Roque de Los Muchachos (ORM), located on the island of La Palma (Spain). \nOSIRIS at the 10.4 m Gran Telescopio Canarias (GTC) and DOLORES at the 3.5 m Telescopio Nazionale Galileo (TNG) were mainly used to obtain MOS, while ACAM at the 4.2 m William Herschel Telescope (WHT) provided some additional spectra that were retrieved through a long-slit setup.\nA detailed description of the instrument characteristics and the configuration adopted for these measurements has been presented in papers I and II.\n\nWhen the MOS technique was used with OSIRIS and DOLORES, we observed each field with a single mask containing between 45 and 60 slitlets on average. In this way, we were able to maximize the number of redshifts per observation, obtaining a median number of galaxy members of about 20 (see details in paper II). \n\nTable~\\ref{table:ITP_masses_flag1} in the appendix contains the full list of the 58 ITP objects within PSZ1-North that we studied in this paper. The first, second, and third columns identify the index number, the \\textit{Planck}\\ name of each cluster in the PSZ1 catalogue, and the signal-to-noise ratio of the SZ detection. Columns 4 and 5 indicate the equatorial coordinates (J2000) of the cluster optical centre. Column 6 indicates the distance from the optical centre to the nominal \\textit{Planck}\\ coordinates. \nColumns 7 and 8 contain our spectroscopic redshift and the number of cluster members with spectroscopic measurements, respectively. Columns 9 to 11 contain the velocity dispersion, the dynamical mass, and the SZ mass estimates. The description of the process with which these numbers were derived is discussed in Sections~\\ref{subsec:POFU_spec} and \\ref{sec:dyn_mass}. Finally, columns 12 and 13 indicate whether the cluster was used to constrain the scaling relation and if it was part of the \\textit{Planck}\\ cosmological sample (hereafter, PlCS), as defined in \\cite{planck13_cat}.\n\nDuring this program, we also characterised 35 bona fide GCs that have not been associated with the SZ signal measured by \\textit{Planck}\\, either because they lie at a distance of more than $5\\hbox{$^{\\prime}$}$ from the nominal \\textit{Planck}\\ pointing or because their velocity dispersion is too low. These objects were labelled {\\tt Flag}$=3$ in our validation process \\citep[see details in ][]{rafa18, rafa20}. These 35 clusters are listed in the column \"Beyond PSZ1\" in Table~\\ref{tab:sample} and are presented in Table~\\ref{table:ITP_masses_flag3}, using the same format as in Table~\\ref{table:ITP_masses_flag1}.\n\nThe location of all these ITP clusters is shown in Figure~\\ref{fig:fullsample} in red. We obtained more than $2000$ individual redshift measurements in this work. The mean number of galaxy members for these ITP clusters is ${N_{\\rm gal}} = 15$. \nAll the spectra associated with these observations that we present here for the first time will be published online and are included in the VO database. \n\n\\subsection{SDSS archival data}\n\\label{sub:sdss}\nWith the aim of characterising the full PSZ1-North sample, we decided to enlarge our ITP sample by searching in the SDSS Data Release 14 \\citep{aihara11_sdss} for additional spectroscopic information. \nFor each of the $401$ PSZ1 sources within the SDSS footprint, we retrieved all spectroscopic redshifts within $15\\hbox{$^{\\prime}$}$ radius around the nominal \\textit{Planck}\\ coordinates. We used the position of the brightest galaxy in the radial velocity catalogue as the cluster centre, within a range of $\\pm 2500\\,{\\rm km}\\:{\\rm s}^{-1}$ around the \\textit{Planck}\\ validation redshift. In most cases, this position corresponded to the BCG. If not, after a visual inspection of SDSS RGB images, we selected the centre as the position of a clear BCG (not observed spectroscopically). Alternatively, we used the cluster members mean coordinates as the optical centre.\n\nFor each individual PSZ1 field, we analysed the retrieved set of spectroscopic redshifts in the same way as we treated the clusters that were observed during the ITP program. However, due to the lack of (enough number of) spectra in some fields, it was only possible to characterise the spectroscopic properties for 212 regions. As in the previous case, we only considered the clusters for which we had  at least ${N_{\\rm gal}} \\text{seven}$ spectroscopic members.\n\nTable~\\ref{table:SDSS_masses} in the appendix contains the final list of PSZ1-North clusters with SDSS information that is discussed in this paper. The table format is identical to that of Table~\\ref{table:ITP_masses_flag1}. \nThe location of these 212 clusters is also shown in Figure~\\ref{fig:fullsample} in blue. We used $\\sim 10000$ spectra in this analysis. The mean number of galaxy members for these clusters is ${N_{\\rm gal}} = 35$. \n\nFinally, the joint ITP and SDSS sample consists of 270 GCs, 226 of which are also included in the PSZ2 catalogue. In our sample, the percentage of PSZ1 clusters that are also detected in the PSZ2 is therefore $85\\%$. This number is slightly higher than the same fraction computed for the whole PSZ1 catalogue, $76\\%$ \\citep[see table 10 in][]{planck15_cat}.\n\n\n\\section{Velocity dispersion estimates}\n\\label{subsec:POFU_spec}\n \nFor each , the final product of either our ITP observations or the SDSS archival data search is a catalogue of radial velocities of galaxies within a certain field. This means that the cluster is detected as an over-density in the radial velocity distribution of at least four galaxies in a range of $\\pm 5000\\,{\\rm km}\\:{\\rm s}^{-1}$ around the velocity of the BCG. \nThe membership selection is based on the galaxy position in the 2D projected phase space $(r, cz)$, where $r$ is the projected distance from the cluster centre, and $cz$ is the galaxy line-of-sight (LoS) velocity. To minimize the presence of interlopers, we defined a region of membership by performing a cut at $r=2.5$\\,Mpc (at the redshift of the cluster) and an iterative $2.5-\\sigma$ clipping in the $cz$ coordinate, taking into account the radial profile of the expected velocity dispersion \\citep{Mamon10}. The first criterion restricts our galaxy sample to a distance between 1.5 and 2 $R_{200}$ (depending on the cluster mass and redshift) and in this way limits the contamination of interlopers at large distances from the centre of the clusters. \n\n\\begin{figure}\n\\centering\n\\includegraphics[trim=0.5cm 0.3cm 0.5cm 1.3cm, clip,width=0.49\\textwidth]{figure\/PSZ1_374_hist_gauss.png}\n\\includegraphics[trim=0.5cm 0.3cm 0.5cm 1cm, clip,width=0.49\\textwidth]{figure\/PSZ1_450_hist_gauss.png}\n\\caption{Two examples of the velocity distribution of cluster members: PSZ1 G103.94+25.81 (top) and PSZ1 G123.55?10.34 (bottom). Both clusters have been observed within the ITP program. The histograms contain 17 and 30 members, respectively. In both cases, the red line corresponds to a Gaussian distribution centred in the mean cluster velocity, and with $\\sigma$ equal to the estimated velocity dispersion $\\sigma_v$. }\n\\label{fig:example_vel_histo}\n\\end{figure}\n\nFigure~\\ref{fig:example_vel_histo} shows two examples of the final velocity histogram of cluster members for two ITP clusters. \nFigure~\\ref{fig:PSZ1_hist} shows the 2D projected phase-space and the velocity distribution of the stack of all cluster members selected using the procedure explained above for the ITP sub-sample (1432 clusters) and for the SDSS sub-sample (3879 clusters). \n\nAfter this member selection was concluded, we estimated the velocity dispersion using the gapper estimator \\citep{gapper}. We followed the procedure outlined in \\cite{Ferragamo20}, and we corrected this velocity dispersion estimate by taking the statistical bias introduced by the under-sampling and sigma clipping into account, as well as the physical bias due to the aperture radius. \nOur velocity dispersion results are listed in column 9 of Tables~\\ref{table:ITP_masses_flag1}, \\ref{table:SDSS_masses}, and \\ref{table:ITP_masses_flag3}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim=0.5cm 0.4cm 0.5cm 0.5cm, clip, width=0.49\\textwidth]{figure\/caustic_stack_members_25clip_rr200.png}\n\\includegraphics[trim=0.5cm 0.4cm 0.5cm 0.5cm, clip, width=0.49\\textwidth]{figure\/caustic_stack_members_25clip_rr200_SDSS.png}\n\\caption{Projected phase-space and velocity histogram of the stack of all ITP clusters members (top) and all SDSS cluster members (bottom). In both cases, velocities are normalised to the cluster velocity dispersion. The dashed black lines indicate the $2.5-\\sigma$ clipping, and the dotted blue lines show the same clipping, but take the velocity dispersion radial profile as shown in \\cite{Mamon10} into account. The red line in the right panel represents a Gaussian fit of the stacked velocity histogram normalised to the total number of members.}\n\\label{fig:PSZ1_hist}\n\\end{figure}\n\n\n\\section{Dynamical mass estimates}\n\\label{sec:dyn_mass}\n\nWe computed the dynamical mass $M_{200}^{\\rm dyn}$ by using the bias-corrected mass estimator described in \\cite{Ferragamo20}. This estimator is based on the AGN fit of the \\cite{munari13} scaling relation, given by\n\\begin{equation}\n\\frac{\\sigma}{\\mathrm{km\\ s^{-1}}} =\nA\\left[\\frac{h(z)\\ M_{200}}{10^{15}\\ \\mathrm{M}_{\\odot}}\\right]^{\\alpha},\n\\label{eq:smg}\n\\end{equation}\nwhere $A=1177.0$ and $\\alpha = 0.364$.\n\nFollowing prescriptions described in \\cite{Ferragamo20}, we corrected\nfor the aperture radius bias. The physical bias correction due to the fraction of massive galaxies (which tends to reduce the mass of the clusters by $5\\,\\%$ at most) cannot be evaluated in this work because it requires a well-constrained luminosity function. This cannot be obtained because the clusters we analysed have only a few spectroscopic members.\n\nIt follows that the dynamical masses reported here are to be considered as a lower limit to the true mass. The derived mass bias is therefore to be considered as an upper limit. In order to compare our results with the SZ flux ($Y_{500}^{\\rm SZ}$) and SZ mass ($M_{500}^{\\rm SZ}$) values derived by the \\textit{Planck}\\ Collaboration, we need to rescale our values from $M_{200}$ to $M_{500}$. \n\nTo do this, we assumed an NFW density profile \\citep{NFW},\n\\begin{equation}\n\\rho(r) = \\frac{\\rho_s}{\\left( r\/r_s \\right)\\left( 1+ r\/r_s \\right)^2}\n\\label{eq:NFW_profile}\n,\\end{equation}\nwhere $\\rho_s$ and $r_s$ are the characteristic density and the scale radius, respectively. These parameters are related to $R_{200}$ and $\\rho_{200} \\equiv 200 \\times \\rho_c$, where $\\rho_c$ is the critical density, through the concentration parameter $c_{200}$, which is defined as\n\\begin{align} \n\\label{eq:c_def}\nc_{200} \\equiv R_{200}\/r_s, \\\\\n\\rho_s = \\frac{\\rho_{200} c_{200}^3 g(c)}{3,}\n\\end{align}\nwhere $g(x) = 1\/ \\left[ \\ln (1+x)-x\/(1+x) \\right]$. The mass enclosed within a spherical over-density of radius $r_\\Delta$ can be written as\n\\begin{equation}\nM(r_\\Delta) = \\frac{g(c_{200})}{g(c_{200}\\;r_\\Delta\/R_{200})}M_{200},\n\\label{eq:M_NFW}\n\\end{equation}\nwhere $\\Delta$ is the new density contrast at which we wish to obtain the mass (in our case, $\\Delta = 500$).\nFinally, we selected a mass-concentration relation to scale our masses \\citep{lokas01}. Following the procedure in \\cite{Komatsu11_wmap7}, we chose the \\cite{duffy08} relation, which is constrained using N-body simulations and WMAP 5-year cosmological parameters,\n\\begin{equation}\nc_{200} = \\frac{5.71}{\\left( 1+z \\right)^{0.47}} \\left( \\frac{M_{200}}{2 \\times 10^{12} h^{-1} \\  \\rm{M_\\odot}} \\right)^{-0.084}.\n\\label{eq:duffy_MC}\n\\end{equation}\n\n\n\\section{Selection of a representative sub-sample }\n\\label{sec:subsample}\n\nHere we describe the sub-sample of selected 207 PSZ1-North clusters that we used to characterise the scaling relation $M_{500}^{\\rm SZ}$--$M_{500}^{\\rm dyn}$. This sub-sample contains all the 58 ITP objects and 149 SDSS clusters (see last column in Table~\\ref{tab:sample}) that were selected as explained below. This sub-sample represents $27.5$\\,\\% of the full PSZ1-North catalogue. \n\n\\subsection{ ITP sub-sample}\nAlthough our observational program has resulted in more confirmed clusters, we considered only the most reliable ones that had an unambiguous association with the \\textit{Planck}\\ SZ signal to characterise the scaling relation. Because the measured SZ signal is the total integrated electron pressure along the line of sight, we cannot separate the fraction of the $Y_{500}$ signal that corresponds to each cluster  in the case of multiple cluster counterparts inside the same\npointing. We therefore discarded these multiple counterparts for this analysis. In practice, this does not affect the ITP sample, and we kept all 58 ITP clusters within PSZ1-North, as listed in Table~\\ref{table:ITP_masses_flag1} (see the appendix). The last column in the table indicates the objects that are included in the PlCS. By construction, this number is very small (only 2) because the ITP sample includes PSZ1-North objects with unknown counterparts at the time of publication of the catalogue. \n\nFigure~\\ref{fig:z_snr_ITP_hists} shows the distribution of these 58 objects as a function of redshift (left panel) and the S\/N of the SZ detection (right panel). The median redshift of this sub-sample is $z=0.31$, and the median S\/N is $4.9$. \n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[trim=0.4cm 0.4cm 0.4cm 0.4cm, clip, width=0.49\\textwidth]{figure\/ITP_redshift_histogram.png}\n\\includegraphics[trim=0.4cm 0.4cm 0.4cm 0.4cm, clip, width=0.49\\textwidth]{figure\/ITP_SNR_histogram.png}\n\\end{center}\n\\caption{Number of ITP clusters as a function of redshift (left panel) and the $S\/N$ of the SZ detection (right panel) we used, normalised to 88, which is the total number of ITP-validated clusters \\citep[for details, see ][]{rafa20}. }\n \\label{fig:z_snr_ITP_hists}\n\\end{figure*}\n\n\n\\subsection{SDSS sub-sample}\nFrom the full list of 212 SDSS objects discussed in Sec.~\\ref{sub:sdss}, we only considered the 149 SZ clusters with a single optical counterpart at the same redshift as the validation counterpart reported in the PSZ1 catalogue. These clusters lie within $5\\hbox{$^{\\prime}$}$ from the nominal \\textit{Planck}\\ pointing. \nAlthough these prescriptions (${N_{\\rm gal}} \\ge 7$ and a single counterpart) leave us with only $37\\%$ of all the 401 PSZ1 clusters within the SDSS footprint, they guarantee a clean  SDSS sample. A detailed and systematic study of the optical counterparts of previously validated clusters of the PSZ1 catalogue is beyond the scope of this work. \n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[trim=0.4cm 0.4cm 0.4cm 0.4cm, clip, width=0.49\\textwidth]{figure\/SDSS_redshift_histogram.png}\n\\includegraphics[trim=0.4cm 0.4cm 0.4cm 0.4cm, clip, width=0.49\\textwidth]{figure\/SDSS_SNR_histogram.png}\n\\end{center}\n\\caption{Number of SDSS clusters as a function of redshift (left panel) and SZ $S\/N$ (right panel) we used, normalised to the total number of PSZ1 sources within the SDSS footprint (green). Red histograms correspond to the subsample of SDSS clusters within the \\textit{Planck}{} cosmological sample.}\n \\label{fig:z_snr_SDSS_hists}\n\\end{figure*}\n\nFigures \\ref{fig:z_snr_ITP_hists} and \\ref{fig:z_snr_SDSS_hists} show the number of GCs in the ITP and our SDSS sample (green) as a function of the redshift (left panel) and of the $S\/N$ (right panel), respectively. The number of PlCS clusters contained in our SDSS sample is plotted in red. Most of the selected clusters are low-redshift systems with a low S\/N. The median redshift and $S\/N$ are $0.17$ and $5.74$, respectively.\nIt is also interesting to note that although the percentage of validated PSZ1 clusters comprised in SDSS is $25.8\\%$, only $18.6\\%$ of the clusters are in the PlCS. However, 42 (28.2\\,\\%) of the 149 SDSS clusters that we selected are part of the PlCS, with a median redshift equal to 0.11 and a median $S\/N$ of 9.39.\nIn our analysis below, we explore whether the measured mass bias differs for clusters within or outside the cosmological sample. \n\nIn Table \\ref{table:SDSS_masses} (see appendix) we list the velocity dispersion and the dynamical and the SZ mass for these $149$ clusters selected from the SDSS sample. The selected objects are marked with a checkmark in the column \"scaling\". We also indicate the clusters that are included in the cosmological sample. \n\n\\section{$M^{\\rm dyn}$--$M^{\\rm SZ}$ relation}\n\\label{subsec:relation2}\n\nHere we describe the results of the $M^{\\rm dyn}_{500}$--$M^{\\rm SZ}_{500}$ scaling relation using the PSZ1 GC samples selected in the previous section. \nFollowing the definition given in \\cite{planck13_count}, the bias of SZ-derived masses is given by\n\\begin{equation}\nM_{500}^{\\rm SZ} \\equiv \\left( 1-b_{\\rm SZ} \\right) M_{500}^{\\emph{true}},\n\\label{eq:(1-bsz)}\n\\end{equation}\nwhere $M_{500}^{\\rm SZ}$ is the cluster SZ mass. This is estimated following the prescriptions in \\cite{planck13_cat} using the equation \n\\begin{equation}\nE^{-2\/3}(z) \\left[\\frac{D_{A}^2\\ Y_{500}^{\\rm SZ}}{10^{-4}\\ \\rm Mpc^2} \\right] = 10^{-0.19 \\pm 0.02} \\left[ \\frac{M_{500}^{\\rm SZ}}{6 \\times 10^{14}\\ \\rm M_{\\odot}} \\right] ^{1.79 \\pm 0.08},\n\\label{eq:Y500_MYx_1}\n\\end{equation}\nwhich is calibrated using X-ray observations of 71 clusters in the PSZ1 cosmological sample. \n\nIn general, because we cannot account for all the possible biases in the dynamical mass estimation, we defined a global bias of the dynamical mass as\n\\begin{equation}\nM_{500}^{\\rm dyn} \\equiv \\left( 1-b_{\\rm dyn} \\right) M_{500}^{\\emph{true}}.\n\\label{eq:(1-bdy)}\n\\end{equation}\nCombining equations~\\ref{eq:(1-bsz)} and \\ref{eq:(1-bdy)}, we obtain the $M_{\\rm dyn}-M_{\\rm SZ}$ relation as \n\\begin{equation}\nM_{500}^{\\rm SZ} = \\left( 1-B \\right) M_{500}^{\\rm dyn},\n\\label{eq:(1-B)}\n\\end{equation}\nwhere the mass bias $(1-B)$ is defined as the ratio of SZ and dynamical bias,\n\\begin{equation}\n    (1-B) \\equiv \\frac{\\left( 1-b_{\\rm SZ} \\right)}{\\left( 1-b_{\\rm dyn} \\right)}.\n\\end{equation}\n\n\\subsection{Linear regression method}\nWe fit for the relation in equation~\\ref{eq:(1-B)} using a linear regression in logarithmic space because the high noise level of our dynamical mass estimates, with typical \nrelative errors of $\\sim 30$ \\%, could affect the estimation of the fitted parameters. In some cases, we also explored the possibility of a mass dependence by letting the exponent vary freely in the relation, fitting for the slope and intercept in\n\\begin{equation}\n\\ln M_{500}^{\\rm SZ} = \\ln \\left( 1-B \\right) + \\alpha \\ln M_{500}^{\\rm dyn}.\n\\label{eq:fit_log}\n\\end{equation}\nThis problem requires carrying out a linear regression with errors in both axes, and accounting for intrinsic scatter. In order to select the most appropriate method for this problem, we tested five methods that are commonly used in literature, namely the orthogonal distance regression (ODR, implemented in the python \\textit{scipy.odr} package), the Nukers \\citep{Tremaine02}, the maximum likelihood estimator with uniform prior (MLEU), the bivariate correlated errors and intrinsic scatter \\citep[BCES,][]{akritas96}, and the complete maximum likelihood estimator \\citep[CMLE,][]{kelly07}, on realistic mock realisations of PSZ1 sample. In a companion paper associated with the characterisation of the PSZ2 sample \\citep{aguado20}, we show that although all these five methods produced biased results in the recovered slope because of the high noise levels in our sample, the Nukers and the ODR present the lowest bias. For this reason, we adopted ODR as the reference regression method, and we tested it using simulations with realistic noise levels mimicking those present in our sample. See Appendix B in \\cite{aguado20} for a detailed description of the other estimators.\n\nFigure~\\ref{fig:odr_cor} shows that after 500 realisations with realistic noise levels, the ODR method retrieves a mean mass bias parameter that is $\\sim 6\\,\\%$ larger than the input bias, $(1-B) = 0.8$, for the case of a fixed slope ($\\alpha=1$). We also obtain the same bias of $\\sim 6\\,\\%$ when the slope was left as a free parameter. \nThis value was used later to correct for the final results.\n\n\\begin{figure}\n        \\centering\n        \\includegraphics[trim=0.5cm 0.4cm 0.5cm 0.5cm, clip, width=0.49\\textwidth]{figure\/ODR_correction_3_3.png}\n        \\caption{Distribution of the mass bias parameter estimates using the ODR method to fit Eq.~\\ref{eq:fit_log} with the slope fixed to 1. The vertical dotted lines represent the mean (recovered) value and the simulated (input) value $(1-B)=0.8$ in blue and red, respectively.}\n        \\label{fig:odr_cor}\n\\end{figure}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[trim=0.4cm 0.4cm 0.4cm 0.4cm, clip, width=0.49\\textwidth]{figure\/ed_bias\/scaling_relation_ITP_ed_biased_boot.png}\n\\includegraphics[trim=0.4cm 0.4cm 0.4cm 0.4cm, clip, width=0.49\\textwidth]{figure\/ed_bias\/scaling_relation_SDSS_ed_biased_cosmo_boot}\n\\end{center}\n\\caption{$M_{500}^{\\rm dyn}-M_{500}^{\\rm SZ}$ scaling relation (Eq.~\\ref{eq:fit_log} with the slope fixed to 1) for the ITP (left panel) and SDSS (right panel) samples. Red and green dots represent the SDSS clusters included in and excluded from the PlCS, respectively. Red, green, and blue lines are the best fit for the SDSS \\textit{cosmo}, the SDSS \\textit{not cosmo,} and the whole SDSS sample, respectively, whereas the dotted line is the $1:1$ relation. The ITP and the SDSS fits are at $2.6-\\sigma$. his difference between these fits suggests a possible Eddington bias contamination.x}\n \\label{fig:relation_ITP_SDSS}\n\\end{figure*}\n\n\\subsection{Mass bias for the ITP and SDSS sub-samples}\nFigure \\ref{fig:relation_ITP_SDSS} shows our fit for the ITP sample (left panel) and the SDSS sample (blue line in the right panel). Apparently, we find a different mass bias for the two sub-samples,\n\\begin{align} \\label{eq:(1-B)_ed_b}\n\\left( 1-B \\right)_{ITP\\phantom{S}} = 1.22 \\pm 0.10;\\\\\n\\left( 1-B \\right)_{SDSS} = 0.89 \\pm 0.08.\n\\end{align}\nBecause the clusters in the two samples were analysed in the same way, this discrepancy cannot be explained by a heterogeneous mass calculation. \nWe ascribe this discrepancy to the different $S\/N$ distribution of cluster samples. The median $S\/N$ for the ITP sample is $4.9$, whereas the SDSS sample median S\/N is $5.7$. Moreover, if we divide the SDSS sample into two parts, the \\textit{cosmo} (the clusters in the PlCS) and the \\textit{not cosmo},  we find that they have a median $S\/N$ $9.4$ and $5.2$, respectively. The fit performed with these two new sub-samples, shown in the right panel of Figure~\\ref{fig:relation_ITP_SDSS}, yields $\\left( 1-B \\right) = 0.83 \\pm 0.11$ (red line) and $\\left( 1-B \\right) = 1.02 \\pm 0.08$ (green line) for the \\textit{cosmo} and \\textit{not cosmo}, respectively. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[trim=0.4cm 0.4cm 0.4cm 0.4cm, clip, width=0.49\\textwidth]{figure\/ed_bias\/scaling_relation_ITP_SDSS_ed_biased_SNR_bins_boot.png}\n\\end{center}\n\\caption{Fit of the scaling relation $M_{500}^{\\rm dyn}-M_{500}^{\\rm SZ}$ in Eq.~\\ref{eq:fit_log} (with the slope fixed to 1) for the whole sample ITP+SDSS (blue line). Green, red, magenta, and yellow dots and lines represents the clusters within the four $S\/N$ bins: $S\/N < 4.81$, $4.81\\leq $S\/N$ < 5.47$, $5.47\\leq $S\/N$ < 6.83,$ and $S\/N \\geq 6.83$ and their best fits, respectively. These fits are only marginally compatible within a maximum difference of $2.9-\\sigma$. The dotted line represents the 1:1 relation.}\n \\label{fig:sc_eb_SNR}\n\\end{figure}\n\nThis evidence led us to investigate the effect of the $S\/N$ of clusters on the $(1-B)$ fits in more detail. \nWe divided the whole catalogue (ITP+SDSS) into four $S\/N$ bins with the same number of clusters, performing the $M_{\\rm dyn}-M_{\\rm SZ}$ fit in each bin and for the whole sample. Figure~\\ref{fig:sc_eb_SNR} shows that the mass bias decreases from lower to higher value of $S\/N$ (see the second column of Table~\\ref{table:SNR_bebc}). This difference suggests that our clusters might be affected (totally or partially) by Eddington bias \\citep[hereafter EB,][]{eddington1913}.\nThe statistical noise scatters above the $S\/N$ threshold some objects with a mass that is lower than the observational limit, resulting in an over-estimation of these masses.\n\n\\begin{table}\n\\small\n\\caption{Value of the mass bias before and after the Eddington bias correction for the S\/N bins.}\n\\label{table:SNR_bebc}\n\\centering\n\\begin{tabular}{ccc}\n\\noalign{\\smallskip}\n\\hline\\hline\n\\noalign{\\smallskip}\nS\/N bin  & \\multicolumn{2}{c}{$(1-B)$} \\\\\n\\cline{2-3}\n\\noalign{\\smallskip}\n  & {EB not corrected} & {EB corrected} \\\\\n\\hline\n\\noalign{\\smallskip}\n$ \\qquad \\quad S\/N < 4.82$     & $1.17 \\pm 0.09$ & $0.99 \\pm 0.07$ \\\\\n$4.82\\leq S\/N < 5.47$              & $1.22 \\pm 0.07$ & $1.08 \\pm 0.06$ \\\\\n$5.47\\leq S\/N < 6.84$              & $0.96 \\pm 0.07$ & $0.89 \\pm 0.06$ \\\\\n$\\; \\qquad \\quad S\/N \\geq 6.84$ & $0.84 \\pm 0.11$ & $0.84 \\pm 0.11$ \\\\\n\n\\noalign{\\smallskip}\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nWe corrected for this EB effect by using the curves in \\cite{remco16}. These were obtained by constructing a $30.000\\,$deg$^2$ mock SZ catalogue that took the \\cite{tinker08} halo mass function, the redshift-dependent comoving volume element, and the \\textit{Planck}{} noise maps properties into account. They reproduced the effect of the noise on the \\emph{\\textup{true}} $S\/N$ as a random Gaussian variable added to the \\emph{\\textup{true}} $S\/N$. They showed that the effect of the bias is more severe the lower the threshold, and it becomes extremely strong for $S\/N \\leq 4.5$. Moreover, they found that the effect of the EB also depends on the redshift. It affected the high-redshift clusters more because the halo mass function at a given S\/N is steeper.\nAlthough these curves were created using the PSZ2 noise maps, they are a good approximation for the PSZ1 as well because they only depend on the mean S\/N level and not on the particular noise level on the PSZ2 maps.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[trim=0.4cm 0.4cm 0.4cm 0.4cm, clip, width=0.49\\textwidth]{figure\/scaling_relation_ITP_SDSS_eb_corrected_SNR_bins_boot.png}\n\\end{center}\n\\caption{Scaling relation $M_{500}^{\\rm dyn}-M_{500}^{\\rm SZ}$ (fit of Eq.~\\ref{eq:fit_log} with a fixed slope equal to 1) after correction for Eddington bias. Magenta, red, yellow, and green dots represent the clusters within the four $S\/N$ bins: $S\/N < 4.82$, $4.82\\leq $S\/N$ < 5.47$, $5.47\\leq $S\/N$ < 6.84,$ and $S\/N \\geq 6.84$, respectively. After correcting for the Eddington bias, performed following the prescriptions by \\cite{remco16}, the best fits within the $S\/N$ bins (green, red, magenta, and yellow lines) are now compatible within $1.9-\\sigma$ at most with each other and with respect to the best fit of our whole sample (blue line). The dotted line represents the 1:1 relation. }\n\\label{fig:sc_eb_cor}\n\\end{figure}\n\nFigure~\\ref{fig:sc_eb_cor} shows the $M_{500}^{\\rm dyn}-M_{500}^{\\rm SZ}$ fit for the whole sample (blue line) and for the $S\/N$ bins (green, yellow, red, and magenta circles and lines), and after the EB correction, all the $\\left( 1-B \\right)$ values are clearly statistically compatible with each other within the $2-\\sigma$ level. The corrected values are listed in the last column of Table~\\ref{table:SNR_bebc}.\n\nAs shown in \\cite{remco16}, the magnitude of the EB depends on redshift. We quantified this by analysing the behaviour of the mass bias parameter as a function of redshift. Table~\\ref{table:z_bebc} shows that the uncorrected SZ masses lead to different values of $\\left( 1-B \\right)$. In particular,  the highest redshift bin differs at almost $3-\\sigma$ from the lowest bin. \nOn the other hand, the $M_{500}^{\\rm dyn}$--$M_{500}^{\\rm SZ}$ relations after the EB correction show more comparable values of the $\\left( 1-B \\right)$: all redshift bins are consistent within the $2-\\sigma$ level. \nHere we note that the bin containing the clusters at the redshift interval $[0.11$, $0.19]$ yields the highest $(1-B)$ value. \n\nAfter the EB correction, the mass bias estimated for the two samples became\n\\begin{align} \\label{eq:(1-B)_ed_b_cor}\n\\left( 1-B \\right)_{ITP\\phantom{S}} = 1.06 \\pm 0.08;\\\\\n\\left( 1-B \\right)_{SDSS} = 0.87 \\pm 0.08,\n\\end{align}\nwhich is compatible within the $2-\\sigma$ level.\n\nWe also investigated the possible dependence of the estimated mass bias on the number of galaxies ${N_{\\rm gal}}$ used to extract the dynamical masses after correction for EB. \nTo do this, each sub-sample (ITP and SDSS) was sub-divided into two bins with the same number of s,  according to the ${N_{\\rm gal}}$ values. We labelled the bin containing lower (LN) and higher (HN) values ${N_{\\rm gal}}$ than the median. We repeated our analysis for each of these bins. For the ITP sample, we obtained\\begin{align} \\label{eq:(1-B)_ITP_ngal}\n\\left( 1-B \\right)_{ITP\\phantom{S}}^{LN} = 1.38 \\pm 0.23 \\\\ \\left( 1-B \\right)_{ITP\\phantom{S}}^{HN} = 0.98 \\pm 0.08,\n\\end{align}\nwhereas the fit for the SDSS sample gave\n\\begin{align} \\label{eq:(1-B)_SDSS_ngal}\n\\left( 1-B \\right)_{SDSS}^{LN} = 1.06 \\pm 0.07 \\\\\n\\left( 1-B \\right)_{SDSS}^{HN} = 0.85 \\pm 0.08.\n\\end{align}\nThe bins for LN and HN are compatible within the $2-\\sigma$ level for both ITP and SDSS samples. However, when we consider the whole sample, the results for the LN and HN sub-samples differ slightly (at the $2.3-\\sigma$ level),\n\\begin{align} \\label{eq:(1-B)_ngal}\n(1-B)^{LN} = 1.11 \\pm 0.07 \\\\\n(1-B)^{HN} = 0.86 \\pm 0.08.\n\\end{align}\nAlthough the residual effects due to ${N_{\\rm gal}}$ should be accounted for in our method \\citep{Ferragamo20}, this result suggests that there might be some marginal residual dependence on ${N_{\\rm gal}}$ in our velocity and mass estimators.\n\n\\begin{table}\n\\small\n\\caption{Value of the mass bias before and after correction for Eddigton bias for the redshift bins. }\n\\label{table:z_bebc}\n\\centering\n\\begin{tabular}{ccc}\n\\noalign{\\smallskip}\n\\hline\\hline\n\\noalign{\\smallskip}\nredshift bin  & \\multicolumn{2}{c}{$(1-B)$} \\\\\n\\cline{2-3}\n\\noalign{\\smallskip}\n  & {EB not corrected} & {EB corrected} \\\\\n\\hline\n\\noalign{\\smallskip}\n$  \\qquad \\quad z < 0.11$     & $0.84 \\pm 0.10$ & $0.82 \\pm 0.09$ \\\\\n$0.11\\leq z < 0.19$               & $1.17 \\pm 0.12$ & $1.10 \\pm 0.11$ \\\\\n$0.19 \\leq z < 0.34$               & $1.05 \\pm 0.08$ & $0.96 \\pm 0.08$ \\\\\n$\\; \\qquad \\quad z \\geq 0.34$  & $1.19 \\pm 0.07$ & $1.06 \\pm 0.06$ \\\\\n\\noalign{\\smallskip}\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Final result for the mass bias}\n\\label{subsec:mass_bias_our}\n\nAfter the EB correction, by fitting the $M_{500}^{\\rm dyn}$--$M_{500}^{\\rm SZ}$ relation for the whole GCs sample and fixing the slope to unity, we obtain\n\\begin{equation}\n\\left( 1-B \\right) = 0.88 \\pm 0.07.\n\\label{eq:(1-B)_final1}\n\\end{equation}\nEach of the parameters and related errors shown so far was estimated as the mean and standard deviation of 10000 bootsrap resamples, respectively.\nHowever, taking the expected $\\sim 6\\%$ bias introduced by the linear regression estimator into account, the corrected value of the mass bias should be\n\\begin{equation}\n\\left( 1-B \\right) = 0.83 \\pm 0.07 \\ (stat) \\pm 0.02 \\ (syst),\n\\label{eq:(1-B)_final}\n\\end{equation}\nwhere the first error is the statistical error, and the second error is a systematic error associated with the bias of the ODR method.\n\n\nThe \\textit{Planck}{} analysis showed that the mass bias could be a function of the cluster mass \\citep{planck13_count}. For this reason, we repeated our fit with the ODR method, but letting the slope free as well. In this case, after correcting for the bias due to the ODR method, we obtain\n\\begin{equation}\n\\frac{M_{500}^{\\rm SZ}}{6\\times10^{14}\\;\\hbox{M$_{\\odot}$}} = \\left(0.84\\pm0.12\\pm0.02 \\right)\n\\left(\\frac{M_{500}^{\\rm dyn}}{6\\times10^{14}\\;\\hbox{M$_{\\odot}$}} \\right)^{1.00\\pm0.23\\pm0.08}\n\\label{eq:(1-B)_final_slope_pivot222}\n\\end{equation}\nat the pivot mass $6\\times 10^{14}\\hbox{M$_{\\odot}$}$. This result, which is compatible within $1-\\sigma$ with the result in Eq. \\ref{eq:(1-B)_final}, does not give indications about a mass dependence for the $M^{\\rm dyn}$--$M^{\\rm SZ}$ relation.\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[trim=0.4cm 0.4cm 0.2cm 0.4cm, clip, width=0.8\\textwidth]{figure\/scaling_relation_ITP_SDSS_eb_corrected_boot.png}\n\\end{center}\n\\caption{Scaling relation between $M_{500}^{\\rm SZ}$ and $M_{500}^{\\rm dyn}$ for the combined whole sample. The blue line represents the relation obtained by fixing the slope to $\\alpha = 1$ (blue line), and the red line represents the relation obtained when the slope was free to vary. The correction of the bias introduced by the ODR estimator is not applied here.}\n \\label{fig:scal_rel_final}\n\\end{figure*}\nFigure \\ref{fig:scal_rel_final} shows the two fits of the $M_{500}^{\\rm dyn}$--$M_{500}^{\\rm SZ}$ scaling relation, with the slope fixed to 1 (blue line) and with the slope free to vary (red line). \n\n\n\\section{Comparison with other works}\n\\label{sec:comparison}\nThe multi-component nature of GCs and multi-wavelength studies allows us to assess the relative bias of mass proxies based on different observables. Here we compare our results on $(1-B)$ with those obtained by other authors using different mass proxies and methods. The literature about this argument is very rich. We restrict our comparison to some of the recent works.  Table~\\ref{table:(1-b)_literature} and Figure~\\ref{fig:comp_literarure_final} summarise all values obtained by all the surveys described below. \n\n\\subsection{Mass bias from \\textit{Planck}\\ Collaboration analysis}\n\\label{subsec:planck_bias}\n\nThe \\textit{Planck}{} Collaboration calibrated the PSZ1 masses using X-ray observations of nearby relaxed clusters. For this reason, the mass bias they found is homologous to the hydrostatic equilibrium mass bias, $M_{500}^{\\rm HE}$. \nTo find the value of the mass bias $M_{500}^{\\rm HE}=(1-b)M_{500}^{\\emph{true}}$, they compared the $Y_{500}$--$M_{500}$ relation derived using data from seven  simulations including different types of input physics \\citep{nagai06, yang10, sehgal10, krause12, battaglia12, kay12, sembolini13}, with the relation obtained by comparing $Y_{500}$ and the hydrostatic mass \\citep{planck13_count}.\nThe main conclusion is that the dependence of bias on mass,  $b\\left(M_{500}^{\\emph{true}}\\right)$, is strong, which also translates into different slopes for the observed and true $Y_{500}$--$M_{500}$ relations. \nBecause of this mass dependence, the \\textit{Planck}{} collaboration selected a representative value for $(1-b)$ as the median value obtained for a mass pivot point $M_{500}=6 \\times 10^{14}\\, \\rm{M_\\odot}$ \\citep{planck13_count},\n\\begin{equation}\n\\left(1-b\\right) = 0.8_{-0.1}^{+0.2}.\n\\label{eq:(1-b)_planck}\n\\end{equation}\nThe reported confidence interval $[0.7, 1]$ accounts for the scatter of the different simulations and measurements. \n\n\\subsection{Bias from dynamical mass estimates}\n\\label{subsec:dynamical_bias}\n\nSeveral groups have studied the relation $M^{\\rm dyn}$--$M^{\\rm SZ}$ using SZ data from the \\textit{Planck}{} PSZ2 catalogue \\citep{amodeo17}, the Atacama Cosmology Telescope (ACT) \\citep{sifon16}, the South Pole Telescope (SPT) \\citep{ruel14}, and spectroscopic data from the Gemini Multi-Object Spectrograph (GMOS), installed at the Gemini telescopes. \nAlthough these  studies also compared the SZ and dynamical masses, each shows important differences in the method with respect to our analysis. \n\n\\subsubsection{South Pole Telescope cluster sample}\n\\label{subsubsec:spt_bias}\nThe first of these analyses was published by \\cite{ruel14}. Their analysis used 43 SZ-selected GCs within the SPT catalogues \\citep{Vanderlinde10, Williamson11, Reichardt13}. The sample consists of massive ($2.7 \\leq M_{500}^{\\rm SPT} \/ 10 ^{14}\\; \\hbox{M$_{\\odot}$}  \\leq 18.0)$ s at $z \\geq 0.3$ with more than 15 members.\nThe most important differences with respect to our study are the scaling relation between observables and masses. From the SZ point of view, $M_{500}^{\\rm SPT}$ were calculated using the $Y$--$M$ scaling in \\cite{Reichardt13}. This relation presents a slightly fainter slope than the slope from \\cite{planck13_count}. This might result in some differences in the mass estimate, especially for the most massive clusters. From the dynamical mass point of view, the SPT group used the \\cite{saro13} relation, which, as shown in \\cite{sifon16}, tends to overestimate the dynamical masses (especially for massive clusters) when compared to the \\cite{munari13} relation we used here.\nAnother difference is that following the prescription in \\cite{beers90}, they used the biweight as the velocity dispersion estimator applied on cluster members selected by an iterative $3-\\sigma$ clipping around the mean velocity. \nIn contrast to what we did here, Ruel and collaborators decided to scale the SZ masses from $M_{500}^{\\rm SPT}$ to $M_{200}^{\\rm SPT}$. However, in order to perform this transformation, they also assumed the same NFW density profile and the \\cite{duffy08} mass-concentration relation as in our method. Finally, \\cite{ruel14} found that \n\\begin{equation}\n\\exp\\Bigg\\{ \\left\\langle \\ln\\left(\\frac{M_{200}^{\\rm SPT}}{M_{200}^{\\rm dyn}} \\right) \\right\\rangle \\Bigg\\} = \\left( 1-B \\right) = 0.72 \\pm 0.57,\n\\label{eq:(1-b)_ruel_SPT}\n\\end{equation}\nwhich is compatible with our result, except for its large error.\n\nWe would like to remark that we report the mass bias \\citep[Eq.~16 in][]{ruel14} in Eq.~\\ref{eq:(1-b)_ruel_SPT}, according to our definition of $(1-B)$ and after propagating the uncertainties.\n\n\\subsubsection{Atacama Cosmology Telescope cluster sample}\n\\label{subsubsec:act_bias}\n\nThe second scaling relation, performed using dynamical masses, was obtained by \\cite{sifon16}.\nFor their cosmological analysis, they used a subsample of ACT clusters composed of 21 objects detected with an $S\/N > 5.1$ at redshifts $0.2 \\leq z \\leq 1.06$ in a mass interval $1.0 < M_{200}^{\\rm dyn} < 13.0$ (in units of $10^{14}$\\,$\\hbox{M$_{\\odot}$}$), which constitute their \\textit{\\textup{cosmological sample}}. The 21 GCs were observed spectroscopically using two different spectrographs: GMOS \\citep{Hook04} at the Gemini-South telescope (Chile), and RSS \\citep{Burgh03} at the SALT telescope (South Africa). They obtained a median number of observed members of ${N_{\\rm gal}}  55$.\nIn this case, there are fewer differences between our method and the method of the ACT group. They used the same scaling relations $Y_{500}$--$M_{500}$ and $\\sigma_{200}$--$M_{200}$, the pressure profile by \\cite{Arnaud10}, and the AGN fit by \\cite{munari13}. The main difference is on the mass-concentration relation that was used to scale $M_{200}^{\\rm dyn}$ to $M_{500}^{\\rm dyn}$. While we used the \\cite{duffy08} relation, they considered the \\cite{dutton14} relation.\nIn addition to the member selection algorithm, they also used the biweight as a velocity dispersion estimator. It is also important to remark that \\cite{sifon16} applied an aperture correction to the velocity dispersion estimate and also took the Eddington bias into account when they corrected the $Y_{500}$ estimation, as explained in \\cite{hasselfield13}.\nIn this case, they did not fit for the relation $M_{\\rm dyn}$ to $M_{\\rm SZ}$, but they defined the mass bias as the ratio of the mean SZ and the mean dynamical mass. They obtained \n\\begin{equation}\n\\frac{\\left \\langle M_{\\rm SZ} \\right \\rangle}{\\left \\langle M_{\\rm dyn} \\right \\rangle}= \\frac{\\left( 1-b_{\\rm SZ} \\right)}{\\beta_{\\rm dyn}}= 1.10 \\pm 0.13^{stat} \\pm 0.14 ^{syst},\n\\label{eq:(1-b)_sifon_ACT}\n\\end{equation}\nwhere $\\beta_{\\rm dyn}$ is the bias of dynamical masses, defined as $\\left \\langle M_{\\rm dyn} \\right \\rangle = \\beta_{\\rm dyn} \\left \\langle M_{\\emph{true}} \\right \\rangle$.\nOur result (Eq. \\ref{eq:(1-B)_final}) and that obtained by \\cite{sifon16} are compatible within $1.7-\\sigma$.\n\n\n\\subsubsection{\\textit{Planck}{} PSZ2 sample}\n\\label{subsubsec:amodeo_bias}\n\nThe most recent published scaling relation was obtained by \\cite{amodeo17} using 17 low-redshift ($z<0.5$) clusters from the PSZ2 catalogue in the mass interval $3.0 < M_{200}^{\\rm dyn} < 14.0$ (units of $10^{14}\\,\\hbox{M$_{\\odot}$}$) and with a median ${N_{\\rm gal}} = 20$.\nIn this case, the SZ masses were estimated following the same prescription as we considered here. However, they selected a different velocity dispersion-mass relation. This choice was dictated by the method they used to determine the bias. Similarly to the \\textit{Planck}{} Collaboration, they decided not to compare the masses themselves, but to do the comparison with the $\\sigma_{200}$--$M_{200}$ scaling relation normalisation parameter by assuming a self-similar slope ($\\alpha=1\/3$). To do this, they rescaled the \\textit{Planck}{} $M_{500}^{\\rm SZ}$ to $M_{200}^{\\rm SZ}$ by assuming the mass-concentration relation by \\cite{dutton14}, and fitted the $\\sigma_{200}^{\\rm obs}-M_{200}^{\\rm SZ}$ relation with a fixed slope $\\alpha=1\/3$. They obtained the normalisation parameter $A=1158 \\pm 61$ in this way. \nThey compared this result with the \\cite{evrard08} relation, which was constrained with an N-body DM-only simulation with a normalisation parameter $A_{DM}=1082.9 \\pm 4.0$, from which they obtained \n\\begin{equation}\n\\left( 1-b \\right) = \\left( \\frac{A_{DM}}{A} \\right)^3\\ b_v^3\\ f_{EB}\\ f_{corr} = 0.64 \\pm 0.11,\n\\label{eq:(1-b)_amodeo}\n\\end{equation}\nwhere $b_v$, $f_{EB}$ , and $f_{corr}$ are the velocity bias, the Eddington bias correction, and the correction for the correlated scatter between velocity dispersion and\\textit{Planck}{} masses, respectively.\nThe velocity bias is defined as $b_v= A_{gal}\/A_{DM}$, which arises from assuming that the DM and the galaxies of a cluster can have different velocities. \\cite{munari13} used a hydrodynamical simulation and fitted the $\\sigma$--$M$ relation considering both the DM particles and galaxies. They obtained two different values for the normalisation parameters, which leads to a velocity bias $b_v=1.08$. In this paper, we accounted for the velocity bias using the AGN fit by \\cite{munari13}. Instead, the \\cite{evrard08} fit was made with DM-only simulations. \nThe Eddington bias correction in this case is a global correction quantified in $f_{EB}=0.84 \\pm 0.027$ and the correlated scatter correction arises from the \\cite{stanek10} study on the covariance between observables, which is estimated using the Millennium Gas Simulation \\citep{hartley08}. \\cite{stanek10} found a significant correlation between velocity dispersion and SZ signal due to non-gravitational processes in GCs. \\cite{amodeo17} quantified this bias as $f_{corr}=1.01$. \n\nComparing our result ($\\left( 1-B \\right) = 0.83 \\pm 0.07 \\pm 0.02$) with equation~\\ref{eq:(1-b)_amodeo}, we see that they are consistent at $\\sim 1.3-\\sigma$.\n\n\n\\subsection{Weak-lensing mass bias }\n\\label{subsec:WL_bias}\n\nSeveral research groups have studied the SZ mass bias, assuming the weak-lensing (WL) mass as the reference \\emph{\\textup{true}} mass. Here we describe seven of them. \n\n\\subsubsection{WtG project}\n\\label{subsubsec:WtG_bias}\n\nIn 2014, the group of the Weighing the Giants (WtG) project studied the relation of the $M_{500}^{\\rm SZ}$ and the $M_{500}^{\\rm WL}$ using a sample of 38 GCs in common with the PSZ1 catalogue \\citep{von_der_Linden14}.\nFor a complete description of the WL mass estimate and for the GCs catalogue they used to constrain the scaling relation, see \\cite{applegate14}. \n\\cite{von_der_Linden14} found through a bootstrap realisation of the unweighted mean of the ratio between WL and SZ masses the following value:\n\\begin{equation}\n\\left \\langle \\frac{M_{500}^{\\rm SZ}}{M_{500}^{\\rm WtG}} \\right \\rangle = \\left( 1-b \\right) = 0.698 \\pm 0.062.\n\\label{eq:(1-b)_WtG}\n\\end{equation}\nThis result is compatible at $1.25-\\sigma$ with our estimate of the mass bias.\n\nA sub-sample of 22 GCs of the WtG catalogue is part of the PSZ1 PlCS. Repeating the analysis with this sub-sample alone, they obtained \n\\begin{equation}\n\\left( 1-b \\right) = 0.688 \\pm 0.072.\n\\label{eq:(1-b)_WtG_cosmo}\n\\end{equation}\nThis result is slightly smaller than the previous one, but they are perfectly compatible with each other. This value was used by the \\textit{Planck}{} Collaboration in its 2015 cosmological analysis based on the PSZ2 catalogue as one of the priors on the value of the mass bias.\nThe WtG group also performed the fit by letting the slope of the power law vary freely. By using the Bayesian linear regression method developed by \\cite{kelly07}, they found \n\\begin{equation}\n\\frac{M_{500}^{\\rm SZ}}{10^{15}\\;\\hbox{M$_{\\odot}$}} = \\pl0.699_{-0.060}^{+0.059}\\right) \\left( \\frac{M_{500}^{WtG}}{10^{15}\\;\\hbox{M$_{\\odot}$}}\\right)^{0.68^{+0.15}_{-0.11}}\n\\label{eq:(1-b)_WtG_alpha1}\n\\end{equation}\nand\n\\begin{equation}\n\\frac{M_{500}^{\\rm SZ}}{10^{15}\\;\\hbox{M$_{\\odot}$}} = \\pl0.697_{-0.095}^{+0.077}\\right) \\left( \\frac{M_{500}^{WtG}}{10^{15}\\;\\hbox{M$_{\\odot}$}}\\right)^{0.76^{+0.39}_{-0.20}},\n\\label{eq:(1-b)_WtG_alpha2}\n\\end{equation}\nfor the whole and the cosmological sample, respectively. \nBoth results are compatible within the errors with our fit in Eq~\\ref{eq:(1-B)_final_slope_pivot222}. \n\n\n\\subsubsection{Canadian Cluster Comparison Project}\n\\label{subsubsec:CCCP_bias}\n\n\\cite{hoekstra15} presented the comparison between the WL masses, estimated by the Canadian Cluster Comparison Project (CCCP), and the masses from the PSZ1 catalogue of 37 GCs, 20 of them with $S\/N \\geq 7$.\nThrough a linear fit, they found\n\\begin{align} \\label{eq:(1-b)_CCCP}\n\\left( 1-b \\right) = 0.76 \\pm 0.05\\\\\n\\left( 1-b \\right) = 0.78 \\pm 0.07\n\\end{align}\nfor the whole sample of 37 clusters and for the 20 with $S\/N \\geq 7$, respectively.\nThese two results are perfectly compatible with our estimate of $\\left( 1-b \\right)$.\n\nDuring the comparison of their results with those of WtG, \\cite{hoekstra15} fitted a power-law function to investigate whether the bias depended on the mass in their data as well. Using the entire sample, they found\n\\begin{equation}\n\\frac{M_{500}^{\\rm SZ}}{10^{15}\\;h_{70}^{-1} \\  \\rm{M_\\odot}} = \\left(0.76 \\pm 0.04 \\right) \\times \\left(\\frac{M_{500}^{CCCP}}{10^{15}\\;h_{70}^{-1} \\  \\rm{M_\\odot}} \\right)^{0.64\\pm0.17}.\n\\label{eq:(1-b)_CCCP_slope}\n\\end{equation}\nThe compatibility between the slope obtained by \\cite{von_der_Linden14}, \\cite{hoekstra15} and this paper supports the conclusion by \\cite{von_der_Linden14} that the bias of \\textit{Planck}{} masses might depend on the cluster mass. \n\n\\subsubsection{Local Cluster Substructure Survey}\n\\label{subsubsec:LoCuSS_bias}\n\n\\cite{smith16} analysed the mass bias with a sample of 44 clusters in common with the Local Cluster Substructure Survey (LoCuSS) and the PSZ2 catalogue. They obtained \n\\begin{equation}\n\\left( 1-b \\right) = 0.95 \\pm 0.04.\n\\label{eq:(1-b)_LoCuSS}\n\\end{equation}\nThe LoCuSS sample consists of clusters at redshift $0.15 \\leq z \\leq 0.3$ with WL masses, estimated in \\cite{okabe16}, between $2.12 \\leq M_{200}^{WL} \/ 10^{14} \\  \\rm{M_\\odot} \\leq 12.75$. In this case, the mass bias is defined as the weighted mean of the logarithmic ratio of SZ and WL masses, as\n\\begin{equation}\n\\left( 1-b \\right) =  \\rm{exp} \\left[ \\frac{\\sum_{i=1}^{n}\\ \\omega_i\\ \\rm{ln}\\left( \\frac{M_{SZ,i}}{M_{WL,i}} \\right)}{\\sum_{i=1}^{n}\\ \\omega_i} \\right],\n\\label{eq:(1-b)_LoCuSS_2}\n\\end{equation}\nwith weights defined by \n\\begin{equation}\n\\omega_i = \\left[ \\left( \\frac{\\delta M_{SZ,i}}{\\langle \\delta M_{SZ} \\rangle}\\right)^2 + \\left( \\frac{\\delta M_{WL}}{\\langle \\delta M_{WL,i}\\rangle} \\right)^2 \\right]^{-1}.\n\\label{eq:(1-b)_LoCuSS_3}\n\\end{equation}\n\nThis LoCuSS measurement ($0.95\\pm 0.04$) is higher than our value of $\\left( 1-B \\right)$. Because no Eddington bias correction is cited in \\cite{smith16} or in \\cite{okabe16}, we decided to repeat the fit by limiting our sample to the clusters within the range $0.15 \\leq z \\leq 0.3$. As a result, we obtained $\\left( 1-B \\right) = 1.17 \\pm 0.11$. However, this result might be biased by the clusters within the $0.11 \\leq z < 0.19$ (see Table~\\ref{table:z_bebc} and the discussion in Section~\\ref{subsec:relation2}).\n\n\\subsubsection{Canada-France-Hawaii Telescope Stripe 82 Survey}\n\\label{subsubsec:CS82_bias}\nThe sample used by \\cite{battaglia16} to determine the mass bias consists of 19 GCs from the ACT equatorial sample \\citep{hasselfield13} observed during the Canada-France-Hawaii Telescope Stripe 82 Survey (CS82). They divided their sample into two $S\/N$ bins. The first contains nine clusters with $S\/N > 5$, $\\left< M_{500}^{\\rm SZ} \\right>=4.7 \\pm 1.0 \\ (\\times 10^{14}\\; \\hbox{M$_{\\odot}$})$, and the second consists of ten clusters within the range $4< S\/N<5$, with $\\left< M_{500}^{\\rm SZ} \\right>=2.7 \\pm 1.0 \\ (\\times 10^{14}\\; \\hbox{M$_{\\odot}$})$. The mass bias they found, defined as the ratio of the mean SZ and WL masses, is\n\\begin{align} \\label{eq:(1-b)_strip82}\n\\left( 1-b \\right)_{S\/N > 5} = 0.87 \\pm 0.27,\\\\\n\\left( 1-b \\right)_{S\/N < 5} = 0.82 \\pm 0.36.\n\\end{align}\nFor a detailed description of the WL mass estimation, see \\cite{battaglia16} and the references therein, and for SZ mass estimates, see \\cite{hasselfield13}.\nIt is important to remark that the ACT SZ masses are corrected for Eddington bias as explained in \\cite{hasselfield13}. Furthermore, \\cite{battaglia16} estimated that by comparing $31$ clusters in common between ACT and \\textit{Planck}{}, the corrected ACT masses are lower by $0.89$ times than those of \\textit{Planck}{}on average. \nThese results are clearly compatible with our estimate of the bias.\n\n\\subsubsection{PSZ2LenS}\n\\label{subsubsec:PSZ2LenS_bias}\n\n\\cite{sereno17} used 32 clusters from the PSZ2LenS sample with a published SZ signal in \\textit{Planck}{} catalogues to estimate the $\\left( 1-b \\right)$ parameter. The PSZ2LenS sample consists of 35 GCs detected by \\textit{Planck}{} and within the sky coverage of the Canada France Hawaii Telescope Lensing Survey \\citep[CFHTLenS,][] {heymans12} and the Red Cluster Sequence Lensing Survey \\citep[RCSLenS,][]{hildebrandt16}. The PSZ2Lens clusters lie in a wide mass range, $M_{500}^{WL}=0.9-14.8 \\times 10^{14}\\; \\hbox{M$_{\\odot}$}$.\nSZ masses are taken from \\textit{Planck}{} catalogues \\citep{planck13_cat, planck15_cat}, and were therefore calculated as in this paper. For a detailed description of the WL analysis, see \\cite{sereno17} and references therein. \nThe PSZ2LenS research group, after assuming the Eddington bias correction from \\cite{battaglia16} and \\cite{sereno15}, found\n\\begin{equation}\n\\exp \\left(\\frac{\\ln\\left \\langle M_{\\rm SZ} \\right \\rangle}{\\ln\\left \\langle M_{\\rm WL} \\right \\rangle} \\right) = \\left( 1-b \\right) = 0.76 \\pm 0.08.\n\\label{eq:(1-b)_PSZ2LenS}\n\\end{equation}\nThis value is statistically consistent with our result.\nThey also used a sub-sample of 15 PSZ2LenS clusters within the \\textit{Planck}{} cosmological sample, finding $\\left( 1-b \\right) = 0.67 \\pm 0.09$. This result is lower but still compatible with our result at $1.2-\\sigma$. \n\n\\subsubsection{Cluster Lensing And Supernova survey with Hubble}\n\\label{subsubsec:CLASH_bias}\n\nThe WL masses of 21 GCs from the Cluster Lensing And Supernova survey with Hubble \\citep[CLASH,][]{postman12}) were used by \\cite{penna17} to compare with the respective SZ masses from \\textit{Planck}{} catalogues. They performed a Bayesian analysis to constrain the mass bias $\\left( 1-b \\right)$. After the correction of $M_{\\rm SZ}$ for Eddington bias from \\cite{battaglia16}, they found\n\\begin{equation}\n\\left( 1-b \\right) = 0.73 \\pm 0.10,\n\\label{eq:(1-b)_CLASH}\n\\end{equation}\nwhich is fully compatible with our result.\n\n\\subsubsection{Hyper Suprime-Cam Subaru Strategic Program}\n\\label{subsubsec:HSC-SSP_bias}\n\nThe Hyper Suprime-Cam Subaru Strategic Program \\citep[HSC-SSP,][]{aihara18b, aihara18a} group published two studies for which they compared their WL masses with SZ masses from \\textit{Planck}{} PSZ2 and ACTPol (Atacama Cosmology Telescope Polarimeter experiment).\nIn the first study, \\cite{medezinski18} used five clusters from the HSC-SSP first-year catalogue \\citep{planck15_cat} in common with the PSZ2 catalogue. They chose the mean $M_{500}^{\\rm WL}$, defined as the weighted stack of the WL signal of these five clusters. The mean SZ mass is the weighted mean of the masses retrieved from the PSZ2 corrected for Eddington bias using the prescription given by \\cite{battaglia16}. The weight for the WL stack and for the SZ mean is the same and depends on the errors on the galaxy shape measurement for each cluster. The ratio of the mean WL and SZ mass is the bias\n\\begin{equation}\n\\frac{\\left \\langle M_{\\rm SZ} \\right \\rangle}{\\left \\langle M_{\\rm WL} \\right \\rangle}  = \\left( 1-b \\right) = 0.80 \\pm 0.14,\n\\label{eq:(1-b)_HSC_planck}\n\\end{equation}\nwhich is compatible with our result. \n\nIn a second study, \\cite{miyatake18} used eight GCs in common with the ACTPol \\citep{hilton18} sample. As in the previous analysis, the mean WL mass is the result of the weighted stack of all the clusters taken into account, whereas $\\left \\langle M_{500}^{\\rm SZ} \\right \\rangle$ is the weighted mean of the corrected SZ masses from the ACTPol catalogue. The mass bias for this sample is\n\\begin{equation}\n\\frac{\\left \\langle M_{SZ} \\right \\rangle}{\\left \\langle M_{WL} \\right \\rangle}  = \\left( 1-b \\right) = 0.74_{-0.12}^{+0.13}.\n\\label{eq:(1-b)_HSC_ACTPol}\n\\end{equation}\nIn this case, the value of the mass bias is also compatible with the value obtained in our analysis.\n\n\\begin{table*}\n        \\tiny\n        \\caption{Summary of mass bias measurements from the literature according to each of the particular definitions in section~\\ref{sec:comparison}.}\n        \\label{table:(1-b)_literature}\n        \\centering\n        \\begin{tabular}{c c c c c}\n        \\noalign{\\smallskip}\n        \\hline\\hline\n        \\noalign{\\smallskip}\n        \\, SURVEY & REFERENCE SAMPLE & N. CLUSTERS & Mass bias & reference \\\\ \n        \\noalign{\\smallskip}\n        \\hline\n        \\hline\n        \\noalign{\\smallskip}\n        \\multicolumn{5}{c}{X-RAY } \\\\\n        \\noalign{\\smallskip}\n        \\hline\n        \\noalign{\\smallskip}\n         & \\textit{Planck}{} PSZ1 & 189  & $0.8_{-0.2}^{+0.1} $ & {\\cite{planck13_cat}}\\\\\n        \\noalign{\\smallskip}\n        \\hline\n        \\hline\n        \\noalign{\\smallskip}\n        \\multicolumn{5}{c}{VELOCITY DISPERSION } \\\\\n        \\noalign{\\smallskip}\n        \\hline\n        \\noalign{\\smallskip}\n         & SPT & 44 & $0.72 \\pm 0.57$ & {\\cite{ruel14}}  \\\\\n         & ACT & 21 & $1.10 \\pm 0.13$ & {\\cite{sifon16}} \\\\\n         & \\textit{Planck}{} PSZ2& 17 & $0.64 \\pm 0.11$ & {\\cite{amodeo17}} \\\\\n         & \\textit{Planck}{} PSZ1 & 207 & $0.83 \\pm 0.07 \\pm 0.02 $ & {This work} \\\\\n        \\noalign{\\smallskip}\n        \\hline\n        \\hline\n        \\noalign{\\smallskip}\n        \\multicolumn{5}{c}{WEAK LENSING } \\\\\n        \\noalign{\\smallskip}\n        \\hline\n        \\noalign{\\smallskip}\n        WtG    & \\textit{Planck}{} PSZ1 & 38 & $0.688 \\pm 0.072$ & {\\cite{von_der_Linden14}} \\\\\n        CCCP & \\textit{Planck}{} PSZ1 & 37 & $0.76 \\pm 0.05$ & {\\cite{hoekstra15}} \\\\\n        LoCuSS  & \\textit{Planck}{} PSZ2 & 44   & $0.95 \\pm 0.04$ & {\\cite{smith16}} \\\\\n        CS82  & ACT & 19 & $0.87 \\pm 0.27$ & {\\cite{battaglia16}} \\\\\n        PSZ2LenS  & \\textit{Planck}{} PSZ2 & 32 & $0.76 \\pm 0.08$ & {\\cite{sereno17}} \\\\\n        CLASH  & \\textit{Planck}{} PSZ1 & 21 & $0.73 \\pm 0.10$ & {\\cite{penna17}} \\\\\n        HSC-SSP  & \\textit{Planck}{} PSZ2 & 5 & $0.80 \\pm 0.15$ & {\\cite{medezinski18}} \\\\\n        HSC-SSP  & ACTPol & 8 & $0.74_{-012}^{+0.13}$ & {\\cite{miyatake18}} \\\\\n        \\noalign{\\smallskip}\n        \\hline\n        \\end{tabular}\n\\end{table*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[trim=0.4cm 0.35cm 0.35cm 0.35cm, clip, width=\n0.8 \\textwidth]{figure\/planck_mass_bias_literature_3.png}\n\\end{center}\n\\caption{Value of the mass bias from previous studies. In blue we show the result from \\cite{planck13_count}, using a scaling relation from X-ray observations; in green we plot the mass bias from $M_{\\rm dyn}$-$M_{\\rm SZ}$ scaling relations, and in red we show those from weak-lensing studies. All these values are listed in table \\ref{table:(1-b)_literature}. The grey shaded region represents the mass bias values that reconciles the tension between CMB and SZ number counts from \\cite{planck13_count}. The green star represents the mass bias value we found here.}\n \\label{fig:comp_literarure_final}\n\\end{figure*}\n\n\n\n\\section{Conclusions}\n\\label{sec:cosmo}\n\nThis is the third (and last) paper in a series describing the results of the ITP13 observational program, dedicated to the characterisation of the PSZ1 sources in the northern sky without known optical counterparts at the time the catalogue was published. Here we presented for the first time the velocity dispersion and mass estimates for 58 clusters in the PSZ1-North sample and for 35 clusters that are not associated with the PSZ1 sample. \n\nUsing SDSS archival data, we also studied 212 clusters with known counterparts, and we also extracted the velocity dispersion and mass estimation, using the same method as for the ITP sample. This paper presents dynamical masses for 270 s within the PSZ1-North sample.\n\n\nA sub-sample of 207 clusters was used to explore the mass bias between the dynamical mass and the SZ mass estimates.  \nGalaxy cluster number counts are extremely sensitive to $\\Omega_m$ and $\\sigma_8$ through the mass function. However, the cosmological analyses performed by the \\textit{Planck}{} Collaboration result in a tension between the constraints from the primordial CMB power spectrum and those derived from the cluster number counts \\citep{planck13_count, planck15_count}. The main cause of this tension was originally ascribed to the value of the mass bias. The \\textit{Planck}{} Collaboration, through the joint analysis of SZ, X-ray data, and simulations, constrained this parameter to the value of $(1-b)=0.8_{-0.1}^{+0.2}$. \nIn this paper, we used the largest sub-sample of PSZ1 clusters (207) observed both in microwave and optical wavelengths to date, and we obtained $(1-B)=0.83\\pm 0.07 \\ (stat) \\pm 0.02 \\ (syst)$. This measurement presents the lowest statistical error obtained so far using dynamical masses as reference,  and it is statistically compatible with the \\textit{Planck}{} one.\n\nWe have also compared our results with those in the literature. We find that although it is slightly higher than average, our measurement is compatible within $1-\\sigma$ in most cases, both with those obtained from dynamical mass analyses and those obtained from comparison with weak-lensing masses. \nThe final mass bias from this study, as well as the mean value of all other previous results in the literature, is still higher than the value that is required to reconcile the tension between CMB and SZ number counts \\citep{planck13_count, planck15_count}. \n\n\n\n\n\\begin{acknowledgements}\nWe thank R. van der Burg and J.-B. Melin for useful comments.\nThis article is based on observations made with the Gran\nTelescopio Canarias operated by the Instituto de Astrofisica de Canarias, the Isaac Newton Telescope, and the William Herschel Telescope operated by the Isaac Newton Group of Telescopes, and the Italian Telescopio Nazionale Galileo operated by the Fundacion Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica). All these facilities are located at the Spanish Roque de los Muchachos Observatory of the Instituto de Astrofisica de Canarias on the island of La Palma. \nPart of this research has been carried out with telescope time awarded by the CCI International Time Programme at the Canary Islands Observatories (programmes ITP13B-15A).\nFunding for the Sloan Digital Sky Survey (SDSS) has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. \nThis work has been partially funded by the Spanish\nMinistry of Science and Innovation (MICINN) under the projects ESP2013-48362-C2-1-P, AYA2014-60438-P and AYA2017-84185-P. %\nAS and RB acknowledge financial support from MICINN under the Severo Ochoa Programs SEV-2011-0187 and SEV-2015-0548.\nHL is supported by ETAg grants PUT1627 and PRG1006 and by EU through the ERDF CoE grant TK133.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa} \n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}