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Study of an entropy dissipating finite volume scheme for a nonlocal cross-diffusion system

Maxime Herda, Antoine Zurek

TL;DR

This work develops a fully implicit TPFA finite-volume scheme for a nonlocal SKT cross-diffusion system on a periodic domain and proves that it preserves mass and satisfies a discrete entropy-dissipation inequality. It establishes existence, positivity, and stability at each time step, and then shows convergence of the discrete solutions to distributional solutions as the mesh is refined, via a discrete Kolmogorov/duality framework and Kruzhkov-type compactness. The paper also provides a thorough numerical validation in 1D and 2D, including nonlocal-to-local localization limits and Turing instabilities, illustrating the scheme's robustness across nonlocal and local regimes. Together, these results demonstrate a reliable, entropy-preserving numerical approach for nonlocal cross-diffusion systems with practical relevance to population dynamics and pattern formation.

Abstract

In this paper we analyse a finite volume scheme for a nonlocal version of the Shigesada-Kawazaki-Teramoto (SKT) cross-diffusion system. We prove the existence of solutions to the scheme, derive qualitative properties of the solutions and prove its convergence. The proofs rely on a discrete entropy-dissipation inequality, discrete compactness arguments, and on the novel adaptation of the so-called duality method at the discrete level. Finally, thanks to numerical experiments, we investigate the influence of the nonlocality in the system: on convergence properties of the scheme, as an approximation of the local system and on the development of diffusive instabilities.

Study of an entropy dissipating finite volume scheme for a nonlocal cross-diffusion system

TL;DR

This work develops a fully implicit TPFA finite-volume scheme for a nonlocal SKT cross-diffusion system on a periodic domain and proves that it preserves mass and satisfies a discrete entropy-dissipation inequality. It establishes existence, positivity, and stability at each time step, and then shows convergence of the discrete solutions to distributional solutions as the mesh is refined, via a discrete Kolmogorov/duality framework and Kruzhkov-type compactness. The paper also provides a thorough numerical validation in 1D and 2D, including nonlocal-to-local localization limits and Turing instabilities, illustrating the scheme's robustness across nonlocal and local regimes. Together, these results demonstrate a reliable, entropy-preserving numerical approach for nonlocal cross-diffusion systems with practical relevance to population dynamics and pattern formation.

Abstract

In this paper we analyse a finite volume scheme for a nonlocal version of the Shigesada-Kawazaki-Teramoto (SKT) cross-diffusion system. We prove the existence of solutions to the scheme, derive qualitative properties of the solutions and prove its convergence. The proofs rely on a discrete entropy-dissipation inequality, discrete compactness arguments, and on the novel adaptation of the so-called duality method at the discrete level. Finally, thanks to numerical experiments, we investigate the influence of the nonlocality in the system: on convergence properties of the scheme, as an approximation of the local system and on the development of diffusive instabilities.
Paper Structure (25 sections, 12 theorems, 148 equations, 3 figures, 1 table)

This paper contains 25 sections, 12 theorems, 148 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Numerical scheme and main results
  3. Notations and definitions
  4. Numerical scheme
  5. Main results
  6. Existence of solution and entropy dissipation estimate
  7. Entropy dissipation and mass conservation
  8. Proof of Theorem \ref{['thm.SKTexi']}
  9. Estimates on the discrete Kolmogorov equation
  10. Well-posedness of the scheme and $L^\infty$ estimates
  11. Study of the dual problem
  12. Proof of Theorem \ref{['thm.prop']}
  13. Step 1: Maximum principle
  14. Step 2: Duality estimate
  15. Convergence of the scheme
  16. ...and 10 more sections

Key Result

Theorem 1

Let the assumptions (H1)--(H5) hold. Then, for every $1 \leq k \leq N_T$ there exists (at least) one nonnegative solution $(u^k_1,u^k_2)$ to scheme 5.ic--5.mu2. Moreover, this solution satisfies the following properties: Finally for all $j\in\{1,2\}$, if $d_j$ is positive then $u_j^k$ is positive for all $0<k\leq N_T$.

Figures (3)

  • Figure 1: Distance between solution of the nonlocal and local cross-diffusion system at final time versus $\delta/L$. Left: Initial data is indicator function \ref{['eq:CI1']}; Right: Initial data is the smooth function \ref{['eq:CI2']}
  • Figure 2: Turing patterns at final time for (left) linear diffusion for predators and preys ($d_1 = 0.05$, $d_2 = 2$ and $d_{21} = 0$) and (right) cross-diffusion for predators and linear diffusion for preys ($d_1 = 0.05$, $d_2 = 0$ and $d_{21} = 1$).
  • Figure 3: Turing patterns in prey density $u_1$ at final time: (top left) linear diffusion for predators and preys ($d_1 = 0.001$, $d_2 = 4$, $d_{21} = 0$); (top right) nonlocal cross-diffusion for predators with symmetric kernel and linear diffusion for preys ($d_1 = 0.001$, $d_2 = 0$, $d_{21} = 2/5$, $\rho_2 = \rho_2^\text{sym}$); (bottom) nonlocal cross-diffusion for predators with non-symmetric kernel and linear diffusion for preys ($d_1 = 0.001$, $d_2 = 0$, $d_{21} = 2/5$, $\rho_2 = \rho_2^\text{nonsym}$).

Theorems & Definitions (25)

  • Definition 1
  • Theorem 1: Existence of solutions
  • Theorem 2: Qualitative properties of the solutions
  • Theorem 3: Convergence of the scheme
  • Remark 4
  • Proposition 5
  • proof
  • Remark 6: Uniqueness under CFL
  • Lemma 7
  • proof
  • ...and 15 more