A Finite-Volume Scheme for a Cross-Diffusion Model Arising from Interacting Many-Particle Population Systems

Jüngel, Ansgar; Zurek, Antoine

doi:10.1007/978-3-030-43651-3_19

Cited by 5 publications

(6 citation statements)

References 12 publications

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“…Because of the convergences ( 27), (30), and (31), we can apply Lemma 9 in Appendix A to infer that z (30) shows that the convergence holds in L q (Q T ). This convergence as well as (28) and ( 29) allow us to perform the limit ε → 0 in the weak formulation of (1), proving that u solves (1) and (3).…”

Section: Localization Limitmentioning

confidence: 59%

“…Because of the nonlocality, we cannot conclude L 2 (T d ) estimates for u i and ∇u i like in the local case; see [28] and Appendix B. We deduce from (4) only bounds for…”

Section: Introductionmentioning

confidence: 91%

“…Since B has compact support in R, we can apply the proof of [8,Theorem 4.22] to infer that the first term on the right-hand side, formulated as the convolution K ε ij * φ − a ij φ (slightly abusing the notation), converges to zero strongly in L q ′ (R d ) as ε → 0. Thus, taking into account the weak convergence (28), convergence (31) follows.…”

Section: Localization Limitmentioning

confidence: 99%

“…The existence of global weak solutions to the local system (1) and (3) in any bounded polygonal domain was shown in [28] by analyzing a finite-volume scheme. For completeness, we state the assumptions and the theorem and indicate how the result can be proved using the techniques of Section 3.…”

Section: Appendix B Local Cross-diffusion Systemmentioning

confidence: 99%

See 3 more Smart Citations

Nonlocal cross-diffusion systems for multi-species populations and networks

Jüngel¹,

Portisch²,

Zurek³

2021

Preprint

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Nonlocal cross-diffusion systems on the torus, arising in population dynamics and neural networks, are analyzed. The global existence of weak solutions, the weak-strong uniqueness, and the localization limit are proved. The kernels are assumed to be positive definite and in detailed balance. The proofs are based on entropy estimates coming from Shannon-type and Rao-type entropies, while the weak-strong uniqueness result follows from the relative entropy method. The existence and uniqueness theorems hold for nondifferentiable kernels. The associated local cross-diffusion system is also discussed.

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Section: Localization Limitmentioning

confidence: 59%

“…Because of the nonlocality, we cannot conclude L 2 (T d ) estimates for u i and ∇u i like in the local case; see [28] and Appendix B. We deduce from (4) only bounds for…”

Section: Introductionmentioning

confidence: 91%

Section: Localization Limitmentioning

confidence: 99%

Section: Appendix B Local Cross-diffusion Systemmentioning

confidence: 99%

See 2 more Smart Citations

Nonlocal cross-diffusion systems for multi-species populations and networks

Jüngel¹,

Portisch²,

Zurek³

2021

Preprint

Self Cite

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show abstract

“…Thus, Hypotheses (H4)-(H5) are fulfilled. A finite-volume scheme for this system has been already analyzed in [28]. However, the design and the analysis of the scheme are based on a weighted quadratic entropy, i.e.…”

Section: 2mentioning

confidence: 99%

A convergent structure-preserving finite-volume scheme for the Shigesada-Kawasaki-Teramoto population system

Zurek¹,

Jüngel²

2020

Preprint

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An implicit Euler finite-volume scheme for an n-species population crossdiffusion system of Shigesada-Kawasaki-Teramoto-type in a bounded domain with no-flux boundary conditions is proposed and analyzed. The scheme preserves the formal gradientflow or entropy structure and preserves the nonnegativity of the population densities. The key idea is to consider a suitable mean of the mobilities in such a way that a discrete chain rule is fulfilled and a discrete analog of the entropy inequality holds. The existence of finite-volume solutions, the convergence of the scheme, and the large-time asymptotics to the constant steady state are proven. Furthermore, numerical experiments in one and two space dimensiona for two and three species are presented. The results are valid for a more general class of cross-diffusion systems satisfying some structural conditions.

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Convergence of a finite-volume scheme for a degenerate-singular cross-diffusion system for biofilms

Daus

Jüngel

Zurek

2020

IMA Journal of Numerical Analysis

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An implicit Euler finite-volume scheme for a cross-diffusion system modeling biofilm growth is analyzed by exploiting its formal gradient-flow structure. The numerical scheme is based on a two-point flux approximation that preserves the entropy structure of the continuous model. Assuming equal diffusivities the existence of non-negative and bounded solutions to the scheme and its convergence are proved. Finally, we supplement the study by numerical experiments in one and two space dimensions.

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A Finite-Volume Scheme for a Cross-Diffusion Model Arising from Interacting Many-Particle Population Systems

Cited by 5 publications

References 12 publications

Nonlocal cross-diffusion systems for multi-species populations and networks

Nonlocal cross-diffusion systems for multi-species populations and networks

A convergent structure-preserving finite-volume scheme for the Shigesada-Kawasaki-Teramoto population system

Convergence of a finite-volume scheme for a degenerate-singular cross-diffusion system for biofilms

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