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Convergence of a finite volume scheme and dissipative measure-valued-strong stability for a hyperbolic-parabolic cross-diffusion system

Katharina Hopf, Ansgar Jüngel

TL;DR

The paper develops a framework for global dissipative measure-valued solutions to a class of hyperbolic–parabolic cross-diffusion systems with possible rank-deficient diffusion. It introduces a dissipative measure-valued solution concept using Young measures and two Lyapunov entropies $H_S$ and $H_R$, and proves weak–strong uniqueness. A fully discrete implicit Euler finite-volume scheme is shown to generate dissipative measure-valued solutions in the limit, with entropy dissipation and an artificial diffusion vanishing as the mesh refines. When a strong solution exists, convergence is strong; long-time behavior leads to convergence to a steady state characterized by $ abla(B ilde u^*)=0$. The approach combines compactness with entropy methods and yields results on existence, stability, and asymptotics.

Abstract

This article is concerned with the development of a theoretical framework of global measure-valued solutions for a class of hyperbolic-parabolic cross-diffusion systems, and its application to the convergence analysis of a fully discrete finite-volume scheme. After introducing an appropriate notion of dissipative measure-valued solutions to the PDE system, a numerical scheme is proposed which is shown to generate, in the continuum limit, a dissipative measure-valued solution. The parabolic density part of the limiting measure-valued solution is atomic and converges to its constant state for long times. Furthermore, it is proved that whenever the PDE system possesses a strong solution, the convergence of the approximation scheme holds in the strong sense. The results are based on Young measure theory and a weak-strong stability estimate combining Shannon and Rao entropies. The convergence of the numerical scheme is achieved by means of discrete entropy dissipation inequalities and an artificial diffusion, which vanishes in the continuum limit.

Convergence of a finite volume scheme and dissipative measure-valued-strong stability for a hyperbolic-parabolic cross-diffusion system

TL;DR

The paper develops a framework for global dissipative measure-valued solutions to a class of hyperbolic–parabolic cross-diffusion systems with possible rank-deficient diffusion. It introduces a dissipative measure-valued solution concept using Young measures and two Lyapunov entropies and , and proves weak–strong uniqueness. A fully discrete implicit Euler finite-volume scheme is shown to generate dissipative measure-valued solutions in the limit, with entropy dissipation and an artificial diffusion vanishing as the mesh refines. When a strong solution exists, convergence is strong; long-time behavior leads to convergence to a steady state characterized by . The approach combines compactness with entropy methods and yields results on existence, stability, and asymptotics.

Abstract

This article is concerned with the development of a theoretical framework of global measure-valued solutions for a class of hyperbolic-parabolic cross-diffusion systems, and its application to the convergence analysis of a fully discrete finite-volume scheme. After introducing an appropriate notion of dissipative measure-valued solutions to the PDE system, a numerical scheme is proposed which is shown to generate, in the continuum limit, a dissipative measure-valued solution. The parabolic density part of the limiting measure-valued solution is atomic and converges to its constant state for long times. Furthermore, it is proved that whenever the PDE system possesses a strong solution, the convergence of the approximation scheme holds in the strong sense. The results are based on Young measure theory and a weak-strong stability estimate combining Shannon and Rao entropies. The convergence of the numerical scheme is achieved by means of discrete entropy dissipation inequalities and an artificial diffusion, which vanishes in the continuum limit.
Paper Structure (26 sections, 13 theorems, 150 equations, 1 figure)

This paper contains 26 sections, 13 theorems, 150 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Equations
  3. State of the art
  4. Key tools, definitions, and overview
  5. Numerical scheme and main results
  6. Spatial domain and mesh
  7. Function spaces
  8. Discrete gradient
  9. Numerical scheme
  10. Main results
  11. Discrete problem
  12. Definition and continuity of the fixed-point operator
  13. Existence of a fixed point
  14. Limit
  15. Discrete Rao entropy inequality
  16. ...and 11 more sections

Key Result

Theorem 4

Let Hypotheses (H1)--(H3) hold, $k\in{\mathbb N}$, $\eta\ge 0$, and let $u^{k-1}\in V_{\mathcal{T}}^n$ be given. Then there exists a solution $u^k=(u_1^k,\ldots,u_n^k) \in V_{\mathcal{T}}^n$ to scheme 2.init--2.flux satisfying $u_{i,K}^k > 0$ for $i=1,\ldots,n$, $K\in{\mathcal{T}}$. Inductively, let Moreover, $H_R(u^k) \le H_R(u^{k-1})$.

Figures (1)

  • Figure 1: Triangulation of a curved domain.

Theorems & Definitions (26)

  • Definition 1: Dissipative measure-valued solution
  • Remark 1: Consistency of the definition
  • Remark 2: Discrete gradient-flow property for upwind scheme
  • Remark 3: Discrete gradient flow property for logarithmic mean
  • Theorem 4
  • Theorem 5: Convergence of the scheme
  • Remark 6: Full-rank approximation
  • Theorem 7: Weak--strong uniqueness
  • Corollary 8
  • Theorem 9: Long-time behavior
  • ...and 16 more