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Discrete $H$-theorem for a finite volume discretization of a nonlinear kinetic system: application to hypocoercivity

Marianne Bessemoulin-Chatard, Tino Laidin, Thomas Rey

TL;DR

The paper analyzes the long-time behavior of a fully implicit finite-volume discretization for a nonlinear two-species kinetic reaction system in 1D position-velocity space, extending nonlinear entropy-based hypocoercivity results to the discrete setting. It adapts the nonlinear Boltzmann entropy approach to the scheme, proving that the discrete solution converges exponentially fast to the discrete equilibrium with rates and constants that do not depend on the grid, while leveraging $L^ty$ bounds and the stabilizing effect of time-discrete dissipation. A discrete analogue of the continuous hypocoercivity framework is developed, including a discrete Poisson problem and a modified entropy functional $mma^n$, ensuring norm equivalence and providing quantitative decay. These results validate the reliability of the numerical scheme for capturing the correct long-time dynamics and steady-state behavior of the nonlinear kinetic model, with potential applications to hypocoercive discretizations of similar reaction-transport systems.

Abstract

In this article, we study the long-time behavior of a finite-volume discretization for a nonlinear kinetic reaction model involving two interacting species. Building upon the seminal work of [Favre, Pirner, Schmeiser, ARMA, 2023], we extend the discrete exponential convergence to equilibrium result established in [Bessemoulin-Chatard, Laidin, Rey, IMAJNA, 2025], which was obtained in a perturbative framework using weighted $L^2$ estimates. The analysis applies to a broader class of exponentially decaying initial data, without requiring proximity to equilibrium, by exploiting the properties of the Boltzmann entropy. The proof relies on the propagation of the initial $L^\infty$ bounds, derived from monotonicity properties of the scheme, allowing controlled linearizations within the nonlinear entropy estimates. Moreover, we show that the time-discrete dissipation inherent to the numerical scheme plays a crucial stabilizing role, providing control over the nonlinear terms.

Discrete $H$-theorem for a finite volume discretization of a nonlinear kinetic system: application to hypocoercivity

TL;DR

The paper analyzes the long-time behavior of a fully implicit finite-volume discretization for a nonlinear two-species kinetic reaction system in 1D position-velocity space, extending nonlinear entropy-based hypocoercivity results to the discrete setting. It adapts the nonlinear Boltzmann entropy approach to the scheme, proving that the discrete solution converges exponentially fast to the discrete equilibrium with rates and constants that do not depend on the grid, while leveraging bounds and the stabilizing effect of time-discrete dissipation. A discrete analogue of the continuous hypocoercivity framework is developed, including a discrete Poisson problem and a modified entropy functional , ensuring norm equivalence and providing quantitative decay. These results validate the reliability of the numerical scheme for capturing the correct long-time dynamics and steady-state behavior of the nonlinear kinetic model, with potential applications to hypocoercive discretizations of similar reaction-transport systems.

Abstract

In this article, we study the long-time behavior of a finite-volume discretization for a nonlinear kinetic reaction model involving two interacting species. Building upon the seminal work of [Favre, Pirner, Schmeiser, ARMA, 2023], we extend the discrete exponential convergence to equilibrium result established in [Bessemoulin-Chatard, Laidin, Rey, IMAJNA, 2025], which was obtained in a perturbative framework using weighted estimates. The analysis applies to a broader class of exponentially decaying initial data, without requiring proximity to equilibrium, by exploiting the properties of the Boltzmann entropy. The proof relies on the propagation of the initial bounds, derived from monotonicity properties of the scheme, allowing controlled linearizations within the nonlinear entropy estimates. Moreover, we show that the time-discrete dissipation inherent to the numerical scheme plays a crucial stabilizing role, providing control over the nonlinear terms.

Paper Structure

This paper contains 15 sections, 19 theorems, 188 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The continuous setting
  3. Notations and functional setting
  4. Moment dynamics and estimates
  5. Boltzmann entropy dissipation
  6. Nonlinear hypocoercivity
  7. The discrete setting
  8. Notations
  9. Mesh
  10. Discrete velocity profiles
  11. Discrete functional setting
  12. Definition of the numerical scheme
  13. Definition and estimates of the discrete moments
  14. Discrete nonlinear hypocoercivity
  15. Proof of the existence and maximum principle

Key Result

Proposition 1

Under assumptions hyp_chi on $\chi_k$, we further assume that the initial data $\mathbf{F}^{in}$ satisfies for all $x\in\mathbb{T}$, $v\in\mathbb{R}$, with constants $0<\rho_m\leq \rho_M<\infty$. Then there exists a unique global solution $\mathbf{F}$ of eq_1_nonlin--eq_2_nonlin such that for all $x\in\mathbb{T}$, $v\in\mathbb{R}$, $t\geq 0$,

Figures (1)

  • Figure 1: Comparison between the exponential decay of the relative Boltzmann entropy $H$ and of the weighted $L^2$ distance to equilibrium for the numerical scheme studied in BCLR2025.

Theorems & Definitions (31)

  • Proposition 1: Global existence and maximum principle
  • Lemma 1: Moments estimates
  • proof
  • Lemma 2: Microscopic coercivity
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • Lemma 4
  • proof
  • ...and 21 more