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Discrete hypocoercivity for a nonlinear kinetic reaction model

Marianne Bessemoulin-Chatard, Tino Laidin, Thomas Rey

TL;DR

The paper develops a fully discrete finite-volume method for a one-dimensional nonlinear kinetic generation–recombination model and proves exponential convergence to equilibrium in a discrete $L^2$ framework. By adapting the $L^2$ hypocoercivity approach of Dolbeault–Mouhot–Schmeiser to the linearized discrete problem and extending it to the nonlinear scheme via a discrete maximum principle and monotone transport fluxes, it obtains a robust local decay result for small perturbations. A discrete modified entropy $H_\delta^\Delta$ is introduced to establish exponential decay with rate $\kappa>0$, accompanied by micromacro coercivity and discrete moment estimates. Numerical experiments validate the theoretical findings for linear and nonlinear schemes across different equilibrium profiles and flux choices, and demonstrate that monotone fluxes are essential to maintain decay in the nonlinear regime and to preserve the mass-difference structure of the model.

Abstract

In this article, we propose a finite volume discretization of a one dimensional nonlinear reaction kinetic model proposed in [Neumann, Schmeiser, Kint. Rel. Mod. 2016], which describes a 2-species recombination-generation process. Specifically, we establish the long-time convergence of approximate solutions towards equilibrium, at exponential rate. The study is based on an adaptation for a discretization of the linearized problem of the $L^2$ hypocoercivity method introduced in [Dolbeault, Mouhot, Schmeiser, 2015]. From this, we can deduce a local result for the discrete nonlinear problem. As in the continuous framework, this result requires the establishment of a maximum principle, which necessitates the use of monotone numerical fluxes.

Discrete hypocoercivity for a nonlinear kinetic reaction model

TL;DR

The paper develops a fully discrete finite-volume method for a one-dimensional nonlinear kinetic generation–recombination model and proves exponential convergence to equilibrium in a discrete framework. By adapting the hypocoercivity approach of Dolbeault–Mouhot–Schmeiser to the linearized discrete problem and extending it to the nonlinear scheme via a discrete maximum principle and monotone transport fluxes, it obtains a robust local decay result for small perturbations. A discrete modified entropy is introduced to establish exponential decay with rate , accompanied by micromacro coercivity and discrete moment estimates. Numerical experiments validate the theoretical findings for linear and nonlinear schemes across different equilibrium profiles and flux choices, and demonstrate that monotone fluxes are essential to maintain decay in the nonlinear regime and to preserve the mass-difference structure of the model.

Abstract

In this article, we propose a finite volume discretization of a one dimensional nonlinear reaction kinetic model proposed in [Neumann, Schmeiser, Kint. Rel. Mod. 2016], which describes a 2-species recombination-generation process. Specifically, we establish the long-time convergence of approximate solutions towards equilibrium, at exponential rate. The study is based on an adaptation for a discretization of the linearized problem of the hypocoercivity method introduced in [Dolbeault, Mouhot, Schmeiser, 2015]. From this, we can deduce a local result for the discrete nonlinear problem. As in the continuous framework, this result requires the establishment of a maximum principle, which necessitates the use of monotone numerical fluxes.
Paper Structure (20 sections, 22 theorems, 193 equations, 8 figures)

This paper contains 20 sections, 22 theorems, 193 equations, 8 figures.

Table of Contents

  1. Introduction
  2. On a kinetic generation-recombination model
  3. Hypocoercivity
  4. The continuous setting
  5. The linearized case
  6. Extension to the nonlinear setting
  7. The discrete setting
  8. Notations
  9. Mesh
  10. Discrete velocity profiles
  11. Discrete gradients and functional setting in space
  12. Definition of the numerical scheme for the nonlinear system
  13. Definition of the numerical scheme for the linearized problem
  14. Numerical hypocoercivity for the linearized problem
  15. The nonlinear problem
  16. ...and 5 more sections

Key Result

Lemma 1

Let hyp_chi hold and let $F=(f,g)$ be the solution to the linearized system eq_f_lin--eq_g_lin with initial data $F_I=(f_I,g_I)\in\mathcal{H}$ satisfying $\int_{\mathbb{T}\times\mathbb{R}}(f_I-g_I)\,\,\mathrm{d} x\,\,\mathrm{d} v=0$. Then for every $t\geq 0$,

Figures (8)

  • Figure 1: Test 1. Large time behavior of the linearized scheme, same equilibria $\chi_\mathcal{P}$. Snapshots of the densities of the two species at time $t=0$, $0.8$, $1.2$, $1.6$, $2.5$, $50$, using the Lax-Friedrichs fluxes \ref{['flux_F']}-\ref{['flux_G']}.
  • Figure 2: Test 1. Large time behavior of the linearized scheme, same equilibria $\chi_\mathcal{P}$. Time evolution of the weighted $L^2$ norm of the solution $F$ to the linearized problem \ref{['scheme_f_lin']}-\ref{['scheme_g_lin']} (left) and $L^2$ norm of the densities, for three different fluxes function (right).
  • Figure 3: Test 2. Large time behavior of the linearized scheme, different equilibria. Snapshots of the distribution function of each species at time $t=0$, $0.8$, $1.2$, $1.6$, $2.5$, $50$, using the Lax-Friedrichs fluxes \ref{['flux_F']}-\ref{['flux_G']}.
  • Figure 4: Test 2. Large time behavior of the linearized scheme, different equilibria. Time evolution of the weighted $L^2$ norm of the solution $F$ to the linearized problem \ref{['scheme_f_lin']}-\ref{['scheme_g_lin']} (left) and $L^2$ norm of the densities, for three different fluxes.
  • Figure 5: Test 3. Large time behavior of the nonlinear scheme. Snapshots of the distribution function of each species at time $t=0$, $0.83$, $2.25$, $3.35$, $9.67$, $100$, using the Lax-Friedrichs fluxes \ref{['flux_F']}-\ref{['flux_G']}.
  • ...and 3 more figures

Theorems & Definitions (42)

  • Lemma 1: Microscopic coercivity
  • proof
  • Lemma 2: Moments estimates
  • proof
  • Proposition 1
  • Lemma 3
  • proof
  • Lemma 4: Equivalent norm
  • proof
  • proof
  • ...and 32 more