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Relative Entropy for the Numerical Diffusive Limit of the Linear Jin-Xin System

Marianne Bessemoulin-Chatard, Hélène Mathis

TL;DR

This work analyzes the diffusive relaxation limit of the linear Jin–Xin system using a relative entropy framework to quantify convergence to the convection–diffusion limit given by $\partial_t \bar u + a\partial_x \bar u = \lambda^2 \partial_{xx}\bar u$ with $\bar v = a\bar u - \lambda^2\partial_x \bar u$. It derives a continuous relative-entropy identity that yields a convergence rate of $O(\varepsilon^2)$ under regularity assumptions, and develops a semi-discrete finite-volume scheme that preserves the asymptotics and achieves a discrete relative-entropy bound of $O(\varepsilon^4)$ in $L^2$. The discrete analysis mirrors the continuous approach, incorporating numerical residuals and an HLL flux to obtain the same rate improvement at the discrete level, which is supported by numerical experiments on linear and nonlinear variants. The results confirm the efficacy of the relative-entropy method for both continuous and discrete diffusive limits and motivate extensions to more general nonlinear relaxation terms with potential applications to asymptotic-preserving computations. $\,$

Abstract

This paper deals with the diffusive limit of the Jin and Xin model and its approximation by an asymptotic preserving finite volume scheme. At the continuous level, we determine a convergence rate to the diffusive limit by means of a relative entropy method. Considering a semi-discrete approximation (discrete in space and continuous in time), we adapt the method to this semi-discrete framework and establish that the approximated solutions converge towards the discrete convection-diffusion limit with the same convergence rate.

Relative Entropy for the Numerical Diffusive Limit of the Linear Jin-Xin System

TL;DR

This work analyzes the diffusive relaxation limit of the linear Jin–Xin system using a relative entropy framework to quantify convergence to the convection–diffusion limit given by with . It derives a continuous relative-entropy identity that yields a convergence rate of under regularity assumptions, and develops a semi-discrete finite-volume scheme that preserves the asymptotics and achieves a discrete relative-entropy bound of in . The discrete analysis mirrors the continuous approach, incorporating numerical residuals and an HLL flux to obtain the same rate improvement at the discrete level, which is supported by numerical experiments on linear and nonlinear variants. The results confirm the efficacy of the relative-entropy method for both continuous and discrete diffusive limits and motivate extensions to more general nonlinear relaxation terms with potential applications to asymptotic-preserving computations.

Abstract

This paper deals with the diffusive limit of the Jin and Xin model and its approximation by an asymptotic preserving finite volume scheme. At the continuous level, we determine a convergence rate to the diffusive limit by means of a relative entropy method. Considering a semi-discrete approximation (discrete in space and continuous in time), we adapt the method to this semi-discrete framework and establish that the approximated solutions converge towards the discrete convection-diffusion limit with the same convergence rate.
Paper Structure (7 sections, 2 theorems, 53 equations, 2 figures)

This paper contains 7 sections, 2 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The continuous setting
  3. Semi-discrete finite volume scheme and numerical convergence rate
  4. Definition of the semi-discrete scheme
  5. Convergence rate
  6. Numerical experiments
  7. Conclusion and prospects

Key Result

Lemma 1

Let $(\bar{u}_i, \bar{v}_i)_{i \in \mathbb Z}$ be a smooth solution of eq:Pd and let $(\tau_i,u_i)_{i \in \mathbb Z}$ be a solution of eq:Hd. The relative entropy $E_i$, defined by (eq:ERD), verifies the following evolution law: where $F_{i+1/2}$ corresponds to an approximation of the relative entropy flux $F(w|\bar{w})$ at the interface $x_{i+1/2}$ given by and the quantities $R_i^j$, $j=1,\ldo

Figures (2)

  • Figure 1: Linear test case. Top: profile of $(u,v)$ in space, compared to $(\bar{u},\bar{v})$. Bottom: $L^2$ norm of the error $\|(u,v)-(\bar{u}, \bar{v})\|_{L^2(Q_T)}^2$ with respect to $\varepsilon$ in log scale.
  • Figure 2: Nonlinear test case. Top: profile of $(u,v)$ in space, compared to $(\bar{u},\bar{v})$. Bottom: $L^2$ norm of the error $\|(u,v)-(\bar{u}, \bar{v})\|_{L^2(Q_T)}^2$ with respect to $\varepsilon$ in log scale.

Theorems & Definitions (7)

  • proof
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of Theorem \ref{['thm:2']}