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

TL;DR

This paper investigates the diffusive limit of the Jin-Xin model using a relative entropy approach, establishing convergence rates for both continuous and semi-discrete approximations to the convection-diffusion limit.

Contribution

It introduces a relative entropy method to analyze the diffusive limit and adapts it to semi-discrete schemes, providing convergence rates for numerical approximations.

Findings

Convergence rate to the diffusive limit established at the continuous level.

Semi-discrete approximation converges to the discrete convection-diffusion limit.

Method applies to both continuous and semi-discrete frameworks.

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.