Analysis of an Asymptotic Preserving Scheme for Relaxation Systems

TL;DR

This paper analyzes the convergence and accuracy of an asymptotic preserving numerical scheme for nonlinear relaxation systems, demonstrating uniform convergence and providing error estimates through theoretical analysis and numerical tests.

Contribution

It provides a rigorous convergence analysis and error estimates for a class of asymptotic preserving schemes applied to nonlinear relaxation systems.

Findings

Uniform convergence with respect to epsilon and discretization parameter h.

Error estimates for the approximation scheme.

Numerical tests confirming accuracy and efficiency.

Abstract

We study the convergence of a class of asymptotic preserving numerical schemes initially proposed by F. Filbet & S. Jin \cite{filb1} and G. Dimarco & L. Pareschi \cite{DimarcoP} in the context of nonlinear and stiff kinetic equations. Here, our analysis is devoted to the approximation of a system of transport equations with a nonlinear source term, for which the asymptotic limit is given by a conservation laws. We investigate the convergence of the approximate solution $(\ueps_h,\veps_h)$ to a nonlinear relaxation system, where $\eps>0$ is a physical parameter and $h$ represents the discretization parameter. Uniform convergence with respect to $\eps$ and $h$ is proven and error estimates are also obtained. Finally, several numerical tests are performed to illustrate the accuracy and efficiency of such a scheme.