Analysis of an Asymptotic Preserving Scheme for Relaxation Systems

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 $({\text{ueps}}_h,{\text{veps}}_h)$ to a nonlinear relaxation system, where $\text{eps}>0$ is a physical parameter and $h$ represents the discretization parameter. Uniform convergence with respect to $\text{eps}$ 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.