Numerical Convergence Rate for a Diffusive Limit of Hyperbolic Systems: p-System with Damping

TL;DR

This paper analyzes the convergence rate of solutions for a damped hyperbolic p-system towards the porous media equation, using an Asymptotic Preserving scheme and relative entropy methods, supported by numerical experiments.

Contribution

It provides the first rigorous convergence rate analysis for the diffusive limit of the p-system with damping using an AP scheme.

Findings

Convergence rate of classical solutions towards porous media solutions established.

AP scheme preserves the convergence rate in the numerical approximation.

Numerical experiments confirm theoretical results.

Abstract

This paper deals with diffusive limit of the p-system with damping and its approximation by an Asymptotic Preserving (AP) Finite Volume scheme. Provided the system is endowed with an entropy-entropy flux pair, we give the convergence rate of classical solutions of the p-system with damping towards the smooth solutions of the porous media equation using a relative entropy method. Adopting a semi-discrete scheme, we establish that the convergence rate is preserved by the approximated solutions. Several numerical experiments illustrate the relevance of this result.