Convergent and conservative schemes for nonclassical solutions based on kinetic relations

TL;DR

This paper introduces a conservative finite difference scheme that accurately captures nonclassical shock waves in hyperbolic conservation laws by incorporating kinetic relations through a reconstruction technique, validated by stability and convergence tests.

Contribution

The paper presents a novel finite difference scheme that preserves nonclassical shocks sharply and ensures consistency with kinetic relations, advancing numerical methods for hyperbolic conservation laws.

Findings

Scheme accurately captures nonclassical shocks

Method is stable and consistent with kinetic relations

Numerical experiments confirm convergence to physical solutions

Abstract

We propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main challenge is to achieve, at the discretization level, a consistency property with respect to a prescribed kinetic relation. The latter is required for the selection of physically meaningful nonclassical shocks. Our method is based on a reconstruction technique performed in each computational cell that may contain a nonclassical shock. To validate this approach, we establish several consistency and stability properties, and we perform careful numerical experiments. The convergence of the algorithm toward the physically meaningful solutions selected by a kinetic relation is demonstrated numerically for several test cases, including concave-convex as well as convex-concave flux-functions.