Capturing nonclassical shocks in nonlinear elastodynamic with a conservative finite volume scheme

TL;DR

This paper introduces a conservative finite volume scheme capable of accurately capturing nonclassical shocks in nonlinear elastodynamics, preserving their structure without diffusion and validated through multiple test cases.

Contribution

A novel finite volume scheme that exactly captures nonclassical shocks in elastodynamics without introducing artificial diffusion, based on a kinetic relation and entropy considerations.

Findings

Scheme accurately captures nonclassical shocks

No artificial diffusion near shocks

Enables large time asymptotic analysis

Abstract

For a model of nonlinear elastodynamics, we construct a finite volume scheme which is able to capture nonclassical shocks (also called undercompressive shocks). Those shocks verify an entropy inequality but are not admissible in the sense of Liu. They verify a kinetic relation which describes the jump, and keeps an information on the equilibrium between a vanishing dispersion and a vanishing diffusion. The scheme pre-sented here is by construction exact when the initial data is an isolated nonclassical shock. In general, it does not introduce any diffusion near shocks, and hence nonclas-sical solutions are correctly approximated. The method is fully conservative and does not use any shock-tracking mesh. This approach is tested and validated on several test cases. In particular, as the nonclassical shocks are not diffused at all, it is possible to obtain large time asymptotics.