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

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