Interfaces and Free Boundaries


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Volume 18, Issue 2, 2016, pp. 137–159
DOI: 10.4171/IFB/360

Published online: 2016-09-13

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

Nina Aguillon[1]

(1) Université Pierre et Marie Curie (Paris VI), France

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 presented 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 nonclassical 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.

Keywords: Nonclassical shocks, undercompressive shocks, kinetic relation, finite volume schemes

Aguillon Nina: Capturing nonclassical shocks in nonlinear elastodynamic with a conservative finite volume scheme. Interfaces Free Bound. 18 (2016), 137-159. doi: 10.4171/IFB/360