Capturing nonclassical shocks in nonlinear elastodynamic with a conservative finite volume scheme
For researchers in nonlinear wave propagation and elastodynamics, this provides a numerical method to simulate undercompressive shocks that satisfy a kinetic relation, which are not captured by standard schemes.
The authors developed a conservative finite volume scheme that accurately captures nonclassical (undercompressive) shocks in nonlinear elastodynamics without artificial diffusion, enabling correct approximation of nonclassical solutions and large-time asymptotics.
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.