Semi-Discrete Central-Upwind Schemes for Elasticity in Heterogeneous Media

We develop new central-upwind schemes for nonlinear elasticity equations in a heterogeneous medium. Finite volume central-upwind schemes consist of three steps: reconstruction, evolution, and projection onto the original grid. In our new method, the evolution is performed in the standard way by integrating the system over the space-time control volumes. However, the reconstruction and projection are performed in a special manner by taking into account the fact that the conservative variables (strain and momentum) are discontinuous across the material interfaces, while the flux variables (velocity and strain) are continuous across these material interfaces. The new reconstruction and projection procedures lead to the central-upwind scheme with extremely small numerical diffusion so that in long time calculations, the new scheme outperforms existing upwind alternatives. In addition, the proposed scheme can be made positivity preserving. To achieve this goal, the system is rewritten in terms of auxiliary variables and the local propagation speeds of the system are adjusted accordingly. Our numerical experiments demonstrate that the developed scheme is capable of accurately resolving waves with dispersive behavior that over a long period of time evolve into solitary waves while remaining nonnegative.