Low-Dissipation Central-Upwind Schemes for Elasticity in Heterogeneous Media (Communications in Computational Physics, 2025)
We develop new low-dissipation central-upwind (LDCU) schemes for nonlinear elasticity equations in heterogeneous media. In general, central-upwind schemes belong to the class of finite-volume Godunov-type schemes, which consist of threesteps: reconstruction, evolution, and projection onto the original grid. In our newmethod, the evolution is performed in the standard way by integrating the systemover the space-time control volumes. However, the reconstruction and projection areperformed in a special manner. First, we take 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 there: we therefore reconstruct the flux variables. Second, we use a special projection recently introduced in[A. Kurganov and R. Xin, J. Sci. Comput., 96, 2023] to complete the derivation of theLDCU scheme. Our numerical experiments demonstrate that the developed schemesare capable of accurately resolving waves with dispersive behavior that over a longperiod of time evolve into solitary waves.
Keywords: Hyperbolic systems of conservation laws; low-dissipation central-upwind schemes; nonlinear elasticity; heterogeneous media; waves with dispersive behavior; solitary waves.