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Local Time Stepping Applied to Implicit-Explicit Methods for Hyperbolic Systems

In the context of systems of nonlinear conservation laws it can be crucial to use adaptive grids in order to correctly simulate the singularities of the solution over long time ranges while keeping the computing time within acceptable bounds. The adaptive space grid must vary in time according to the local smoothness of the solution. More sophisticated and recent methods also adapt the time-step locally to the space discretization according to the stability condition. We present here such a method designed for an explicit-implicit Lagrange-projection scheme, addressing physical problems where slow kinematic waves coexist with fast acoustic ones. Numerical simulations are presented to validate the algorithms in terms of robustness and efficiency.