Stability of Discontinuous Galerkin Spectral Element Schemes for Wave Propagation when the Coefficient Matrices have Jumps

We use the behavior of the L2 norm of the solutions of linear hyperbolic equations withdiscontinuous coefficient matrices as a surrogate to infer stability of discontinuous Galerkinspectral element methods (DGSEM). Although the L2 norm is not bounded in terms of theinitial data for homogeneous and dissipative boundary conditions for such systems, the L2norm is easier to work with than a norm that discounts growth due to the discontinuities. Weshow that the DGSEM with an upwind numerical flux that satisfies the Rankine–Hugoniot(or conservation) condition has the same energy bound as the partial differential equationdoes in the L2 norm, plus an added dissipation that depends on how much the approximatesolution fails to satisfy the Rankine–Hugoniot jump.Funding: Linkoping University; Simons Foundation [426393]; Klaus-Tschira Stiftung; European Research Council through the ERC Starting Grant "An Exascale aware and Un-crashable Space-Time-Adaptive Discontinuous Spectral Element Solver for NonLinear Conservation Laws" (EXTREME) [71448]; Vetenskapsradet, SwedenSwedish Research Council [2018-05084 VR]; Swedish e-Science Research Center (SeRC)