Abstract:<\/p>\n
We use the behavior of the L<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span> norm of the solutions of linear hyperbolic equations with discontinuous coefficient matrices as a surrogate to infer stability of discontinuous Galerkin spectral element methods (DGSEM). Although the L<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span> norm is not bounded by the initial data for homogeneous and dissipative boundary conditions for such systems, the L<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span> norm is easier to work with than a norm that discounts growth due to the discontinuities. We show 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 equation does in the L<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span> norm, plus an added dissipation that depends on how much the approximate solution fails to satisfy the Rankine-Hugoniot jump.<\/p>\n Accepted in Journal of Scientific Computing.<\/p>\n arXiv link:\u00a0https:\/\/arxiv.org\/abs\/2011.11746<\/a><\/p>\n","protected":false},"excerpt":{"rendered":" Abstract: We use the behavior of the L2 norm of the solutions of linear hyperbolic equations with discontinuous coefficient matrices as a surrogate to infer stability of discontinuous Galerkin spectral element methods (DGSEM). Although the L2 norm is not bounded … Continue reading