Abstract
We use the behavior of the L_{2} 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_{2} norm is not bounded in terms of the initial data for homogeneous and dissipative boundary conditions for such systems, the L_{2} 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_{2} norm, plus an added dissipation that depends on how much the approximate solution fails to satisfy the Rankine–Hugoniot jump.
Highlights
In wave propagation problems, it is natural to find interfaces where material properties like the wave propagation speeds or density abruptly change
We show that the discontinuous Galerkin spectral element methods (DGSEM) with an upwind numerical flux that satisfies the Rankine–Hugoniot condition behaves as the partial differential equation (PDE) does in the L2 norm, plus an added dissipation that depends on how much the approximate solution fails to satisfy the Rankine–Hugoniot jump
As noted in [13], systems of the form (1) with discontinuous coefficient matrices do not necessarily have energy bounded by the initial data when measured in the L2 norm, and we present an example here to motivate the situation
Summary
It is natural to find interfaces where material properties like the wave propagation speeds or density abruptly change. Examples include interfaces between two dielectrics in electromagnetic wave propagation problems, or different rock layers in geophysics. At such interfaces the solutions can make discontinuous jumps, causing difficulties for many numerical methods. We propose a procedure where we use the L2 norm as a surrogate to infer stability of discontinuous Galerkin spectral element methods (DGSEM) for the approximation of hyperbolic equations with discontinuous coefficient matrices. We show that the DGSEM with an upwind numerical flux that satisfies the Rankine–Hugoniot (or conservation) condition behaves as the partial differential equation (PDE) does in the L2 norm, plus an added dissipation that depends on how much the approximate solution fails to satisfy the Rankine–Hugoniot jump
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