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

TL;DR

This paper investigates the stability of discontinuous Galerkin spectral element methods for wave equations with discontinuous coefficients by analyzing the $L_{2}$ norm behavior, showing stability under certain flux conditions.

Contribution

It demonstrates that DGSEM with an upwind flux satisfying the Rankine-Hugoniot condition maintains the PDE's energy bounds in the $L_{2}$ norm despite coefficient jumps.

Findings

DGSEM with upwind flux satisfies the PDE's energy bounds in $L_{2}$ norm.

The $L_{2}$ norm growth is controlled by the Rankine-Hugoniot jump condition.

Stability is achieved despite discontinuous coefficient matrices.

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 by 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.