  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.

Numerical Analysis;Numerical Analysis