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.

翻译：我们使用用不连续系数矩阵解决线性双曲方程式的美元标准作为替代,推断不连续的Galerkin光谱元件方法(DGSEM)的稳定性。 虽然美元标准不受这些系统单一和分散边界条件初始数据的约束,但美元标准比由于不连续而使增长减低的规范更容易发挥作用。 我们显示,具有满足Rancine-Hugoniot(或保护)条件的上风数字通量的DGSEM具有与部分差异方程式在$L+2}标准中相同的能量约束,加上一个取决于近似解决方案在多大程度上不能满足Rancine-Hugoniot跳跃的增量。