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.

我们使用具有不连续系数矩阵的线性双曲方程解的\(L_{2}\)范数的行为作为替代来推断不连续伽辽金谱元方法 (DGSEM) 的稳定性。尽管\(L_{2}\)范数在此类系统的齐次和耗散边界条件的初始数据方面不受限制，但\(L_{2}\)范数比折扣的范数更容易使用不连续性导致的增长。我们表明，具有满足 Rankine-Hugoniot（或守恒）条件的逆风数值通量的 DGSEM 具有与偏微分方程在\(L_{2}\) 范数，加上一个附加的耗散，这取决于近似解在多大程度上无法满足兰金-雨果跳跃。