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arXiv 2011.11746math.NAcs.NA
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系数矩阵有跳时波动问题间断Galerkin谱元格式的稳定性

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

David A. Kopriva, Gregor J. Gassner, Jan Nordström

摘要:

利用具有间断系数矩阵的线性双曲型方程解的$L_{2}$范数的性质作为替代,来推断间断Galerkin谱元方法(DGSEM)的稳定性.虽然$L_{2}$范数不受此类系统的齐次和耗散边界条件的初始数据的约束,但$L_{2}$范数比由于不连续性而折扣增长的范数更容易使用。我们发现,DGSEM与迎风数值通量,满足朗肯-Hugoniot(或守恒)条件具有相同的能量约束的偏微分方程在$L_{2}$规范,加上额外的耗散,这取决于有多少的近似解未能满足朗肯-Hugoniot跳跃。

英文摘要:

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.

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