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当系数矩阵存在跳跃时波传播的间断伽辽金谱元格式的稳定性

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

作者信息

Kopriva David A, Gassner Gregor J, Nordström Jan

机构信息

Department of Mathematics, The Florida State University, Tallahassee, FL 32306 USA.

Computational Science Research Center, San Diego State University, San Diego, CA USA.

出版信息

J Sci Comput. 2021;88(1):3. doi: 10.1007/s10915-021-01516-w. Epub 2021 May 20.

DOI:10.1007/s10915-021-01516-w
PMID:34776602
原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC8550655/
Abstract

We use the behavior of the 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 norm is not bounded in terms of the initial data for homogeneous and dissipative boundary conditions for such systems, the 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 norm, plus an added dissipation that depends on how much the approximate solution fails to satisfy the Rankine-Hugoniot jump.

摘要

我们利用具有间断系数矩阵的线性双曲方程解的范数行为作为替代，来推断间断伽辽金谱元法（DGSEM）的稳定性。尽管对于此类系统的齐次和耗散边界条件，该范数在初始数据方面无界，但与一个消除因间断而导致增长的范数相比，该范数更易于处理。我们表明，具有满足兰金 - 于戈尼奥（或守恒）条件的迎风数值通量的DGSEM在该范数下具有与偏微分方程相同的能量界，再加上一个额外的耗散项，该项取决于近似解未能满足兰金 - 于戈尼奥跳跃条件的程度。

https://cdn.ncbi.nlm.nih.gov/pmc/blobs/1162/8550655/d85002ec6090/10915_2021_1516_Fig5_HTML.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/1162/8550655/7afd1d8a1e03/10915_2021_1516_Fig1_HTML.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/1162/8550655/d95560aabaad/10915_2021_1516_Fig2_HTML.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/1162/8550655/48013aa97c5e/10915_2021_1516_Fig3_HTML.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/1162/8550655/5699a545f13d/10915_2021_1516_Fig4_HTML.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/1162/8550655/d85002ec6090/10915_2021_1516_Fig5_HTML.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/1162/8550655/7afd1d8a1e03/10915_2021_1516_Fig1_HTML.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/1162/8550655/d95560aabaad/10915_2021_1516_Fig2_HTML.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/1162/8550655/48013aa97c5e/10915_2021_1516_Fig3_HTML.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/1162/8550655/5699a545f13d/10915_2021_1516_Fig4_HTML.jpg
https://cdn.ncbi.nlm.nih.gov/pmc/blobs/1162/8550655/d85002ec6090/10915_2021_1516_Fig5_HTML.jpg

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本文引用的文献

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Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides.微圆柱耦合谐振器光波导中基于回音壁模式的光传播数值研究
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