The explicit time-domain spectral-element method (SEM) for synthesizing seismograms has gained tremendous credibility within the seismological community at all scales. Although the recent introduction of nonperiodic homogenization has addressed the spatial meshing difficulty of the mechanical discontinuities, the Courant-Friedrichs-Lewy (CFL) stability criterion strictly constrains the maximum time step, which still puts a great burden on the numerical simulation. In the explicit time-domain SEM, the source of instability of using a time step beyond the stability criterion is that some unstable eigenvalues of the updated matrix are larger than what can be accurately simulated. We have succeeded in removing the CFL stability condition in explicit time-domain SEM by combining the forward time dispersion-transform method, the eigenvalue perturbation, and the inverse time dispersion-transform method. Our theoretical analyses and numerical experiments in the homogeneous, moderate, and strong heterogeneous models show that this combination can accurately simulate waveforms with time steps several times the size of the CFL limit even toward the Nyquist limit especially for the efficient very high degree SEM, which abundantly saves iteration times without suffering from time-dispersion error. It demonstrates a potential application prospect in some situations such as full-waveform inversion that require multiple numerical simulations for the same model.

中文翻译：

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去除具有特征值摄动的显式时域甚高阶谱元法的 Courant-Friedrichs-Lewy 稳定性判据

用于合成地震图的显式时域谱元法 (SEM) 在所有尺度的地震学界都获得了巨大的可信度。尽管最近引入的非周期均匀化解决了机械不连续性的空间网格划分困难，但 Courant-Friedrichs-Lewy (CFL) 稳定性准则严格限制了最大时间步长，这仍然给数值模拟带来了很大的负担。在显式时域 SEM 中，使用超出稳定性准则的时间步长的不稳定来源是更新矩阵的一些不稳定特征值大于可以准确模拟的值。通过结合前向时间色散变换方法、特征值扰动，我们成功地消除了显式时域 SEM 中的 CFL 稳定性条件，和反时差色散变换方法。我们在同质、中等和强异质模型中的理论分析和数值实验表明，这种组合可以准确地模拟时间步长数倍于 CFL 极限甚至接近奈奎斯特极限的波形，特别是对于高效的非常高的 SEM，大量节省迭代时间而不会遭受时间分散错误。它在某些情况下具有潜在的应用前景，例如需要对同一模型进行多次数值模拟的全波形反演。强大的异构模型表明，这种组合可以准确地模拟波形，其时间步长是 CFL 限制大小的几倍，甚至接近奈奎斯特限制，特别是对于高效的非常高的 SEM，这大大节省了迭代时间而不会出现时间色散误差。它在某些情况下具有潜在的应用前景，例如需要对同一模型进行多次数值模拟的全波形反演。强大的异构模型表明，这种组合可以准确地模拟波形，其时间步长是 CFL 限制大小的几倍，甚至接近奈奎斯特限制，特别是对于高效的非常高的 SEM，这大大节省了迭代时间而不会出现时间色散误差。它在某些情况下具有潜在的应用前景，例如需要对同一模型进行多次数值模拟的全波形反演。