Removing the Courant-Friedrichs-Lewy stability criterion of the explicit time-domain very high degree spectral-element method with eigenvalue perturbation

Abstract

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.