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 hasgained tremendous credibility within the seismological community at all scales. Althoughthe recent introduction of non-periodic homogenization has addressed the spatial meshing difficulty of the mechanical discontinuities, the Courant-Friedrichs-Lewy (CFL) stability criterionstrictly constrains the maximum time step, which still puts a great burden on the numericalsimulation. In the explicit time-domain SEM, the source of instability of using a time stepbeyond the stability criterion is that some unstable eigenvalues of the updated matrix are largerthan what can be accurately simulated. We succeed in removing the CFL stability condition inthe 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 both in the homogeneous, moderate and strong heterogeneous models, show that this combination can precisely simulate waveforms with timesteps dozens of the CFL limit even towards the Nyquist limit especially for the efficient veryhigh degree SEM, which abundantly saves the iteration times without suffering from the time-dispersion error. It demonstrates a potential application prospect in some situations such as thefull waveform inversion which requires multiple numerical simulations for the same model.

Novel Hybrid Numerical Simulation of the Wave Equation by Combining Physical and Numerical Representation Theorems and a Review of Hybrid Methodologies

“…If different global and local time steps are adopted, different temporal dispersion errors will be introduced into the local simulation, resulting in inaccurate waveforms to an extent. Note that the spatial and temporal dispersion errors have been proven to be irrelevant (Koene et al., 2018; Lyu, Capdeville, Lu, & Zhao, 2021), which provides a theoretical basis for eliminating the temporal dispersion errors in the global and hybrid simulations.…”

Introduction to the Distributional Finite Difference Method for 3D Seismic Wave Propagation and Comparison With the Spectral Element Method

“…To ensure a fair benchmark, we compare the DFDM simulations with those from the SEM using the same centered second‐order time integration scheme with a displacement formulation to remove any errors caused by differences in time formats. As we know, the spatial and temporal dispersion errors have been proven to not be relevant (Gao et al., 2018; Koene et al., 2018; Lyu, Capdeville, Lu, & Zhao, 2021) under the framework of the FDM, pseudospectral method, and SEM. The spatial and temporal dispersion errors of DFDM also follow this rule because of the successful analytical removal of the time‐dispersion error (Masson & Virieux, 2023).…”