Abstract
In this paper, we present a stable and accurate high-order methodology for the symmetric matrix form (SMF) of the elastic wave equation. We use an accurate high-order upwind finite difference method to define spatial discretization. Then, an efficient complex frequency-shifted (CFS) unsplit multi-axis perfectly matched layer (MPML) is implemented using the auxiliary differential equation (ADE) that is used to build higher-order time schemes for elastodynamics in the unbounded curve domain. It is derived to be compatible with SMF. The SMF framework has a general form of a hyperbolic partial differential equation (PDE) that can be expanded to different dimensions (2D, 3D) or different wave modal (SH, P-SV) without requiring significant modifications owing to a simplified process of derivation and programming. Subsequently, an energy analysis on the framework combined with initial boundary value problems is conducted, and the stability analysis can be extended to a semi-discrete approximation similarly. Thus, we propose a semi-discrete approximation based on ADE CFS-MPML in which the curve domain is discretized using the upwind summation-by-parts (SBP) operators, and where the boundary conditions are enforced weakly using the simultaneous approximation terms (SAT). The proposed method’s robustness and adequacy are illustrated by conducting several numerical simulations.
Highlights
In seismology, as well as in many other applications, such as acoustics, oceanography, and electromagnetics, a number of numerical methods to simulate elastic wave propagation have been developed
The adoption of perfectly matched layer (PML) is in line with the stability requirements in terms of it is flexibility to use the symmetric matrix form (SMF) framework for the elastic wave equation combined with PML/ multi-axis perfectly matched layer (MPML)/non-PML
auxiliary differential equation (ADE) complex frequency-shifted (CFS)-MPML is extended to the curve domain to better suppress the reflected wave generated by the elastic wave propagated to the truncated boundary
Summary
As well as in many other applications, such as acoustics, oceanography, and electromagnetics, a number of numerical methods to simulate elastic wave propagation have been developed. The SBP-SAT methodology has many useful properties, and one of them is to facilitate the derivation of higher-order spatial discretizations that are provably time-stable using a rather straightforward approach based on the energy method, in which the procedure is similar to the energy method in the continuous case. Another property is that the SBP-SAT methodology can handle the boundary condition in a flexible way, and the boundaries contain those of the computational domain and of interfaces between blocks, in which the blocks are obtained by dividing the computational domain [36].
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