Stable Difference Approximations for the Elastic Wave Equation in Second Order Formulation

Abstract

We consider the three-dimensional elastic wave equation for an isotropic heterogeneous material subject to a stress-free boundary condition. Building on our recently developed theory for difference methods for second order hyperbolic systems [H.-O. Kreiss, N. A. Petersson, J. Yström, SIAM J. Numer. Anal., 40 (2002), pp. 1940-1967, we develop an explicit, second order accurate technique which is stable for all ratios of longitudinal over transverse phase velocities. The spatial discretization is self-adjoint, and the stability is obtained through an energy estimate. Seismic events are often modeled using singular source terms, and we devise a technique to place sources independently of the grid while retaining second order accuracy away from the source. Several numerical examples are given.

Simulation of Dynamic Earthquake Ruptures in Complex Geometries Using High-Order Finite Difference Methods

“…Within seismic wave propagation there has been work on the use of unstaggered grids for the first and second order form of the equations of elasticity (Bayliss et al, 1986;Zhang and Chen, 2009;Nilsson et al, 2007). Coordinate transforms have been applied to both forms of the equations to handle complex geometries (Fornberg, 1988;Tessmer et al, 1992;Appelö and Petersson, 2009).…”

The role of topography and lateral velocity heterogeneities on near-source scattering and ground-motion variability

“…In this study, we conduct numerical simulations with the secondorder finite-difference code WPP of Nilsson et al (2007). The numerical method is based on summation by part stencils, featuring efficient absorbing boundaries, vertical mesh refinement and Table 1.…”

“…We set the gridstep to 40 and 26 m for the homogeneous and the heterogeneous models, respectively. Since accurate results are expected when at least 12-15 points per minimum wavelength are observed (Nilsson et al 2007), we low-pass filter all output synthetics at 5 Hz.…”

Three-Dimensional Wavefield Simulation in Heterogeneous Transversely Isotropic Medium with Irregular Free Surface

“…The situation has improved recently. For rectangular domains, a stable and explicit discretization of the free surface boundary conditions has been presented by Nilsson et al (2007). By using boundary-modified difference operators, Nilsson et al (2007) introduce a discretization of the mixed derivatives in the governing equations; they also show that the method is second order accurate for problems with smoothly varying material properties and stable under standard Courant-Friedrichs-Lewy (CFL) constraints, for arbitrarily varying material properties.…”