Elastic wave propagation in anisotropic solids using energy-stable finite differences with weakly enforced boundary and interface conditions

Abstract

Summation-by-parts (SBP) finite difference methods have several desirable properties for second-order wave equations. They combine the computational efficiency of narrow-stencil finite difference operators with provable stability on curvilinear multiblock grids. While several techniques for boundary and interface conditions exist, weak imposition via simultaneous approximation terms (SATs) is perhaps the most flexible one. Although SBP methods have been applied to elastic wave equations many times, an SBP-SAT method for general anisotropic elastic wave equations has not yet been presented in the literature. We fill this gap by deriving energy-stable self-adjoint SBP-SAT methods for general anisotropic materials on curvilinear multiblock grids. The methods are based on fully compatible SBP operators. We demonstrate the stability and accuracy properties of a particular set of fully compatible SBP-SAT schemes using the method of manufactured solutions. We also demonstrate the usefulness of the new method in elastodynamic cloaking and seismic imaging in mountainous regions.

Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics

“…Spectral element methods can be very efficient for large-scale computations [5,27,32,34,77]. Because stability is a challenging issue for these schemes, a lot of effort has been devoted to developing energy stable (linearly stable) [3,54,76], and entropy stable (nonlinearly stable) spatial discretizations [13,15,20,22,23,66,70,71]. Stable fully-discrete schemes can be obtained from these semidiscretizations by using a slight modification of classical time integration schemes, based on the relaxation approach [40,62,64,65].…”