Galerkin methods for numerical solutions of acoustic, elastic and viscoelastic wave equations

Matt A. D. McDonald, Michael P. Lamoureux, Gary F. Margrave

The numerical modelling of wave equations is a common theme in many seismic applications, and is an important tool in understanding how the physical systems of interest react in the process of a seismic experiment. We apply state-of-the-art numerical methods based on domain-decomposition combined with local pseudospectral spatial discretization, to three physically realistic models of seismic waves, namely their propagation in acoustic, elastic, and viscoelastic media. The Galerkin formulation solves the weak form of the partial differential equation representing wave propagation and naturally includes boundary integral terms to represent free surface, rigid, and absorbing boundary effects. Stability, accuracy, and computation issues are discussed in this context along with direct comparison with finite difference methodologies.

This short paper is a summary of highlights from the 2012 MSc thesis of the first author, which aims to bridge the gap between the development of accurate physical models to represent the real world, as seen in seismic modelling, and the implementation of modern numerical techniques for the accurate solutions of partial differential equations.