Upwind methods for second-order wave equations on overlapping grids

Abstract

A normal mode analysis from the theory of Gustafsson, Kreiss, and Sundstrm (GKS) shows that these schemes are robustly stable against numerical instabilities that are known to arise from overlapping grid interpolation. A reformulated upwind scheme is also presented that considers several optimizations to the first-order in time system. This reformulated scheme discretizes the second-order wave equation directly while still retaining the essential stability properties of the first-order in time scheme. We give a formally exact differential-difference equation which is used to derive high-order predictor-corrector schemes. Numerical experiments show this scheme is typically an order of magnitude faster that the original upwind scheme. The thesis concludes by considering future work that may extend the application of the upwind method to the elastic wave equation of solid mechanics and the harmonic coordinate formulation of general relativity

This thesis is concerned with novel high-order upwind schemes for the second-order wave equation. A first-order in time but second-order in space formulation of the wave equation is considered first. Following an approach similar to the original idea of Godunov, an exact solution to a local Riemann problem that incorporates upwinding is embedded into a numerical flux function. Using these upwind numerical flux functions we provide an explicit, conservative, and formally exact single-step update formula. We derive high-order accurate approximations from this exact formula which defines our upwind method. This scheme is applied to Maxwells equations written in second-order form. Several scattering problems are presented consisting of domains with multiple nondispersive materials and high-order convergence is observed. Interfaces and physical boundaries are described with efficient composite overlapping structured grids which are coupled by interpolation and high-order interface conditions.