Chapter 18 - Solving Wave Equations on Unstructured Geometries

Abstract

Every wave solver serving the computational study of waves meets a trade-off of two figures of merit—its computational speed and its accuracy. The use of Discontinuous Galerkin (DG) methods on graphical processing units (GPUs) significantly lowers the cost of obtaining accurate solutions. DG methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust. They allow arbitrary unstructured geometries and easy control of accuracy without compromising simulation stability. The resulting locality in memory access is one of the factors that enables DG to run on off-the-shelf, massively parallel graphics processors (GPUs). In addition, DG's high-order nature lets it require fewer data points per represented wavelength and hence fewer memory accesses, in exchange for higher arithmetic intensity. Both of these factors work significantly in favor of a GPU implementation of DG. Discontinuous Galerkin methods are most often used to solve hyperbolic systems of conservation laws in the time domain. Parabolic and elliptic equations can also be solved using DG methods.