How to solve systems of conservation laws numerically using the graphics processor as a high-performance computational engine

Abstract

The paper has two main themes: The first theme is to give the reader an introduction to modern methods for systems of conservation laws. To this end, we start by introducing two classical schemes, the Lax-Friedrichs scheme and the Lax-Wendroff scheme. Using a simple example, we show how these two schemes fail to give accurate approximations to solutions containing discontinuities. We then introduce a general class of semi-discrete finite-volume schemes that are designed to produce accurate resolution of both smooth and nonsmooth parts of the solution. Using this special class we wish to introduce the reader to the basic principles used to design modern high-resolution schemes. As examples of systems of conservation laws, we consider the shallow-water equations for water waves and the Euler equations for the dynamics of an ideal gas. The second theme in the paper is how programmable graphics processor units (GPUs or graphics cards) can be used to efficiently compute numerical solutions of these systems. In contrast to instruction driven micro-processors (CPUs), GPUs subscribe to the data-stream-based computing paradigm and have been optimised for high throughput of large data streams. Most modern numerical methods for hyperbolic conservation laws are explicit schemes defined over a grid, in which the unknowns at each grid point or in each grid cell can be updated independently of the others. Therefore such methods are particularly attractive for implementation using data-stream-based processing. © 2007 Springer-Verlag Berlin Heidelberg.