Dispersion and Dissipation

Abstract

In the previous chapters we have included a large number of problems and examples involving computational results. Generally, the problems involved were theoretically easy enough so that we could compare our results with the exact solutions. There have been times when we have made no mention of these comparisons and/or did not explain why we sometimes obtain bad results. In this chapter we will develop several techniques for analyzing numerical schemes. Sometimes the results will help us choose the correct parameters (Δx, Δt, etc.) for obtaining a satisfactory solution. Sometimes the results will only tell us why our results are bad. In either case, it is hoped that the material will fill a gap in our knowledge about difference scheme approximations to partial differential equations that will make us better practitioners. As we shall see later, the material presented in this chapter applies to both parabolic and hyperbolic partial differential equations. Since the results are more dramatic for hyperbolic equations, we begin our discussion with results from two previous computations involving the solution of hyperbolic equations.