Phase error and stability of second order methods for hyperbolic problems. I

Abstract

Second order schemes for hyperbolic systems are compared with respect to stability properties and minimization of phase errors. Among the schemes that utilize no points beyond a nine point lattice those with the smallest phase errors are a Strang type splitting scheme, Leapfrog, and Lax-Wendroff (written as a two-step scheme). Because of its optimal permissible time step the time splitting scheme is to be preferred. Both the Burstein and MacCormack schemes are found to be weakly unstable and have larger phase errors while the rotated Richtmyer method has a much larger phase error than these other schemes. It is possible, however, to reduce the phase error by using points beyond the eight nearest neighbors. However, the Richtmyer method together with its generalization by Gourlay and Morris still have very large phase errors. The method of Fromm together with the SHASTA code are nonlinear schemes that do have reduced phase errors. However, because of the smaller permissible time steps coupled with additional complexities it would seem more worthwhile to use higher order schemes once one is willing to use data beyond the nine point rectangular mesh. Numerical experiments, with vector equations, are presented that confirm these results.