High order Lax-Wendroff-type schemes for linear wave propagation

Abstract

The second order accurate Lax-Wendroff scheme is based on the first three terms of a Taylor expansion in time in which the time derivatives are replaced by space derivatives using the governing evolution equations. The space derivatives are then approximated by central difference formulas. In this paper we extent this idea and truncate the Taylor expansion at an arbitrary even order. We use then the so called Cauchy-Kovalevskaya procedure to replace all the time derivatives by space derivatives. The linear case is the main focus of this paper, because the proposed high order schemes are good candidates for the approximation of linear wave motion over long distances and times with the important applications in aeroacoustics and electromagnetics. We formulate the scheme for a general system of linear equations with arbitrary order and in two or three space dimensions. The numerical results are compared with a standard scheme for aeroacoustical applications with respect to their quality and the computational effort.