Lax–Wendroff-type schemes of arbitrary order in several space dimensions

摘要

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 differencing. In this paper, we extend this idea and truncate the Taylor expansion at an arbitrary order. One main building block is the so-called Cauchy–Kovalevskaya procedure to replace all the time derivatives by space derivatives which can be formulated for a general system of linear equations with arbitrary order and in two- or three-space dimensions. 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 important applications in aeroacoustics and electromagnetics. The stability and the efficiency of Lax–Wendroff-type schemes are examined. The numerical results are compared with a standard scheme for aeroacoustical applications with respect to their quality and the computational effort. The extensions of the schemes to general grids, nonconstant and nonlinear cases are alsoaddressed.