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IMA Journal of Numerical Analysis Advance Access originally published online on November 3, 2006
IMA Journal of Numerical Analysis 2007 27(3):593-615; doi:10.1093/imanum/drl031
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© The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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

Frieder Lörcher and Claus-Dieter Munz{dagger}

Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, Pfaffenwaldring 21,70550 Stuttgart, Germany

{dagger} Email: munz{at}iag.uni-stuttgart.de

Received on 20 November 2005. Revised on 30 March 2006. Revised on 2 October 2006.


  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 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.

Key Words: Lax–Wendroff scheme; high-order schemes; multidimensional interpolation


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