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

^{The second-order accurate Lax–Wendroff scheme is basedon the first three terms of a Taylor expansion in time in whichthe time derivatives are replaced by space derivatives usingthe governing evolution equations. The space derivatives arethen approximated by central differencing. In this paper, weextend this idea and truncate the Taylor expansion at an arbitraryorder. One main building block is the so-called Cauchy–Kovalevskayaprocedure to replace all the time derivatives by space derivativeswhich can be formulated for a general system of linear equationswith arbitrary order and in two- or three-space dimensions.The linear case is the main focus of this paper because theproposed high-order schemes are good candidates for the approximationof linear wave motion over long distances and times with importantapplications in aeroacoustics and electromagnetics. The stabilityand the efficiency of Lax–Wendroff-type schemes are examined.The numerical results are compared with a standard scheme foraeroacoustical applications with respect to their quality andthe computational effort. The extensions of the schemes to generalgrids, nonconstant and nonlinear cases are alsoaddressed.}