Prefactored optimized compact finite-difference schemes for second spatial derivatives

摘要

We have described systematically the processes of developingprefactored optimized compact schemes for second spatialderivatives. First, instead of emphasizing high resolution of a singlemonochromatic wave, we focus on improving the representationof the compact finite difference schemes over a wide range ofwavenumbers. This leads to the development of the optimizedcompact schemes whose coefficients will be determined by Fourieranalysis and the least-squares optimization in thewavenumberdomain. The resulted optimized compact schemes provide themaximum resolution in spatial directions for the simulation ofwave propagations. However, solving for each spatial derivativeusing these compact schemes requires the inversion of a bandmatrix. To resolve this issue, we propose a prefactorization strategythat decomposes the original optimized compact scheme intoforward and backward biased schemes, which can be solvedexplicitly. We achieve this by ensuring a property that the realnumerical wavenumbers of both the forward and backwardbiased stencils are the same as that of the original central compactscheme, and their imaginary numerical wavenumbers have thesame values but with opposite signs. This property guaranteesthat the original optimized compact scheme can be completelyrecovered after the summation of the forward and backwardfinite difference operators. These prefactored optimized compactschemes have smaller stencil sizes than even those of the originalcompact schemes, and hence, they can take full advantage of thecomputer caches without sacrificing their resolving power. Comparisonswere made throughout with other well-known schemes.