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We study the numerical anisotropy existent in compact difference schemes asapplied to hyperbolic partial differential equations, and propose an approachto reduce this error and to improve the stability restrictions based on aprevious analysis applied to explicit schemes. A prefactorization of compactschemes is applied to avoid the inversion of a large matrix when calculatingthe derivatives at the next time level, and a predictor-corrector time marchingscheme is used to update the solution in time. A reduction of the isotropyerror is attained for large wave numbers and, most notably, the stabilityrestrictions associated with MacCormack time marching schemes are considerablyimproved. Compared to conventional compact schemes of similar order ofaccuracy, the multidimensional schemes employ larger stencils which wouldpresumably demand more processing time, but we show that the new stabilityrestrictions render the multidimensional schemes to be in fact more efficient,while maintaining the same dispersion and dissipation characteristics of theone dimensional schemes.

Numerical anisotropy study of a class of compact schemes