The simplest finite difference approximations for spatial derivatives are centered, explicit, and applied to "regular" equispaced grids. Well-established generalizations include the use of implicit (compact) approximations and staggered grids. We find here that the combination of these two concepts, together with high formal order of accuracy, is very effective for approximating the first derivatives in space that occur in many wave-type PDEs.
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Fornberg et al. (1999) studied this question.
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