Optimized compact finite difference schemes with maximum resolution

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Description/Abstract

Direct numerical simulations and computational aeroacoustics require an accurate finite difference scheme that
has a high order of truncation and high-resolution characteristics in the evaluation of spatial derivatives. Compact
finite difference schemes are optimized to obtain maximum resolution characteristics in space for various spatial
truncation orders. An analytic method with a systematic procedure to achieve maximum resolution characteristics
is devised for multidiagonal schemes, based on the idea of the minimization of dispersive (phase) errors in the wave
number domain, and these are applied to the analytic optimization of multidiagonal compact schemes. Actual
performances of the optimized compact schemes with a variety of truncation orders are compared by means of
numerical simulations of simple wave convections, and in this way the most effective compact schemes are found
for tridiagonal and pentadiagonal cases, respectively. From these comparisons, the usefulness of an optimized
high-order tridiagonal compact scheme that is more efficient than a pentadiagonal scheme is discussed. For the
optimized high-order spatial schemes, the feasibility of using classical high-order Runge-Kutta time advancing
methods is investigated.