Accurate Finite Difference Algorithms

Abstract

Clear examples of the difficulties associated with applying CFD techniques to apparently simple problems are provided by computational aeroacoustics and computational electromagnetics. These applications require accurate wave propagation over long distances for a wide range of frequencies, placing a severe demand on numerical algorithms, and raising issues related to efficiency, accuracy, compatible space and time treatments, high frequency data, propagation along characteristic surfaces, isotropy, stable and accurate artificial boundary treatments, and nonrestrictive stability bounds. This paper briefly presents two methods for the development of finite difference algorithms which are intended to address these issues. High order single step explicit algorithms are possible, and examples with up to eleventh order accuracy will be shown. High resolution algorithms in the sense of (Lele, 1992) are also possible, with amplification factor and relative phase change per time step which are virtually 1 for normal mode frequencies in [0, π] and CFL numbers in [0, 1]. If our most accurate algorithm is used to propagate an initial periodic sine wave, then after five periods with four grid points per wavelength, the maximum error is O[10^-6], and after five hundred thousand periods with eight grid points per wavelength, the maximum error is O[10^-4]. High order algorithms are relatively more efficient (Kreiss & Oliger, 1972), and their relative efficiency tends to increase as the error bound decreases, as the simulation time increases, and as the spatial dimension increases.