Grid-Optimized Dispersion-Relation-Preserving Schemes on General Geometries for Computational Aeroacoustics

Abstract

In this paper, we investigate the dispersion-relation-preserving property of a finite difference scheme on general geometries for computational aeroacoustics, where nondispersive and nondissipative properties are of critical importance. The analysis pertains to the application of the optimization algorithm of the dispersion-relation-preserving (DRP) scheme in wave number space to the general geometries—nonuniform Cartesian and curvilinear grids. In many computational aeroacoustics applications, the DRP schemes have often been favored for their accuracy and efficiency. DRP schemes, however, are implemented only on uniform Cartesian grids. Practical problems in aeroacoustics, however, are seldom confined to uniform Cartesian grids with the associated computing grids often being nonuniform Cartesian or curvilinear. Grid-optimized, dispersion-relation-preserving (GODRP) finite difference schemes are proposed, based on optimization that gives finite difference equations locally the same dispersion relation as the original partial differential equations on the grid points in the nonuniform Cartesian or curvilinear mesh. This local dispersion-relation-preserving property guarantees global accuracy of numerical schemes in the wave number space over the full domain. The basic idea behind mathematical formulations of GODRP schemes is that the optimization in the wave number space is carried out not on the computational domain but on the physical domain. Because the properties of Cartesian and curvilinear grids differ—whether the coupling of the coordinate variables between the physical and computational domains exists or not, different mathematical formulations are developed for each grid type. To investigate the effectiveness of GODRP schemes, a sequence of benchmark problems is executed. Through many numerical test problems, it is shown that the use of GODRP schemes can broaden the application area of conventional DRP schemes to aeroacoustic phenomena, enhancing both the speed and accuracy of the computation using nonuniform Cartesian or curvilinear grids.