Grid-optimized upwind dispersion-relation-preserving scheme on non-uniform Cartesian grids for computational aeroacoustics

Grid-optimized upwind dispersion-relation-preserving (GOUPDRP) finite difference schemes on non-uniform Cartesian grids are proposed in this paper. With the high-order accuracy and the robustness, GOUPDRP schemes cannot only minimize the dispersion error and the dissipation error, but also damp out the spurious short waves automatically without the demand of filter or explicit dissipation terms. So far, some optimized upwind schemes have been implemented only on uniform grids. It is well known that most of aeroacoustic problems are seldom confined to uniform grids. Unlike other optimized upwind schemes, GOUPDRP schemes can be easily applied to the non-uniform meshes, which give finite difference equations locally the same dispersion relation as original partial differential equations on the non-uniform grids. Mathematic formulations of the GOUPDRP schemes are derived on the non-uniform meshes in the paper. In order to investigate the effectiveness of GOUPDRP schemes, benchmark problems are solved by this scheme; numerical solutions are compared with their analytical counterparts. Numerical results suggest that GOUPDRP schemes are highly efficient and relatively potential to solve acoustical problems on non-uniform grids.