Weighted compact nonlinear hybrid scheme based on a family of mapping functions for aeroacoustics problem

The study of differential schemes with high accuracy and low dispersion is of great significance for the numerical simulation of aeroacoustics over complex geometries. The midpoint-and-node-to-node explicit finite difference is employed to solve the flux derivatives of the linear Euler equations for improving the robustness. To obtain the numerical flux at the center of the grid element, we adopted a hybrid interpolation method of center and upwind interpolations, combined with a symmetrical conservative metric method, to achieve high-resolution discretization of the acoustic field variables and geometric variables of the structural grid. To suppress spurious oscillations and improve the resolution of discontinuous regions, a family of mapping functions is developed to establish different smoothness indicators and applied to the sixth-order weighted compact nonlinear hybrid scheme (WCNHS), forming a new WCNHS scheme based on piecewise exponential mapping functions (WCNHS-Pe). The approximate dispersion relation shows that the dispersion error and numerical dissipation of WCNHS-Pe are smaller than those of WCNHS with a simple mapping function to the original weights in Jiang and Shu and other mapping function-weighted WCNHS schemes. We have applied various WCNHS schemes to the numerical simulation of the Shu-Osher problem, propagation of Gaussian impulses on two-dimensional wavy grids, sound transmission at discontinuous interfaces, propagation of Gaussian impulses on three-dimensional wavy grids, etc. Numerical results indicate that the WCNHS-Pe scheme has better discontinuity capture capability, higher resolution, and lower dispersion under the same differential stencil, and is suitable for computational aeroacoustics of complex geometries.