High-order WCNS scheme with a constrained weighted least squares immersed boundary method for inviscid compressible flow simulations

Abstract Robust and accurate numerical methods for numerical simulations of compressible fluid flows featuring discontinuities around complex geometries are challenging and of increasing practical interest. In this paper, a third-order weighted compact nonlinear scheme (WCNS) for solving inviscid compressible flows on Cartesian grids and a third-order constrained weighted least squares immersed boundary (CWLS-IB) method for dealing with irregular boundaries are developed. The numerical fluxes at the cell edges are calculated by a nonlinear combination of an optimal high-order polynomial on a global stencil and two low-order polynomials on two sub-stencils. A new global smoothness indicator of the global stencil is introduced to improve the accuracy of the scheme at both the first- and second-order critical points. The third-order CWLS method is employed to extrapolate the values at ghost cells outside the physical domain and a robust WENO-type extrapolation is adopted to eliminate oscillations when shocks occur near boundaries by introducing a nonlinear weight that can switch automatically between the high-order and the low-order extrapolation. Several numerical experiments are provided to verify the performance of the present WCNS scheme and the CWLS-IB method in terms of the accuracy, robustness and spatial resolution.