On construction of shock-capturing boundary closures for high-order finite difference method

Abstract

High-order finite difference method is widely used in the field of computational fluid dynamics. Since it can be implemented in a dimension-by-dimension fashion, it is much more efficient than some other methods in higher dimensions. However, although interior shock-capturing schemes are mature, shock-capturing boundary closures are only in their infancy due to the issue of insufficient stencils near boundary. To address this issue, we focus on a fifth-order multi-resolution weighted compact nonlinear scheme. Firstly, to enhance the shock-capturing capability of the interior scheme, we propose a new technique by adding a free parameter to the formulae of smoothness indicators. Then we follow the idea of the nonlinear spectral difference method to construct shock-capturing boundary closures when the interior scheme cannot be used. To validate the proposed method, several benchmark numerical examples are presented. The results show that the shock-capturing capability is effectively improved and the proposed nonlinear boundary closures work well for the cases with shock transmission or shock near boundary.