Unconditionally optimal high-order weighted compact nonlinear schemes with sharing function for Euler equations

Abstract

Weighted compact nonlinear schemes (WCNS) are a type of high-order shock-capturing finite difference schemes commonly used in various applications. Their spatial discretizations involve a nonlinear interpolation step and a linear difference step. However, the nonlinear interpolation step requires significantly more computational resources compared to the linear difference step. Therefore, simplifying the interpolation step is an effective way to improve the efficiency of these schemes. In this paper, we propose a new approach to construct WCNS schemes based on the sharing function for Euler equations. This approach uses a set of common nonlinear interpolation weights for different components, resulting in a significant improvement in efficiency compared to the original WCNS schemes that use different sets of weights for each component. To ensure accuracy at critical points of any orders, we employ nonlinear weights that guarantee unconditionally optimal high order. Additionally, this new approach may reduce oscillations caused by non-characteristic interpolation. Furthermore, we also develop a new shock detector using the sharing function, enabling us to employ characteristic interpolation for nonsmooth regions and linear component-wise interpolation for the rest. We validate the proposed schemes based on the sharing function through numerical examples of one- and two-dimensional Euler equations, demonstrating their effectiveness in terms of shock-capturing capability and efficiency.