On Z-type nonlinear weights for shock-capturing schemes with unconditionally optimal high order

Abstract

In this paper, we revisit nonlinear weights for high-order shock-capturing schemes and analyze their requirements for achieving optimal high order of accuracy regardless of the order of critical points, known as the unconditionally optimal high-order (UOHO) property. Specifically, we focus on nonlinear interpolation schemes and demonstrate how this property can be satisfied. By applying the general analysis to a fifth-order nonlinear interpolation scheme with Z-type nonlinear weights, we propose two simple ways to modify the nonlinear weights to satisfy the UOHO property. It is demonstrated numerically that the resulting fifth-order weighted compact nonlinear schemes do possess the UOHO property. Furthermore, several numerical examples are conducted to demonstrate the advantages of these new schemes in terms of shock-capturing capability and resolution. While the analysis is focused on nonlinear interpolation schemes, we also show that the proposed modifications can be directly applied to WENO-Z schemes, resulting in good shock-capturing capability and satisfying the UOHO property.