On Jiang–Shu-type nonlinear weights for shock-capturing schemes with unconditionally optimal high order

Jiang–Shu-type (JS-type) nonlinear weights are commonly used in various applications. However, they may suffer from a loss of accuracy near critical points. This paper aims to address this issue by revisiting nonlinear weights for fifth-order shock-capturing schemes. Without loss of generality, we take a fifth-order nonlinear interpolation scheme as an example, but the results can be applied directly for schemes based on reconstruction. By analyzing the difference between nonlinear weights and optimal linear weights directly, we are able to derive the requirements for the nonlinear weights to satisfy the unconditionally optimal high order (UOHO) property. To ensure the UOHO property and the shock-capturing capability, we develop in this paper a new adaptive parameter for the JS-type nonlinear weights. In addition, the constructed nonlinear interpolation scheme is designed to be scale-invariant, namely, the performance is independent of the magnitude of the considered quantities. To demonstrate the superiority of the proposed nonlinear weights, they are applied to construct a corresponding weighted compact nonlinear scheme and compared to some other nonlinear weights in terms of the UOHO and scale-invariant properties, capturing discontinuities and resolving small-scale structures.