A multi-resolution weighted compact nonlinear scheme with unconditionally optimal high order

In a recent work [J. Comput. Phys. 477 (2023) 111877], an efficient multi-resolution weighted compact nonlinear scheme (WCNS) is proposed for solving compressible flows. This paper mainly focuses on improving the nonlinear interpolation scheme therein in terms of shock-capturing capability. Additionally, to address the issue of the effect of the small parameter employed to avoid division by zero, the nonlinear weights are designed to be of the exponential-type, as proposed in a recent study [J. Comput. Phys. 478 (2023) 111978], such that the order of the scheme is irrelevant to the order of the critical points. To simplify the computation of smoothness indicators, we also propose to only use the ones for traditional Jiang-Shu-type nonlinear weights. Numerical examples are conducted to show the improvement of the proposed scheme in terms of shock-capturing capability.