An Improved High-Accuracy WENO-ZN Scheme

This paper proposed a new high-accuracy WENO scheme, denoted as the WENO-ZNP scheme, which maintains optimal order of convergence in smooth areas containing first- to third-order critical points. Taking the fifth-order scheme as an example, WENO-ZNP achieves fifth-order accuracy at third-order critical points, while WENO-ZN [1] achieves fourth-order accuracy at second-order critical points but fails to improve the order at third-order critical points. The key idea of the new scheme is to redefine the nonlinear weights based on the WENO-ZN scheme. Firstly, we replace the conventional local smoothness indicator β ₖ with the modified indicator η ₖ , which simplifies the computation of the adaptive function θ . To further enhance accuracy at critical points, a parameter s associated with the local smoothness indicator η ₖ is incorporated to adjust the nonlinear weights. In addition, we design a correction term to promote a more balanced and reasonable distribution of weights among the sub-stencils, thereby improving the overall reconstruction quality. Theoretical analysis and accuracy tests indicate that the WENO-ZNP scheme maintains fifth-order accuracy up to third-order critical points. Moreover, the new scheme enhances numerical stability and shock-capturing capability while maintaining high-order accuracy and it can be easily extended to higher-order schemes.