High‐resolution WENO‐Z type scheme with mapped global smoothness indicator for hyperbolic conservation laws

Abstract

To address the shortcomings of the WENO‐Z scheme in terms of reducing accuracy at higher‐order critical points, this paper proposes a new two‐parameter global smoothing factor by introducing a sub‐stencil mapping function and develops a new WENO‐Z scheme, WENO‐ZI. With the chosen parameters and , the new scheme can achieve the desired accuracy at critical points of any order, and the resolution is enhanced as increases. Numerical results indicate that the new scheme performs optimally when and for one‐dimensional problems, and for two‐dimensional problems. The theoretical results verify the high accuracy and high‐resolution of the new WENO‐ZI scheme. Numerical experiments of the new mapping method for one‐ and two‐dimensional benchmark problems are compared with various numerical schemes. Results demonstrate that the new WENO scheme achieves higher resolution compared to other numerical methods. It satisfies the desired accuracy at critical points of arbitrary orders and can be easily extended to higher‐order schemes.