An improved WENO-Z+ scheme for solving hyperbolic conservation laws

Abstract

Increasing the weight of the less-smooth substencil is an effective way to improve the resolution of the WENO scheme. Based on this knowledge, Acker et al. (2016) [14] constructed the WENO-Z+ scheme by introducing a novel term, which has a role of increasing the weight of the less-smooth substencil, into the WENO-Z weight formula. In this paper, we propose a set of modifications to WENO-Z+ scheme to give full play to its potential in spectral properties, resulting the WENO-Z+M scheme. WENO-Z+M shows excellent spectral properties. In the numerical tests, the resolution of the fifth-order WENO-Z+M scheme is significantly higher than those of the ninth-order WENO-Z scheme and the state-of-art fifth-order WENO scheme (Zhu and Hu, 2020 [25]). This paper also reveals the unique performance of the WENO-Z+ type schemes. The previous WENO schemes always have spectral properties and accuracy inferior to those of the underlying linear schemes. In contrast, by assigning weights greater than the corresponding ideal weights to the less-smooth substencils, WENO-Z+ type schemes can achieve spectral properties and accuracy superior to the underlying linear scheme.