A low‐dissipation third‐order weighted essentially nonoscillatory scheme with a new reference smoothness indicator

Abstract

SummaryThe classical third‐order weighted essentially nonoscillatory (WENO) scheme is notoriously dissipative as it loses the optimal order of accuracy at critical points and its two‐point finite difference in the smoothness indicators is unable to differentiate the critical point from the discontinuity. In recent years, modifications to the smoothness indicators and weights of the classical third‐order WENO scheme have been reported to reduce numerical dissipation. This article presents a new reference smoothness indicator for constructing a low‐dissipation third‐order WENO scheme. The new reference smoothness indicator is a nonlinear combination of the local and global stencil smoothness indicators. The resulting WENO‐Rp3 scheme with the power parameter p=1.5 achieves third‐order accuracy in smooth regions including critical points and has low dissipation, but numerical results show this scheme cannot keep the ENO property near discontinuities. The recommended WENO‐R3 scheme (p=1) keeps the ENO property and performs better than several recently developed third‐order WENO schemes.