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.

Improved third-order WENO scheme with new reference smoothness indicator

“…Using the weighted process of WENO-Z scheme, we can improve the resolution and restore the optimal convergence order. With this methodology,recently, a series of third-order WENO schemes [11,12,13,14] with high resolution and accuracy are proposed.…”

“…In this article, we propose a new reference smoothness indicator τ 4Re for the third-order WENO scheme. This reference smoothness indicator is constructed such that it is a square sum of the L 2 -norm of the difference between the linear combination of the first derivative of the reconstruction polynomial of each candidate stencil and the first derivative of the reconstruction polynomial of the global stencil, and has fourth order accuracy in smooth regions with smaller quantities compared with the reference indicator τ 4P [11,14]. Theoretical analysis shows that the resultant WENO-Re3 scheme can attain third order accuracy in smooth region including critical points.A series of one-dimensional and two-dimensional numerical examples show that WENO-Re3 scheme has higher resolution and less dissipation than several existing third-order WENO schemes: WENO-JS3, WENO-Z3 and WENO-P3 [11].…”

A smoothness indicators free third-order weighted ENO scheme for gas-dynamic Euler equations

“…Wu et al [35,36,37] developed three WENO schemes, namely WENO-N, WENO-NP, and WENO-NN, using different nonlinear combinations of local smoothness indicators and local smoothness factors. Wang et al [38] introduced a new reference smoothness indicator to construct a third-order WENO-R scheme with low dissipation. Li et al [39] constructed a new global smoothness indicator by using Taylor expansions to handle local smoothness indicators and developed the WENO-ZF scheme.…”

A smoothness indicators free third-order weighted ENO scheme for gas-dynamic Euler equations

“…Wu et al [35,36,37] developed three WENO schemes, namely WENO-N, WENO-NP, and WENO-NN, using different nonlinear combinations of local smoothness indicators and local smoothness factors. Wang et al [38] introduced a new reference smoothness indicator to construct a third-order WENO-R scheme with low dissipation. Li et al [39] constructed a new global smoothness indicator by using Taylor expansions to handle local smoothness indicators and developed the WENO-ZF scheme.…”