A general improvement in the WENO-Z-type schemes

TL;DR

This paper introduces an improved WENO-Z-type scheme called LOP-GMWENO-X, which enhances resolution and reduces oscillations in hyperbolic problem simulations, especially around discontinuities and high-frequency regions.

Contribution

The paper extends the OP criterion to WENO-Z-type schemes, developing the LOP-GMWENO-X method that offers superior accuracy and stability over existing schemes.

Findings

Reduces spurious oscillations in simulations.

Maintains high resolution at critical points.

Decreases post-shock oscillations in 2D problems.

Abstract

A new type of finite volume WENO schemes for hyperbolic problems was devised in [36] by introducing the order-preserving (OP) criterion. In this continuing work, we extend the OP criterion to the WENO-Z-type schemes. We firstly rewrite the formulas of the Z-type weights in a uniform form from a mapping perspective inspired by extensive numerical observations. Accrodingly, we build the concept of the locally order-preserving (LOP) mapping which is an extension of the order-preserving (OP) mapping and the resultant improved WENO-Z-type schemes are denoted as LOP-GMWENO-X. There are four major advantages of the LOP-GMWENO-X schemes superior to the existing WENO-Z-type schemes. Firstly, the new schemes can amend the serious drawback of the existing WENO-Z-type schemes that most of them suffer from either producing severe spurious oscillations or failing to obtain high resolutions in long calculations of hyperbolic problems with discontinuities. Secondly, they can maintain considerably high resolutions on solving problems with high-order critical points at long output times. Thirdly, they can obtain evidently higher resolution in the region with high-frequency but smooth waves. Finally, they can significantly decrease the post-shock oscillations for simulations of some 2D problems with strong shock waves. Extensive benchmark examples are conducted to illustrate these advantages.