A General Improvement in the WENO-Z-Type Schemes (Communications in Computational Physics, 2022)
A new type of finite volume WENO schemes for hyperbolic problems wasdevised in [33] by introducing the order-preserving (OP) criterion. In this continuingwork, we extend the OP criterion to the WENO-Z-type schemes. We firstly rewrite theformulas of the Z-type weights in a uniform form from a mapping perspective inspiredby extensive numerical observations. Accordingly, we build the concept of the locallyorder-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 amendthe serious drawback of the existing WENO-Z-type schemes that most of them sufferfrom either producing severe spurious oscillations or failing to obtain high resolutionsin long calculations of hyperbolic problems with discontinuities. Secondly, they canmaintain considerably high resolutions on solving problems with high-order criticalpoints at long output times. Thirdly, they can obtain evidently higher resolution in theregion with high-frequency but smooth waves. Finally, they can significantly decreasethe post-shock oscillations for simulations of some 2D problems with strong shockwaves. Extensive benchmark examples are conducted to illustrate these advantages.
Keywords: WENO-Z-type schemes; locally order-preserving mapping; hyperbolic problems.