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Volume 31, Issue 5
A General Improvement in the WENO-Z-Type Schemes

Ruo Li & Wei Zhong

Commun. Comput. Phys., 31 (2022), pp. 1362-1401.

Published online: 2022-05

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  • Abstract

A new type of finite volume WENO schemes for hyperbolic problems was devised in [33] 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. Accordingly, 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.

  • AMS Subject Headings

65M06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-31-1362, author = {Li , Ruo and Zhong , Wei}, title = {A General Improvement in the WENO-Z-Type Schemes}, journal = {Communications in Computational Physics}, year = {2022}, volume = {31}, number = {5}, pages = {1362--1401}, abstract = {

A new type of finite volume WENO schemes for hyperbolic problems was devised in [33] 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. Accordingly, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0202}, url = {http://global-sci.org/intro/article_detail/cicp/20508.html} }
TY - JOUR T1 - A General Improvement in the WENO-Z-Type Schemes AU - Li , Ruo AU - Zhong , Wei JO - Communications in Computational Physics VL - 5 SP - 1362 EP - 1401 PY - 2022 DA - 2022/05 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2021-0202 UR - https://global-sci.org/intro/article_detail/cicp/20508.html KW - WENO-Z-type schemes, locally order-preserving mapping, hyperbolic problems. AB -

A new type of finite volume WENO schemes for hyperbolic problems was devised in [33] 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. Accordingly, 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.

Ruo Li & Wei Zhong. (2022). A General Improvement in the WENO-Z-Type Schemes. Communications in Computational Physics. 31 (5). 1362-1401. doi:10.4208/cicp.OA-2021-0202
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