基于非结构四边形网格的WENO格式

doi:10.19596/j.cnki.1001-246x.7811

计算物理 ›› 2018, Vol. 35 ›› Issue (5): 525-534.DOI: 10.19596/j.cnki.1001-246x.7811

基于非结构四边形网格的WENO格式

赵丰祥^1,2, 潘亮², 王双虎²

1. 中国工程物理研究院研究生院, 北京 100088;
2. 北京应用物理与计算数学研究所, 北京 100088

收稿日期:2017-12-11 修回日期:2017-12-25 出版日期:2018-09-25 发布日期:2018-09-25
通讯作者: 潘亮(1986-),男,博士,E-mail:panliangjlu@sina.com
基金资助:
国家自然科学基金（11701038，915303108，U1630247）及中国博士后科学基金（2016M600065）资助项目

Weighted Essentially Non-oscillatory Schemes on Unstructured Quadrilateral Meshes

ZHAO Fengxiang^1,2, PAN Liang², WANG Shuanghu²

1. The Graduate School of China Academy of Engineering Physics, Beijing 100088, China;
2. Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Received:2017-12-11 Revised:2017-12-25 Online:2018-09-25 Published:2018-09-25

摘要/Abstract

摘要： 基于非结构四边形网格发展求解双曲守恒律的三阶加权基本无振荡（WENO）格式.针对任意非结构四边形网格选取重构模板，并给出基于线性多项式的三阶线性重构.但对于一般的非结构四边形网格，会出现非常大的线性权和负权，使得非线性重构的WENO格式对光滑问题也不稳定.本文给出一个处理非常大的线性权的优化重构方法，对优化后得到的负线性权采用分裂方法进行处理.对于非线性权，提出一种考虑局部网格和物理量间断的新光滑度量因子.采用优化重构方法和新的非线性权，当前的三阶WENO格式在质量很差的网格上也具有很好的稳定性.理论的三阶精度在数值精度测试算例中得到验证，同时一范数和无穷范数的误差绝对值不依赖于网格质量；具有强间断的数值结果证明了当前格式的有效性.

关键词: WENO重构, 非结构四边形网格, 双曲守恒律, 有限体积格式

Abstract: A third-order weighted essentially non-oscillatory (WENO) scheme is developed for hyperbolic conservation laws on unstructured quadrilateral meshes. As starting point of WENO reconstruction, a general stencil is proposed for any local topology on quadrilateral meshes. With selected stencil, a unified linear scheme was constructed. However, very large weights and non-negative may appear, which leads the scheme unstable even for smooth flows. An optimization approach is given to deal with very large linear weights on unstructured meshes. Splitting technique is considered to deal with negative weights obtained by optimization approach. Non-linear weight with a new smooth indicator is proposed as well. With optimization approach for very large weights and splitting technique for negative weights, the current scheme becomes more robust. Numerical tests are presented to validate accuracy. Expected convergence rate of accuracy is obtained. And absolute value of error is not affected by mesh quality. Numerical results for flow with strong discontinuities are presented to validate robustness of the WENO scheme.

Key words: WENO reconstruction, unstructured quadrilateral meshes, hyperbolic conservation laws, finite volume method

中图分类号:

赵丰祥, 潘亮, 王双虎. 基于非结构四边形网格的WENO格式[J]. 计算物理, 2018, 35(5): 525-534.

ZHAO Fengxiang, PAN Liang, WANG Shuanghu. Weighted Essentially Non-oscillatory Schemes on Unstructured Quadrilateral Meshes[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2018, 35(5): 525-534.

参考文献

[1] LAX P, WENDROFF B. Systems of conservation laws[J].Comm Pure Appl Math,1960,13:217-237.
[2] VAN LEER B. Towards the ultimate conservative difference scheme V:A second order sequel to Godunov's method[J]. J Comput Phys,1979,32:101-136.
[3] HARTEN A. High resolution schemes for hyperbolic conservation laws[J]. J Comput Phys, 1983,49:357-393.
[4] HARTEN A, ENGQUIST B, OSHER S. Uniformly high order accurate essentially non-oscillatory schemes, Ⅲ[J]. J Comput Phys, 1987,71:231-303.
[5] SHU C W, OSHER S. Efficient implementation of essentially non-oscillatory shock capturing schemes[J]. J Comput Phys,1988, 77:439-471.
[6] LIU X D, OSHER S, CHAN T. Weighted essentially non-oscillatory schemes[J]. J Comput Phys, 1994,115:200-12.
[7] JIANG G S, SHU C W. Efficient implementation of weighted ENO schemes[J]. J Comput Phys,1996,126:202-28.
[8] COCKBURN B, SHU C W. The Runge-Kutta discontinuous Galerkin method for conservation laws V:Multidimensional systems[J]. J Comput Phys, 1998,141:199-224.
[9] HENRICK A K, ASLAM T D, Powers J M. Mapped weighted essentially non-oscillatory schemes:Achieving optimal order near critical points[J]. J Comput Phys,2005,207(2):542-567.
[10] BORGES R, CARMONA M, COSTA B. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws[J]. J Comput Phys, 2008, 227:191-211.
[11] ZHAO F, PAN L, LI Z, WANG S. A new class of high-order weighted essentially non-oscillatory schemes for hyperbolic conservation laws[J]. Computers and Fluids, 2017, 159:81-94.
[12] HARTEN A, CHAKRAVARTHY S R. Multidimensional ENO schemes for general geometries[R]. ICASE Report No.91-76, 1991.
[13] ABGRALL R. On essentially non-oscillatory schemes on unstructured meshes:Analysis and implementation[J]. J Comput Phys, 1994,114:45-58.
[14] SONAR T. On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations:Polynomial recovery, accuracy and stencil selection[J]. Comput Methods Appl Mech Engrg, 1997, 140:157-181.
[15] FRIEDRICH O. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids[J]. J Comput Phys,1998, 144:194-212.
[16] DUMBSER M, KASER M. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems[J]. J Comput Phys, 1998, 221:693-723.
[17] HU C, SHU C W. Weighted essentially non-oscillatory schemes on triangular meshes[J]. J Comput Phys,1999,150:97-127.
[18] SHI J, HU C, SHU C W. A technique of treating negative weights in WENO schemes[J]. J Comput Phys, 2002, 175:108-127.
[19] LIU Y, ZHANG Y T. A robust reconstruction for unstructured WENO schemes[J]. J Sci Comput, 2013,54:603-621.
[20] TORO E. Riemann solvers and numerical methods for fluid dynamics[M]. Springer, 1997.
[21] WOODWARD P, COLELLA P. Numerical simulations of two-dimensional fluid flow with strong shocks[J]. J Comput Phys,1984,54:115-73.
[22] WANG Q, REN Y, LI W. Compact high order finite volume method on unstructured grids Ⅱ:Extension to two-dimensional Euler equations[J]. J Comput Phys, 2016,314:883-908.

基于非结构四边形网格的WENO格式

Weighted Essentially Non-oscillatory Schemes on Unstructured Quadrilateral Meshes

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