High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

doi:10.1007/s10483-020-2554-8

Applied Mathematics and Mechanics (English Edition) ›› 2020, Vol. 41 ›› Issue (1): 173-192.doi: https://doi.org/10.1007/s10483-020-2554-8

• 论文 • 上一篇

High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

Lingyan TANG, Songhe SONG, Hong ZHANG

College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China

收稿日期:2019-03-18 修回日期:2019-06-09 发布日期:2019-12-14
通讯作者: Lingyan TANG E-mail:tanglingyan@aliyun.com
基金资助:
Project supported by the National Natural Science Foundation of China (No. 11571366) and the Basic Research Foundation of National Numerical Wind Tunnel Project (No. NNW2018-ZT4A08)

High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

Lingyan TANG, Songhe SONG, Hong ZHANG

College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China

Received:2019-03-18 Revised:2019-06-09 Published:2019-12-14
Contact: Lingyan TANG E-mail:tanglingyan@aliyun.com
Supported by:
Project supported by the National Natural Science Foundation of China (No. 11571366) and the Basic Research Foundation of National Numerical Wind Tunnel Project (No. NNW2018-ZT4A08)

摘要/Abstract

摘要： In this paper, the maximum-principle-preserving (MPP) and positivitypreserving (PP) flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes (WCNSs) for scalar conservation laws and the compressible Euler systems in both one and two dimensions. The main idea of the present method is to rewrite the scheme in a conservative form, and then define the local limiting parameters via case-by-case discussion. Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy. Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.

关键词: hyperbolic conservation law, maximum-principle-preserving (MPP), positivity-preserving (PP), weighted compact nonlinear scheme (WCNS), finite difference scheme

Abstract: In this paper, the maximum-principle-preserving (MPP) and positivitypreserving (PP) flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes (WCNSs) for scalar conservation laws and the compressible Euler systems in both one and two dimensions. The main idea of the present method is to rewrite the scheme in a conservative form, and then define the local limiting parameters via case-by-case discussion. Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy. Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.

Key words: hyperbolic conservation law, maximum-principle-preserving (MPP), positivity-preserving (PP), weighted compact nonlinear scheme (WCNS), finite difference scheme

中图分类号:

65N12
76M20

Lingyan TANG, Songhe SONG, Hong ZHANG. High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws[J]. Applied Mathematics and Mechanics (English Edition), 2020, 41(1): 173-192.

参考文献

[1] DENG, X. G. and ZHANG, H. X. Developing high-order weighted compact nonlinear schemes. Journal of Computational Physics, 165, 22-44(2000)
[2] DENG, X. G., LIU, X., MAO, M. L., and ZHANG, H. X. Advances in high-order accurate weighted compact nonlinear schemes. Advances in Mechanics, 37(3), 417-427(2007)
[3] DENG, X. G. High-order accurate dissipative weighted compact nonlinear schemes. Science in China (Series A), 45(3), 356-370(2002)
[4] NONOMURA, T. and FUJII, K. Effects of difference scheme type in high-order weighted compact nonlinear schemes. Journal of Computational Physics, 228, 3533-3539(2009)
[5] NONOMURA, T., GOTO, Y., and FUJII, K. Improvements of efficiency in seventh-order weighted compact nonlinear scheme. 6th Asia Workshop on Computational Fluid Dynamics, Tokyo, Japan, AW6-16(2010)
[6] NONOMURA, T., IIZUKA, N., and FUJII, K. Freestream and vortex preservation properties of high-order WENO and WCNS on curvilinear grids. Computers and Fluids, 39(2), 197-214(2010)
[7] DENG, X. G., MAO, M. L., TU, G. H., LIU, H. Y., and ZHANG, H. X. Geometric conservation law and applications to high-order finite difference schemes with stationary grids. Journal of Computational Physics, 230(4), 1100-1115(2011)
[8] DENG, X. G., MIN, Y. B., MAO, M. L., LIU, H. Y., TU, G. H., and ZHANG, H. X. Further studies on geometric conservation law and applications to high-order finite difference schemes with stationary grids. Journal of Computational Physics, 239, 90-111(2013)
[9] ZHAO, G. Y., SUN, M. B., XIE, S. B., and WANG, H. B. Numerical dissipation control in an adaptive WCNS with a new smoothness indicator. Applied Mathematics and Computation, 330, 239-253(2018)
[10] ZHANG, X. X. and SHU, C. W. On maximum-principle-satisfying high order schemes for scalar conservation laws. Journal of Computational Physics, 229, 3091-3120(2010)
[11] ZHANG, X. X. and SHU, C. W. Maximum-principle-satisfying and positivity-preserving highorder schemes for conservation laws:survey and new developments. Proceedings of the Royal Society A, 467, 2752-2776(2011)
[12] ZHANG, X. X., XIA, Y. H., and SHU, C. W. Maximum-principle-satisfying and positivitypreserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes. Journal of Scientific Computing, 50, 29-62(2012)
[13] ZHANG, X. X. and SHU, C. W. On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. Journal of Computational Physics, 229, 8918-8934(2010)
[14] ZHANG, X. X. and SHU, C. W. Positivity-preserving high order finite difference WENO schemes for compressible Euler equations. Journal of Computational Physics, 231, 2245-2258(2012)
[15] XU, Z. F. Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws:one-dimensional scalar problem. Mathematics of Computation, 83, 2213-2238(2014)
[16] XIONG, T., QIU, J. M., and XU, Z. F. A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows. Journal of Computational Physics, 252, 310-331(2013)
[17] XIONG, T., QIU, J. M., and XU, Z. F. Parametrized positivity preserving flux limiters for the high order finite difference WENO scheme solving compressible Euler equations. Journal of Scientific Computing, 67, 1066-1088(2016)
[18] YANG, P., XIONG, T., QIU, J. M., and XU, Z. F. High order maximum principle preserving finite volume method for convection dominated problems. Journal of Scientific Computing, 67(2), 795-820(2016)
[19] CHRISTLIEB, A. J., LIU, Y., TANG, Q., and XU, Z. F. High order parametrized maximumprinciple-preserving and positivity-preserving WENO schemes on unstructured meshes. Journal of Computational Physics, 281, 334-351(2015)
[20] SHU, C. W. and OSHER, S. Efficient implementation of essentially non-oscillatory shock capturing schemes. Journal of Computational Physics, 77, 439-471(1988)
[21] XIONG, T., QIU, J. M., XU, Z. F., and CHRISTLIB, A. High order maximum principle preserving semi-Lagrangian finite difference WENO schemes for the Vlasov equation. Journal of Computational Physics, 273, 618-639(2014)
[22] GUO, Y., XIONG, T., and SHI, Y. A positivity preserving high order finite volume compactWENO scheme for compressible Euler equations. Journal of Computational Physics, 274, 505-523(2014)

High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

RichHTML

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 9

编辑推荐

Metrics

本文评价

[1]	Shichao ZHENG, Xiaogang DENG, Dongfang WANG. New optimized flux difference schemes for improving high-order weighted compact nonlinear scheme with applications[J]. Applied Mathematics and Mechanics (English Edition), 2021, 42(3): 405-424.
[2]	Yanli QIAO, Xiaoping WANG, Huanying XU, Haitao QI. Numerical analysis for viscoelastic fluid flow with distributed/variable order time fractional Maxwell constitutive models[J]. Applied Mathematics and Mechanics (English Edition), 2021, 42(12): 1771-1786.
[3]	Shitang CUI, Xiaojun NI. Propagation of combined longitudinal and torsional stress waves in a functionally graded thin-walled tube[J]. Applied Mathematics and Mechanics (English Edition), 2021, 42(12): 1717-1732.
[4]	Xiao-dong LI;Min JIANG;Jun-hui GAO;Da-kai LIN;Li LIU;Xiao-yan LI. Recent advances of computational aeroacoustics[J]. Applied Mathematics and Mechanics (English Edition), 2015, 36(1): 131-140.
[5]	吴春秀张鹏 S. C. WONG 乔殿梁戴世强. Solitary wave solution to Aw-Rascle viscous model of traffic flow[J]. Applied Mathematics and Mechanics (English Edition), 2013, 34(4): 523-528.
[6]	罗振东欧秋兰谢正辉. Reduced finite difference scheme and error estimates based on POD method for non-stationary Stokes equation[J]. Applied Mathematics and Mechanics (English Edition), 2011, 32(7): 847-858.
[7]	张红娜宇波王艺魏进家李凤臣. Comments on Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD[J]. Applied Mathematics and Mechanics (English Edition), 2009, 30(5): 669-676.
[8]	罗振东;朱江;谢正辉;张桂芳. DIFFERENCE SCHEME AND NUMERICAL SIMULATION BASED ON MIXED FINITE ELEMENT METHOD FOR NATURAL CONVECTION PROBLEM[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(9): 1100-1110.
[9]	刘儒勋;周朝晖. THE REMAINDER-EFFECT ANALYSIS OF FINITE DIFFERENCE SCHEMES AND THE APPLICATIONS[J]. Applied Mathematics and Mechanics (English Edition), 1995, 16(1): 87-96.