Entropy convergence of new two-value scheme with slope relaxation for conservation laws

doi:10.1007/s10483-016-2109-8

Applied Mathematics and Mechanics (English Edition) ›› 2016, Vol. 37 ›› Issue (11): 1551-1570.doi: https://doi.org/10.1007/s10483-016-2109-8

Entropy convergence of new two-value scheme with slope relaxation for conservation laws

Yue WANG¹, Jiequan LI²

1. Institute of Applied Physics and Computational Mathematics, Beijing 100094, China;
2. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100094, China

收稿日期:2016-01-25 修回日期:2016-07-11 出版日期:2016-11-01 发布日期:2016-11-01
通讯作者: Yue WANG E-mail:wangyue@iapcm.ac.cn
基金资助:
Project supported by the National Natural Science Foundation of China (Nos.11371063,11501040,and 91530108) and the Doctoral Program from the Education Ministry of China (No.20130003110004)

Entropy convergence of new two-value scheme with slope relaxation for conservation laws

Yue WANG¹, Jiequan LI²

1. Institute of Applied Physics and Computational Mathematics, Beijing 100094, China;
2. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100094, China

Received:2016-01-25 Revised:2016-07-11 Online:2016-11-01 Published:2016-11-01
Supported by:
Project supported by the National Natural Science Foundation of China (Nos.11371063,11501040,and 91530108) and the Doctoral Program from the Education Ministry of China (No.20130003110004)

摘要/Abstract

摘要：

This paper establishes the entropy convergence of a new two-value high resolution finite volume scheme with slope relaxation for conservation laws.This scheme,motivated by the general method of high resolution schemes that have high-order accuracy in smooth regions of solutions and are free of oscillations near discontinuities,unifies and evolves slopes directly with a slope relaxation equation that governs the evolution of slopes in both smooth and discontinuous regions.Proper choices of slopes are realized adaptively via a relaxation parameter.The scheme is shown to be total-variation-bounded (TVB) stable and satisfies cell-entropy inequalities.

关键词: conservation law, two-value scheme, slope relaxation

Abstract:

Key words: two-value scheme, slope relaxation, conservation law

中图分类号:

Yue WANG, Jiequan LI. Entropy convergence of new two-value scheme with slope relaxation for conservation laws[J]. Applied Mathematics and Mechanics (English Edition), 2016, 37(11): 1551-1570.

参考文献

[1] Toro, E. F. Riemann Solvers and Numerical Methods for Fluid Dynamics (3rd edition), Springer, Berlin (2009)
[2] LeVeque, R. J. Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge (2003)
[3] Nessyahu, H. and Tadmor, E. Non-oscillatory central differencing for hyperbolic conservation laws. Journal of Computational Physics, 87, 408-463(1990)
[4] Cockburn, B. and Shu, C. W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation law. Mathematics of Computation, 52, 411-435(1989)
[5] Ben-Artzi, M. and Falcovitz, J. Generalized Riemann Problems in Computational Gas Dynamics, Cambridge University Press, Cambridge (2003)
[6] Van Leer, B. Towards the ultimate conservative difference scheme V:a second-order sequel to Godunov's method. Journal of Computational Physics, 32, 101-136(1979)
[7] LeVeque, R. J. Numerical Methods for Conservation Laws, Birkhäuser, Basel (2002)
[8] Godlewski, E. and Raviart, P. A. Hyperbolic Systems of Conservation Laws, Mathematics and Applications, Ellipses, Paris (1991)
[9] Lax, P. D. and Wendroff, B. Systems of conservation laws. Communications on Pure and Applied Mathematics, 13, 217-237(1960)
[10] Cauchy, A. Mèmoire surl'emploi du calcul des limites dans l'intègration des èquations aux dèrivèes partielles. Comptes Rendus, 1, 17-58(1842)
[11] Capdeville, G. Towards a compact high-order method for nonlinear hyperbolic systems Ⅱ:the Hermite-HLLC scheme. Journal of Computational Physics, 227, 9428-9462(2008)
[12] Yang, H. On wavewise entropy inequalities for high-resolution schemes I:the semidiscrete case. Mathematics of Computation, 65, 45-67(1996)
[13] Yang, H. On wavewise entropy inequality for high-resolution schemes Ⅱ:fully discrete MUSCL schemes with exact evolution in small time. SIAM Journal on Numerical Analysis, 36, 1-31(1998)
[14] Jiang, G. and Shu, C. On a cell entropy inequality for discontinuous Galerkin methods. Mathematics of Computation, 62, 531-538(1994)
[15] Kurganov, A. and Tadmor, E. New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. Journal of Computational Physics, 160, 241-282(2000)
[16] Li, H., Wang, Z., and Mao, D. K. Numerical neither dissipative nor compressive scheme for linear advection equation and its application to the Euler system. Journal of Scientific Computing, 36, 285-331(2008)
[17] Smoller, J. Shock Waves and Reaction-Diffusion Equations (2rd edition), Springer-Verlag, New York (1994)
[18] Ben-Artzi, M., Falcovitz, J., and Li, J. The convergence of the GRP schemes. Discrete and Continuous Dynamical Systems, 23, 1-27(2009)
[19] Diperna, R. Measure-valued solutions to conservation laws. Archive for Rational Mechanics and Analysis, 88, 223-270(1985)
[20] Osher, S. and Tadmor, E. On the convergence of difference approximations to scalar conservation laws. Mathematics of Computation, 50, 19-51(1988)
[21] Sod, G. A. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. Journal of Computational Physics, 27, 1-31(1978)
[22] Einfeldt, B., Munz, C. D., Roe, P. L., and Sjogreen, B. P. L. On Godunov-type methods near low densities. Journal of Computational Physics, 92, 273-295(1991)
[23] Woodward, P. and Colella, P. The numercial simulation of two-dimensional fluid flow with strong shocks. Journal of Computational Physics, 54, 115-173(1984)

Entropy convergence of new two-value scheme with slope relaxation for conservation laws

Entropy convergence of new two-value scheme with slope relaxation for conservation laws

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	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.
[2]	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.
[3]	M. NADJAFIKHAH R. BAKHSHANDEH CHAMAZKOTI F. AHANGARI. Potential symmetries and conservation laws for generalized quasilinear hyperbolic equations[J]. Applied Mathematics and Mechanics (English Edition), 2011, 32(12): 1607-1614.
[4]	胡伟鹏邓子辰韩松梅范玮. Multi-symplectic Runge-Kutta methods for Landau-Ginzburg-Higgs equation[J]. Applied Mathematics and Mechanics (English Edition), 2009, 30(8): 1027-1034.
[5]	戴天民. RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅵ)—CONSERVATION LAWS OF MASS AND INERTIA[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(12): 1369-1374.
[6]	戴天民. RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅳ)——SURFACE COUSERVATION LAWS[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(11): 1245-1252.
[7]	戴天民. RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅲ) —NOETHER’S THEOREM[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(10): 1134-1140.
[8]	曾文平;黄浪扬;秦孟兆. THE MULTI-SYMPLECTIC ALGORITHM FOR “GOOD” BOUSSINESQ EQUATION[J]. Applied Mathematics and Mechanics (English Edition), 2002, 23(7): 835-841.
[9]	方建会. THE CONSERVATION LAW OF NONHOLONOMIC SYSTEM OF SECOND-ORDER NON-CHETAVE’S TYPE IN EVENT SPACE[J]. Applied Mathematics and Mechanics (English Edition), 2002, 23(1): 89-94.
[10]	戴天民. NEW CONSERVATION LAWS OF ENERGY AND C-D INEQUALITIES IN CONTINUA WITH MICROSTRUCTURE[J]. Applied Mathematics and Mechanics (English Edition), 2001, 22(2): 135-143.
[11]	戴天民. NEW CONSERVATION LAWS OF ENERGY AND C-D INEQUALITIES IN CONTINUA WITHOUT MICROSTRUCTURE[J]. Applied Mathematics and Mechanics (English Edition), 2001, 22(2): 127-134.
[12]	王利民;陈浩然;徐世烺. CONSERVATION LAW AND APPLICATION OF J-INTEGRAL IN MULTI-MATERIALS[J]. Applied Mathematics and Mechanics (English Edition), 2001, 22(10): 1216-1224.
[13]	云天铨. BASIC EQUATIONS, THEORY AND PRINCIPLE OF COMPUTATIONAL STOCK MARKET (Ⅲ)-BASIC THEORIES[J]. Applied Mathematics and Mechanics (English Edition), 2000, 21(8): 861-868.
[14]	方建会. THE CONSERVATION LAW OF SECOND-ORDER NON-HOLONOMIC SYSTEM OF NON-CHETAEV’S TYPE[J]. Applied Mathematics and Mechanics (English Edition), 2000, 21(7): 836-840.
[15]	张毅;梅凤翔. NOETHER’S THEORY OF MECHANICAL SYSTEMS WITH UNILATERAL CONSTRAINTS[J]. Applied Mathematics and Mechanics (English Edition), 2000, 21(1): 59-66.