Acta mathematica scientia, Series B >
AN ADAPTIVE VERSION OF GLIMM'S SCHEME
Received date: 2009-11-17
Online published: 2010-03-20
Supported by
The work of H. Kim was supported by a Korea Research Foundation Grant from the Korean Government (MOEHRD) (KRF-2007-331-C00053). The work of M. Laforest was supported by the National Science and Engineering Council of Canada and the Canadian Foundation for Innovation.
This article describes a local error estimator for Glimm's scheme for hyperbolic systems of conservation laws and uses it to replace the usual random choice in Glimm's scheme by an optimal choice. As a by-product of the local error estimator, the procedure provides a global error estimator that is shown numerically to be a very accurate estimate of the error in L1(R) for all times. Although there is partial mathematical evidence for the error estimator proposed, at this stage the error estimator must be considered ad-hoc. Nonetheless, the error estimator is simple to compute, relatively inexpensive, without adjustable parameters and at least as accurate as other existing error estimators. Numerical experiments in 1-D for Burgers' equation and for Euler's system are performed to measure the asymptotic accuracy of the resulting scheme and of the error estimator.
H. Kim , M. Laforest , D. Yoon . AN ADAPTIVE VERSION OF GLIMM'S SCHEME[J]. Acta mathematica scientia, Series B, 2010 , 30(2) : 428 -446 . DOI: 10.1016/S0252-9602(10)60057-4
[1] Adjerid A, Flaherty J E, Krivodonova L. A posteriori discontinuous Galerkin error estimation for hyperbolic problems. Comput Methods Appl Mech Engrg, 2002, 191: 1097--1112
[2] Ainsworth M, Oden J T. A Posteriori Error Estimation in Finite Element Analysis. New York: John Wiley & Sons, 2000
[3] Alouges F, De Vuyst F, Le Coq G, Lorin E. The reservoir technique: a way to make godunov schemes zero or very low diffusive. application to collela-glaz solver. Euro J Mech B, 2008, 27(6): 643--664
[4] Becker R, Rannacher R. A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West J Num Math, 1996, 4: 237--264
[5] Bressan A. Global solutions of systems of conservation laws by wave-front tracking. J Math Anal Appl, 1992, 170: 414--432
[6] Bressan A. Hyperbolic Systems of Conservation Laws: The one-dimensonal Cauchy Problem. Volume 20 of Oxford Lecture Series in Mathematics and its Applications. Oxford: Oxford University Press, 2001
[7] Bressan A, Marson A. Error bounds for a deterministic version of the Glimm scheme. Arch Rat Mech Anal, 1998, 142(2): 155--176
[8] Chorin A J. Random choice solution of hyperbolic systems. J Comput Phys, 1976, 22(4): 517--533
[9] Cockburn B, Gau H. A posteriori error estimates of general numerical methods for scalar conservation laws.
Mat Aplic Comp, 1995, 14(1): 37--47
[10] Colella P. Glimm's method for gas dynamics. SIAM J Sci Statist Comput, 1982, 3(1): 76--110
[11] Crasta G, Bressan A, Piccoli B. Well posedness of the Cauchy problem for $n \times n$ systems of conservation laws. Memoirs Amer Math Soc, 2000, 146
[12] Dafermos C. Hyperbolic Conservation Laws in Continuum Pysics. Volume 325 of Grundlehren der mathematischen Wissenschaften. New York: Springer-Verlag, 2000
[13] Dafermos C M. Polygonal approximation of solution to the initial value problem for a conservation law.
J Math Anal Appl, 1972, 38: 33--41
[14] DiPerna R J. Global existence of solutions to nonlinear hyperbolic systems of conservation laws. J Diff Eq, 1976, 20(1): 187--212
[15] Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math, 1965, 18: 697--715
[16] Gosse L, Makridakis C. Two a posteriori error estimates for one-dimensional scalar conservation laws.
SIAM J Numer Anal, 2000, 38(3): 964--988
[17] Harten A, Lax P D. A random choice finite difference scheme for hyperbolic conservation laws. SIAM J Numer Anal, 1981, 18(2): 289--315
[18] Hoff D, Smoller J. Error bounds for Glimm difference approximations for scalar conservation laws. Trans Amer Math Soc, 1985, 289: 611--642
[19] Holden H, Risebro N H. Front Tracking for Hyperbolic Conservation Laws. Volume 152 of Applied Mathematical Sciences. Berlin: Springer-Verlag, 2002
[20] Houston P, Süli E. Adaptive finite element approximation of hyperbolic problems//Barth T J, Deconinck H, ed.
Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics. Volume 25 of Lecture Notes in Computational Sciences and Engineering. Berlin: Springer Verlag, 2003: 269--344
[21] Hu J, LeFloch P G. L1 continuous dependence property for systems of conservation laws. Arch Ration Mech Anal, 2001, 151(1): 45--93
[22] Jiang G, Shu C W. Eficient implementation of weighted ENO schemes. J Comput Phy, 1996, 126: 202--228
[23] Johnson C, Szepessy A. Adaptive finite element methods for conservation laws based on a posteriori error estimates. Comm Pure Appl Math, 1995, 48: 199--234
[24] Kröner D, Ohlberger M. A posteriori error estimates for upwind finite volume schemes for conservation laws in multi dimensions. Math Comput, 1999, 69(229): 25--39
[25] Laforest M. A posteriori error estimate for front-tracking: systems of conservation laws. SIAM J Math Anal, 2004, 35(5): 1347--1370
[26] Laforest M. Mechanisms for error propagation and cancellation in Glimm's scheme without rarefactions. J Hyp Diff Eq, 2007, 4(3): 501--531
[27] Laforest M. An a posteriori error estimate for Glimm's scheme//Proceedings of the 11th {I}nternational Conference on Hyperbolic {P}roblems: Theory, Numerics and Applications. 2008: 643--651
[28] Laforest M. Error estimators for nonlinear conservation laws and entropy production. 2009. In prepapration.
[29] Lax P. Hyperbolic systems of conservation laws II. Comm Pure Appl Math, 1957, 10: 537--566
[30] LeFloch P G. Propagating phase boundaries: formulation of the problem and existence via the Glimm method. Arch Rational Mech Anal, 1993, 123(2): 153--197
[31] Liska R, Wendroff B. Comparison of several difference schemes on 1D and 2D test problems for the Euler equations. SIAM J Sci Comput, 2003, 25(3): 995--1017
[32] Liu T -P. Decay to {N}-wave solutions of general systems of nonlinear hyperbolic conservation laws. Comm Pure Appl Math, 1977, 30: 585--610
[33] Liu T -P. The deterministic version of the Glimm scheme. Comm Math Phys, 1977, 57: 135--148
[34] Liu T -P, Yang T. L1 stability for $2 \times 2$ systems of hyperbolic conservation laws. J Amer Math Soc, 1999, 12(3): 729--774
[35] Lucier B J. Error bounds for the methods of Glimm, Godunov and LeVeque. SIAM J Numer Anal, 1985, 22(6): 1074--1081
[36] Risebro N H. A front-tracking alternative to the random choice method. Proc Amer Math Soc, 1993, 117(4): 1125--1139
[37] Smoller J.Shock Waves and Reaction-Diffusion Equations. New York: Springer-Verlag, 1983
[38] Tadmor E. Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. SIAM J Numer Anal, 1991, 28: 891--906
[39] Yoon D, Kim H J, Hwang W. Adaptive mesh refinement for weighted essentially non-oscillatory schemes.
Bull Korean Math Soc, 2008, 45(4): 781--795
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