Mathematics of Computation

Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients

Author(s): Laurent Gosse; François James.
Journal: Math. Comp. 69 (2000), 987-1015.
MSC (1991): Primary 65M06, 65M12; Secondary 35F10
Posted: March 1, 2000
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Conservative linear equations arise in many areas of application, including continuum mechanics or high-frequency geometrical optics approximations. This kind of equation admits most of the time solutions which are only bounded measures in the space variable known as duality solutions. In this paper, we study the convergence of a class of finite-difference numerical schemes and introduce an appropriate concept of consistency with the continuous problem. Some basic examples including computational results are also supplied.

1.: F. Bouchut, On zero pressure gas dynamics Advances in Kinetic theory and computing, Selected papers, Ser. Adv. Math. Appl. Sci., 22, 171-190, World Scientific, 1994. MR 96e:76107
2.: F. Bouchut and F. James, Équations de transport unidimensionnelles à coefficients discontinus, C.R. Acad. Sci. Paris, Série I, 320 (1995), 1097-1102. MR 96m:35197
3.: F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients, Nonlinear Analysis, TMA, 32 (1998), n $^\circ$ 7, 891-933. CMP 98:12
4.: F. Bouchut and F. James, Solutions en dualité pour les gaz sans pression, C.R. Acad. Sci. Paris, Série I, 326 (1998), 1073-1078. MR 99e:35180
5.: F. Bouchut and F. James, Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Diff. Eq. 24 (1999), n $^\circ$ 11-12, 2173-2189. CMP 2000:03
6.: F. Bouchut and F. James, Differentiability with respect to initial data for a scalar conservation law, Proceedings of the 7-th Conference on Hyperbolic Problems, Zürich, 1998, M. Fey & R. Jeltsch, Eds., International Series of Numerical Mathematics, 129, Birkhaüser, Bäsel (1999), 113-118. CMP 2000:03
7.: Y. Brenier and S. Osher, The discrete one-sided Lipschitz condition for convex scalar conservation laws, SIAM J. Numer. Anal., 25 (1988), 8-23. MR 89a:65134
8.: E.D. Conway, Generalized Solutions of Linear Differential Equations with Discontinuous Coefficients and the Uniqueness Question for Multidimensional Quasilinear Conservation Laws, J. Math. Anal. Appl., 18 (1967), 238-251. MR 34:6293
9.: G. Dal Maso, P. LeFloch and F. Murat, Definition and Weak Stability of Nonconservative Products, J. Math. Pures Appl., 74 (1995), 483-548. MR 97b:46052
10.: W. E, Y.G. Rykov and Y.G. Sinai, Generalized Variational Principles, Global Weak Solutions and Behavior with Random Initial Data for Systems of Conservation Laws Arising in Adhesion Particle Dynamics, Comm. Math. Phys., 177 (1996), 349-380. MR 98a:82077
11.: E. Fatemi, B. Engquist and S. Osher, Numerical solution of the high frequency asymptotic expansion for the scalar wave equation, J. Comp. Physics, 120 (1995), 145-155. MR 96c:65173
12.: E. Fatemi, B. Engquist and S. Osher, Finite difference methods for geometrical optics and related nonlinear PDEs approximating the high frequency Helmoltz equation, Department report UCLA, March 1995.
13.: B. Engquist and S. Osher, Stable and entropy satisfying approximations for transonic flow calculations, Math. Comp. 34 (1980), 45-75. MR 81b:65082
14.: B. Engquist and O. Runborg, Multi-phase computations in geometrical optics, J. Comput. Appl. Math., 74 (1996), 175-192. MR 97k:78010
15.: E. Godlewski, M. Olazabal and P.-A. Raviart, On the linearization of hyperbolic systems of conservation laws. Application to stability, Équations aux dérivées partielles et applications, articles dédiés à J.-L. Lions, Gauthier-Villars, Paris, 1998, 549-570. MR 99h:35125
16.: E. Grenier, Existence globale pour le système des gaz sans pression C.R. Acad. Sci. Paris, 321 (1995), 171-174. MR 96k:35145
17.: D. Hoff, The Sharp Form of Oleinik's Entropy Condition in Several Space Variables, Trans. Amer. Math. Soc., 276 (1983), 707-714. MR 84b:35080
18.: F. James and M. Sepúlveda, Convergence results for the flux identification in a scalar conservation law, SIAM J. Cont. Opt., 37 (1999), n $^\circ$ 3, 869-891. MR 2000b:65171
19.: H.C. Kranzer and B.L. Keyfitz, A Strictly Hyperbolic System of Conservation Laws Admitting Singular Shocks, Nonlinear Evolution Equations that Change Type, IMA Vol. Math. Appl., 27 (1990), Springer-Verlag. MR 92g:35133
20.: B. Larrouturou, How to preserve the mass fractions positivity when computing multicomponent flows, J. Comp. Phys, 95 (1991), 59-84. MR 92k:76069
21.: P. LeFloch, An existence and uniqueness result for two nonstrictly hyperbolic systems, Nonlinear evolution equations that change type, IMA Vol. Math. Appl., 27 (1990), Springer Verlag. MR 91g:35171
22.: P. LeFloch and Z. Xin, Uniqueness via the Adjoint Problems for Systems of Conservation Laws, Comm. Pure Appl. Math., XLVI (11), 1499-1533 (1993). MR 94h:35153
23.: P.-L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, 69, Pitman, 1982. MR 84a:49038
24.: M. Olazabal, Résolution numérique du système des perturbations linéaires d'un écoulement MHD, Thèse université Paris 6, 1998.
25.: O.A. Oleinik, Discontinuous Solutions of Nonlinear Differential Equations, Amer. Math. Soc. Transl. (2), 26 (1963), 95-172. MR 27:1721
26.: F. Poupaud and M. Rascle, Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients, Comm. Partial Diff. Eq. 22 (1997), 337-358. MR 98e:35111
27.: E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp. 43 (1984), 995-1011. MR 86g:65163
28.: E. Tadmor, Local Error Estimates for Discontinuous Solutions of Nonlinear Hyperbolic Equations, SIAM J. Numer. Anal. 28 (1991), 891-906. MR 92d:35190
29.: D. Tan, T. Zhang and Y. Zheng, Delta shock-waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Diff. Eq. 112 (1994), 1-32. MR 95g:35124
30.: A.I. Volpert, The spaces BV and quasilinear equations, Math. USSR Sb., 2 (1967), n $^\circ$ 2, 225-267. MR 35:7172

Retrieve articles in Mathematics of Computation with MSC (1991): 65M06, 65M12, 35F10

Laurent Gosse
Affiliation: Foundation for Research and Technology Hellas, Institute of applied and Computational Mathematics, P.O. Box 1527, 71110 Heraklion, Crete, Greece
Email: laurent@palamida.math.uch.gr

François James
Affiliation: MAPMO, UMR CNRS 6628, Université d'Orléans, BP 6759, 45067 Orléans Cedex 2, France
Email: james@cmapx.polytechnique.fr

DOI: 10.1090/S0025-5718-00-01185-6
PII: S 0025-5718(00)01185-6
Keywords: Linear conservation equations, duality solutions, finite difference schemes, weak consistency, nonconservative product
Received by editor(s): September 9, 1998
Posted: March 1, 2000
Additional Notes: Work partially supported by TMR project HCL #ERBFMRXCT960033.
Copyright of article: Copyright 2000, American Mathematical Society

Olof Runborg, Some new results in multiphase geometrical optics, M2AN Math. Model. Numer. Anal 34 (2000), 1203-1231. (English)

Stefan Ulbrich, A Sensitivity and Adjoint Calculus for Discontinuous Solutions of Hyperbolic Conservation Laws with Source Terms, SIAM Journal on Control and Optimization 41 (2002), 740-797. (English)

Suncica Canic, Blood flow through compliant vessels after endovascular repair: wall deformations induced by the discontinuous wall properties, Comput. Visual. Sci. 4 (2002), 147-155.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS