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

HTML articles powered by AMS MathViewer

- by Laurent Gosse and François James PDF
- Math. Comp.
**69**(2000), 987-1015 Request permission

## Abstract:

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.## References

- F. Bouchut,
*On zero pressure gas dynamics*, Advances in kinetic theory and computing, Ser. Adv. Math. Appl. Sci., vol. 22, World Sci. Publ., River Edge, NJ, 1994, pp. 171–190. MR**1323183** - François Bouchut and François James,
*Équations de transport unidimensionnelles à coefficients discontinus*, C. R. Acad. Sci. Paris Sér. I Math.**320**(1995), no. 9, 1097–1102 (French, with English and French summaries). MR**1332618** - F. Bouchut and F. James,
*One-dimensional transport equations with discontinuous coefficients*, Nonlinear Analysis, TMA,**32**(1998), n$^\circ$ 7, 891-933. - François Bouchut and François James,
*Solutions en dualité pour les gaz sans pression*, C. R. Acad. Sci. Paris Sér. I Math.**326**(1998), no. 9, 1073–1078 (French, with English and French summaries). MR**1647170**, DOI 10.1016/S0764-4442(98)80065-7 - 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. - 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. - Yann Brenier and Stanley Osher,
*The discrete one-sided Lipschitz condition for convex scalar conservation laws*, SIAM J. Numer. Anal.**25**(1988), no. 1, 8–23. MR**923922**, DOI 10.1137/0725002 - Edward 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**206474**, DOI 10.1016/0022-247X(67)90054-6 - Gianni Dal Maso, Philippe G. Lefloch, and François Murat,
*Definition and weak stability of nonconservative products*, J. Math. Pures Appl. (9)**74**(1995), no. 6, 483–548. MR**1365258** - Weinan E, Yu. G. Rykov, and Ya. 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), no. 2, 349–380. MR**1384139** - E. Fatemi, B. Engquist, and S. Osher,
*Numerical solution of the high frequency asymptotic expansion for the scalar wave equation*, J. Comput. Phys.**120**(1995), no. 1, 145–155. MR**1345031**, DOI 10.1006/jcph.1995.1154 - 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. - Björn Engquist and Stanley Osher,
*Stable and entropy satisfying approximations for transonic flow calculations*, Math. Comp.**34**(1980), no. 149, 45–75. MR**551290**, DOI 10.1090/S0025-5718-1980-0551290-1 - Björn Engquist and Olof Runborg,
*Multi-phase computations in geometrical optics*, J. Comput. Appl. Math.**74**(1996), no. 1-2, 175–192. TICAM Symposium (Austin, TX, 1995). MR**1430373**, DOI 10.1016/0377-0427(96)00023-4 - Edwige Godlewski, Marina Olazabal, and Pierre-Arnaud Raviart,
*On the linearization of hyperbolic systems of conservation laws. Application to stability*, Équations aux dérivées partielles et applications, Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris, 1998, pp. 549–570. MR**1648240** - Emmanuel Grenier,
*Existence globale pour le système des gaz sans pression*, C. R. Acad. Sci. Paris Sér. I Math.**321**(1995), no. 2, 171–174 (French, with English and French summaries). MR**1345441** - David Hoff,
*The sharp form of Oleĭnik’s entropy condition in several space variables*, Trans. Amer. Math. Soc.**276**(1983), no. 2, 707–714. MR**688972**, DOI 10.1090/S0002-9947-1983-0688972-6 - François James and Mauricio Sepúlveda,
*Convergence results for the flux identification in a scalar conservation law*, SIAM J. Control Optim.**37**(1999), no. 3, 869–891. MR**1680830**, DOI 10.1137/S0363012996272722 - Herbert C. Kranzer and Barbara Lee Keyfitz,
*A strictly hyperbolic system of conservation laws admitting singular shocks*, Nonlinear evolution equations that change type, IMA Vol. Math. Appl., vol. 27, Springer, New York, 1990, pp. 107–125. MR**1074189**, DOI 10.1007/978-1-4613-9049-7_{9} - B. Larrouturou,
*How to preserve the mass fractions positivity when computing compressible multi-component flows*, J. Comput. Phys.**95**(1991), no. 1, 59–84. MR**1112315**, DOI 10.1016/0021-9991(91)90253-H - Philippe LeFloch,
*An existence and uniqueness result for two nonstrictly hyperbolic systems*, Nonlinear evolution equations that change type, IMA Vol. Math. Appl., vol. 27, Springer, New York, 1990, pp. 126–138. MR**1074190**, DOI 10.1007/978-1-4613-9049-7_{1}0 - Philippe LeFloch and Zhou Ping Xin,
*Uniqueness via the adjoint problems for systems of conservation laws*, Comm. Pure Appl. Math.**46**(1993), no. 11, 1499–1533. MR**1239319**, DOI 10.1002/cpa.3160461103 - Pierre-Louis Lions,
*Generalized solutions of Hamilton-Jacobi equations*, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR**667669** - M. Olazabal,
*Résolution numérique du système des perturbations linéaires d’un écoulement MHD*, Thèse université Paris 6, 1998. - O. A. Oleĭnik,
*Discontinuous solutions of non-linear differential equations*, Amer. Math. Soc. Transl. (2)**26**(1963), 95–172. MR**0151737**, DOI 10.1090/trans2/026/05 - F. Poupaud and M. Rascle,
*Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients*, Comm. Partial Differential Equations**22**(1997), no. 1-2, 337–358. MR**1434148**, DOI 10.1080/03605309708821265 - Eitan Tadmor,
*Numerical viscosity and the entropy condition for conservative difference schemes*, Math. Comp.**43**(1984), no. 168, 369–381. MR**758189**, DOI 10.1090/S0025-5718-1984-0758189-X - Eitan Tadmor,
*Local error estimates for discontinuous solutions of nonlinear hyperbolic equations*, SIAM J. Numer. Anal.**28**(1991), no. 4, 891–906. MR**1111445**, DOI 10.1137/0728048 - De Chun Tan, Tong Zhang, and Yu Xi Zheng,
*Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws*, J. Differential Equations**112**(1994), no. 1, 1–32. MR**1287550**, DOI 10.1006/jdeq.1994.1093 - A. I. Vol′pert,
*Spaces $\textrm {BV}$ and quasilinear equations*, Mat. Sb. (N.S.)**73 (115)**(1967), 255–302 (Russian). MR**0216338**

## Additional Information

**Laurent Gosse**- Affiliation: Foundation for Research and Technology Hellas, Institute of applied and Computational Mathematics, P.O. Box 1527, 71110 Heraklion, Crete, Greece
- MR Author ID: 611045
- 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
- Received by editor(s): September 9, 1998
- Published electronically: March 1, 2000
- Additional Notes: Work partially supported by TMR project HCL #ERBFMRXCT960033.
- © Copyright 2000 American Mathematical Society
- Journal: Math. Comp.
**69**(2000), 987-1015 - MSC (1991): Primary 65M06, 65M12; Secondary 35F10
- DOI: https://doi.org/10.1090/S0025-5718-00-01185-6
- MathSciNet review: 1670896