This paper deals with the problem of numerical approximation in the Cauchy-Dirichlet problem for a scalar conservation law with a flux function having finitely many discontinuities. The well-posedness of this problem was proved by Carrillo [J. Evol. Eq. 3 (2003) 687-705]. Classical numerical methods do not allow us to compute a numerical solution (due to the lack of regularity of the flux). Therefore, we propose an implicit Finite Volume method based on an equivalent formulation of the initial problem. We show the well-posedness of the scheme and the convergence of the numerical solution to the entropy solution of the continuous problem. Numerical simulations are presented in the framework of Riemann problems related to discontinuous transport equation, discontinuous Burgers equation, discontinuous LWR equation and discontinuous non-autonomous Buckley-Leverett equation (lubrication theory).
@article{M2AN_2008__42_5_699_0,
author = {Martin, S\'ebastien and Vovelle, Julien},
title = {Convergence of implicit finite volume methods for scalar conservation laws with discontinuous flux function},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {42},
year = {2008},
pages = {699-727},
doi = {10.1051/m2an:2008023},
mrnumber = {2454620},
zbl = {1155.65071},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2008__42_5_699_0}
}
Martin, Sébastien; Vovelle, Julien. Convergence of implicit finite volume methods for scalar conservation laws with discontinuous flux function. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) pp. 699-727. doi : 10.1051/m2an:2008023. http://gdmltest.u-ga.fr/item/M2AN_2008__42_5_699_0/
[1] , and , -theory of scalar conservation law with continuous flux function. J. Funct. Anal. 171 (2000) 15-33. | MR 1742856 | Zbl 0944.35048
[2] , and , First order quasilinear equations with boundary conditions. Comm. Partial Diff. Eq. 4 (1979) 1017-1034. | MR 542510 | Zbl 0418.35024
[3] , and , About a generalized Buckley-Leverett equation and lubrication multifluid flow. Eur. J. Appl. Math. 17 (2006) 491-524. | MR 2296026 | Zbl 1127.35044
[4] and , Conservation laws with continuous flux functions. NoDEA Nonlinear Differ. Equ. Appl. 3 (1996) 395-419. | MR 1418588 | Zbl 0961.35088
[5] , Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal. 147 (1999) 269-361. | MR 1709116 | Zbl 0935.35056
[6] , Conservation laws with discontinuous flux functions and boundary condition. J. Evol. Eq. 3 (2003) 687-705. | MR 1980978 | Zbl 1027.35069
[7] , and , Convergence of the finite volume method for multidimensional conservation laws. SIAM J. Numer. Anal. 32 (1995) 775-796. | MR 1335651 | Zbl 0845.65051
[8] and , Convergence of finite difference schemes for scalar conservation laws in several space variables. SIAM J. Numer. Anal. 30 (1993) 675-700. | MR 1220646 | Zbl 0781.65078
[9] , An explicit a priori estimate for a finite volume approximation of linear advection on non-Cartesian grids. SIAM J. Numer. Anal. 42 (2004) 484-504 (electronic). | MR 2084223 | Zbl 1127.65322
[10] , and , Solutions to a scalar discontinuous conservation law in a limit case of phase transitions. J. Math. Fluid Mech. 7 (2005) 153-163. | MR 2177124 | Zbl 1065.35184
[11] , Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal. 88 (1985) 223-270. | MR 775191 | Zbl 0616.35055
[12] , and , Finite volume methods, in Handbook of numerical analysis VII, North-Holland, Amsterdam (2000) 713-1020. | MR 1804748 | Zbl 0981.65095
[13] , and , An implicite finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic Markov processes. J. Comput. Appl. Math. (to appear). | MR 2474631 | Zbl 1158.65008
[14] , and , A Lax-Wendroff type theorem for upwind finite volume schemes in 2D. East-West J. Numer. Math. 4 (1996) 279-292. | MR 1430241 | Zbl 0872.65093
[15] , First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) 228-255. | MR 267257 | Zbl 0215.16203
[16] , Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002). | MR 1925043 | Zbl 1010.65040
[17] and , Large-time behaviour of the entropy solution of a scalar conservation law with boundary conditions. Quart. Appl. Math. 65 (2007) 425-450. | MR 2354881 | Zbl 1142.35512
[18] , Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. Uspehi Mat. Nauk 14 (1959) 165-170. | MR 117408 | Zbl 0132.33303
[19] , Initial boundary value problem for a scalar conservation law. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 729-734. | MR 1387428 | Zbl 0852.35013
[20] , Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions. RAIRO Modél. Math. Anal. Numér. 25 (1991) 749-782. | Numdam | MR 1135992 | Zbl 0751.65061
[21] , Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics, Heriot Watt Symposium 4, Pitman Res. Notes in Math., New York (1979) 136-192. | MR 584398 | Zbl 0437.35004
[22] , Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes. RAIRO Modél. Math. Anal. Numér. 28 (1994) 267-295. | Numdam | MR 1275345 | Zbl 0823.65087
[23] , Spaces and quasilinear equations. Mat. Sb. (N.S.) 73 (115) (1967) 255-302. | MR 216338 | Zbl 0168.07402
[24] , Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains. Num. Math. 90 (2002) 563-596. | MR 1884231 | Zbl 1007.65066