Convergence of implicit finite volume methods for scalar conservation laws with discontinuous flux function

Martin, Sébastien; Vovelle, Julien

ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 5, pp. 699-727.

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).

[1] B.P. Andreianov, P. Bénilan and S.N. Kružkov, $L^1$-theory of scalar conservation law with continuous flux function. J. Funct. Anal. 171 (2000) 15-33. | MR | Zbl

[2] C. Bardos, A.-Y. Leroux and J.-C. Nedelec, First order quasilinear equations with boundary conditions. Comm. Partial Diff. Eq. 4 (1979) 1017-1034. | MR | Zbl

[3] G. Bayada, S. Martin and C. Vázquez, About a generalized Buckley-Leverett equation and lubrication multifluid flow. Eur. J. Appl. Math. 17 (2006) 491-524. | MR | Zbl

[4] Ph. Benilan and S.N. Kružkov, Conservation laws with continuous flux functions. NoDEA Nonlinear Differ. Equ. Appl. 3 (1996) 395-419. | MR | Zbl

[5] J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal. 147 (1999) 269-361. | MR | Zbl

[6] J. Carrillo, Conservation laws with discontinuous flux functions and boundary condition. J. Evol. Eq. 3 (2003) 687-705. | MR | Zbl

[7] B. Cockburn, F. Coquel and P.G. Lefloch, Convergence of the finite volume method for multidimensional conservation laws. SIAM J. Numer. Anal. 32 (1995) 775-796. | MR | Zbl

[8] F. Coquel and P. Le Floch, Convergence of finite difference schemes for scalar conservation laws in several space variables. SIAM J. Numer. Anal. 30 (1993) 675-700. | MR | Zbl

[9] B. Després, 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 | Zbl

[10] J.-P. Dias, M. Figueira and J.-F. Rodrigues, Solutions to a scalar discontinuous conservation law in a limit case of phase transitions. J. Math. Fluid Mech. 7 (2005) 153-163. | MR | Zbl

[11] R.J. Diperna, Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal. 88 (1985) 223-270. | MR | Zbl

[12] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis VII, North-Holland, Amsterdam (2000) 713-1020. | MR | Zbl

[13] R. Eymard, S. Mercier and A. Prignet, 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 | Zbl

[14] D. Kröner, M. Rokyta and M. Wierse, A Lax-Wendroff type theorem for upwind finite volume schemes in 2D. East-West J. Numer. Math. 4 (1996) 279-292. | MR | Zbl

[15] S.N. Kružkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) 228-255. | MR | Zbl

[16] R.J. Leveque, Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002). | MR | Zbl

[17] S. Martin and J. Vovelle, Large-time behaviour of the entropy solution of a scalar conservation law with boundary conditions. Quart. Appl. Math. 65 (2007) 425-450. | MR | Zbl

[18] O. Oleĭnik, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. Uspehi Mat. Nauk 14 (1959) 165-170. | MR | Zbl

[19] F. Otto, Initial boundary value problem for a scalar conservation law. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 729-734. | MR | Zbl

[20] A. Szepessy, 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 | Zbl

[21] L. Tartar, 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 | Zbl

[22] J.-P. Vila, 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 | Zbl

[23] A.I. Vol'Pert, Spaces $\mathrm{b}\mathrm{v}$ and quasilinear equations. Mat. Sb. (N.S.) 73 (115) (1967) 255-302. | MR | Zbl

[24] J. Vovelle, Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains. Num. Math. 90 (2002) 563-596. | MR | Zbl