Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T10:16:31.800Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function

Published online by Cambridge University Press:  04 July 2008

Sébastien Martin  and
Julien Vovelle
Sébastien Martin
Affiliation:
Université Paris-Sud XI / Laboratoire de Mathématiques - CNRS UMR 8628, Bât. 425, 91405 Orsay Cedex, France.
Julien Vovelle
Affiliation:
ENS Cachan Antenne de Bretagne / IRMAR - CNRS UMR 6625, Av. R. Schuman, Campus de Ker Lann, 35170 Bruz, France. [email protected]
Get access

Abstract

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).

Keywords

Finite Volume schemeconservation lawdiscontinuous flux.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 5 , September 2008 , pp. 699 - 727
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andreianov, B.P., Bénilan, P. and Kružkov, S.N., L 1-theory of scalar conservation law with continuous flux function. J. Funct. Anal. 171 (2000) 15–33. CrossRef
Bardos, C., Leroux, A.-Y. and Nedelec, J.-C., First order quasilinear equations with boundary conditions. Comm. Partial Diff. Eq. 4 (1979) 1017–1034. CrossRef
Bayada, G., Martin, S. and Vázquez, C., About a generalized Buckley-Leverett equation and lubrication multifluid flow. Eur. J. Appl. Math. 17 (2006) 491–524. CrossRef
Ph. Benilan, S.N. Kružkov, Conservation laws with continuous flux functions. NoDEA Nonlinear Differ. Equ. Appl. 3 (1996) 395–419. CrossRef
Carrillo, J., Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal. 147 (1999) 269–361. CrossRef
Carrillo, J., Conservation laws with discontinuous flux functions and boundary condition. J. Evol. Eq. 3 (2003) 687–705. CrossRef
Cockburn, B., Coquel, F. and LeFloch, P.G., Convergence of the finite volume method for multidimensional conservation laws. SIAM J. Numer. Anal. 32 (1995) 775–796. CrossRef
Coquel, F. and Le Floch, P., Convergence of finite difference schemes for scalar conservation laws in several space variables. SIAM J. Numer. Anal. 30 (1993) 675–700. CrossRef
Després, B., 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). CrossRef
Dias, J.-P., Figueira, M. and Rodrigues, J.-F., Solutions to a scalar discontinuous conservation law in a limit case of phase transitions. J. Math. Fluid Mech. 7 (2005) 153–163. CrossRef
DiPerna, R.J., Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal. 88 (1985) 223–270. CrossRef
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis VII, North-Holland, Amsterdam (2000) 713–1020.
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).
Kröner, D., Rokyta, M. and Wierse, M., Lax-Wendroff, A type theorem for upwind finite volume schemes in 2D. East-West J. Numer. Math. 4 (1996) 279–292.
Kružkov, S.N., First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) 228–255.
R.J. LeVeque, Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002).
Martin, S. and Vovelle, J., Large-time behaviour of the entropy solution of a scalar conservation law with boundary conditions. Quart. Appl. Math. 65 (2007) 425–450. CrossRef
Oleĭnik, O., Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. Uspehi Mat. Nauk 14 (1959) 165–170.
Otto, F., Initial boundary value problem for a scalar conservation law. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 729–734.
Szepessy, A., Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions. RAIRO Modél. Math. Anal. Numér. 25 (1991) 749–782. CrossRef
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.
Convergence, J.-P. Vila 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.
A.I. Vol'pert, Spaces bv and quasilinear equations. Mat. Sb. (N.S.) 73 (115) (1967) 255–302.
Vovelle, J., Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains. Num. Math. 90 (2002) 563–596. CrossRef