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

Mathematical Modelling and Numerical Analysis

Volumn 42, Issue 5, 2008, Pages 699-727

  Martin, Sébastien ^a   Vovelle, Julien ^b  

^a UNIVERSITÉ PARIS SACLAY   (France)

^b UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

Conservation law; Discontinuous flux; Finite Volume scheme

Indexed keywords

EID: 49949105926     PISSN: 0764583X     EISSN: 12903841     Source Type: Journal    
DOI: 10.1051/m2an:2008023     Document Type: Article

References (24)

1. 1-theory of scalar conservation law with continuous flux function. J. Funct. Anal. 171 (2000) 15-33.
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.
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.
4. Ph. Benilan and S.N. Kružkov, Conservation laws with continuous flux functions. NoDEA Nonlinear Differ. Equ. Appl. 3 (1996) 395-419.
5. J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal. 147 (1999) 269-361.
6. J. Carrillo, Conservation laws with discontinuous flux functions and boundary condition. J. Evol. Eq. 3 (2003) 687-705.
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.
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.
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).
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.
11. R.J. DiPerna, Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal. 88 (1985) 223-270.
12. R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in Handbook of numerical analysis VII, North-Holland, Amsterdam (2000) 713-1020.
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).
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.
15. S.N. Kružkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) 228-255.
16. R.J. LeVeque, Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002).
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.
18. O. Oleǐnik, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. Uspehi Mat. Nauk 14 (1959) 165-170.
19. F. Otto, Initial boundary value problem for a scalar conservation law. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 729-734.
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.
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.
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.
23. A.I. Vol'pert, Spaces bv and quasilinear equations. Mat. Sb. (NS.) 73 (115) (1967) 255-302.
24. J. Vovelle, Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains. Num. Math. 90 (2002) 563-596.