Summary
We study the theoretical and numerical coupling of two general hyperbolic conservation laws. The coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model. In order to analyze the convergence of the coupled numerical scheme, we first revisit the approximation of the boundary value problems. We then prove the convergence and characterize the limit solution of the coupled schemes in a few simple but significative coupling situations. The general coupling problem is analyzed for Riemann initial data and illustrated by numerical simulations. Résumé. Nous nous intéressons à une nouvelle forme de couplage de deux systèmes hyperboliques de lois de conservation. Ce couplage assure de façon faible la continuité de la solution à l’interface sans imposer la conservativité du modèle couplé. Pour étudier la convergence du schéma d’approximation numérique, nous commençons par reprendre les résultats concernant l’approximation du problème aux limites. Nous démontrons ensuite la convergence du schéma couplé dans un certain nombre de cas intéressants. Le cas général du couplage est étudié et illustré numériquement pour une donnée initiale de Riemann.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abgrall, R., Karni, S.: Computations of Compressible multifluids. Journal of Compu- tational Physics, 169, 594–623 (2001)
Bardos, C., Le Roux, A.-Y., Nédélec, J.-C.: First order quasilinear equations with boundary conditions. Comm. in Partial Differential equations, 4(9), 1017–1034 (1979)
Benharbit, S., Chalabi, A., Vila, J.-P.: Numerical viscosity and convergence of finite volume methods for conservation laws with boundary conditions. SIAM J. Numer. Anal. 32(3), 775–796 (1995)
Bouchut, F., James, F.: One-dimensional transport equations with discontinuous coefficients. Nonlinear Analysis, TMA 32, 891–933 (1997)
Chainais-Hillairet, C., Grenier, E.: Numerical boundary layers for hyperbolic systems in 1-D. Math. Mod. Num. Anal., 35, 91–106 (2001)
Crandall, M.G., Majda, A.: Monotone difference approximations for scalar conserva- tion laws. Math. of Comp. Anal., 34, 1–21 (1980)
Dubois, F., Le Floch, P.: Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. of Diff. Equations, 71, 93–122 (1988)
Gisclon, M.: Étude des conditions aux limites pour un système strictement hyperbolique via l’approximation parabolique. J. Math. Pures Appl., 75, 485–508 (1996)
Gisclon, M., Serre, D.: Étude des conditions aux limites pour un système hyperbo- lique, via l’approximation parabolique. C. R. A. S., série I, 319, 377–382 (1994)
Gisclon, M., Serre, D.: Conditions aux limites pour un système strictement hyper- bolique fournies par le schéma de Godunov. M 2 AN 31, 359–380 (1997)
Godlewski, E., Raviart, P.-A.: Hyperbolic systems of conservation laws. Mathématiques & Applications 3/4, Ellipses, Paris 1991
Godlewski, E., Raviart, P.-A.: Numerical approximation of hyperbolic systems of conservation laws. Appl. Math. Science 118, Springer, New York 1996
Joseph, K.T., LeFloch, P.: Boundary layers in weak solutions of hyperbolic conserva- tion laws. Arch. Rat. Mech. Anal., 147, 47–88 (1999)
Kan, P.-T., Santos, M., Xin, Z.: Initial-boundary value problem for conservation laws. Comm. Math. Phys. 186, 701–730 (1997)
Le Roux, A.-Y.: Étude du problème mixte pour une équation quasilinéaire du premier ordre. C.R. Acad. Sc. Paris, 285, 351–354 (1977)
Le Roux, A.-Y.: Approximation de quelques problèmes hyperboliques non linéaires, Thesis University of Rennes, France 1979
Pougeard Dulimbert, T.: Extraction de faisceaux d’ions à partir de plasmas neutres : Modélisation et simulation numérique. Thesis University Paris 6, France 2001
Seguin, N., Vovelle, J.: Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients, Finite Volume for Complex Applications III, R. Herbin and D. Kröner Eds, Hermes Penton Science 429–436 (2002)
Serre, D.: Systèmes de lois de conservation I and II, Diderot éditeur, Paris 1996
Serre, D.: Couches limites non caractéristiques pour les systèmes de lois de conservation, Preprint 2001
Tadmor, E.: Numerical viscosity and entropy condition for conservative difference schemes. Mathematics of computations, 43, 369–381 (1984)
Tang, H.S., Zhou, T.: On nonconservative algorithms for grid interfaces. SIAM J. Numer. Anal., 37, 173–193 (1999)
Vasseur, A.: Strong traces for solutions to multidimensional scalar conservation laws. Arch. Rat. Mech. Anal. 160, 181–194 (2001)
Vasseur, A.: Well posedness of scalar conservation laws with singular sources. Preprint 2001
Vignal, M.-H.: Schémas volumes finis pour des équations elliptiques ou hyperboliques avec conditions aux limites, convergence et estimation d’erreur. Thesis ENS of Lyon, France 1997
Vovelle, J.: Convergence of finite volume monotone scheme for scalar conservation law on bounded domain (preprint 2000) accepted for publication in Numerische Mathematik.
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000):65M12, 65M30, 76M12, 35L50, 35L65
24 December 2001
Rights and permissions
About this article
Cite this article
Godlewski, E., Raviart, PA. The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: I. The scalar case. Numer. Math. 97, 81–130 (2004). https://doi.org/10.1007/s00211-002-0438-5
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-002-0438-5