The interface coupling of the gas dynamics equations

Chalons, Christophe; Raviart, Pierre-Arnaud; Seguin, Nicolas

doi:10.1090/S0033-569X-08-01087-X

Authors: Christophe Chalons, Pierre-Arnaud Raviart and Nicolas Seguin
Journal: Quart. Appl. Math. 66 (2008), 659-705
MSC (2000): Primary 76N15, 35F25, 35L65
DOI: https://doi.org/10.1090/S0033-569X-08-01087-X
Published electronically: October 1, 2008
MathSciNet review: 2465140
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Abstract: We investigate the one-dimensional coupling of two systems of gas dynamics at a fixed interface. The coupling constraints consist in requiring the continuity of a system of nonconservative variables at the interface. Since we are dealing with hyperbolic systems, weak coupling conditions are proposed. The existence and the uniqueness of the solutions of the coupled Riemann problem are investigated. Several examples of solutions satisfying the weak coupling conditions are contructed, either continuous or discontinuous with respect to the nonconservative variables at the interface.

References

A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P.-A. Raviart, and N. Seguin. Coupling of multiphase flow models. In Proceedings of the Eleventh International Meeting on Nuclear Thermal-Hydraulics (NURETH), 2005.

A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P.-A. Raviart, and N. Seguin. Homogeneous models with phase transition: Coupling by finite volume methods. In Finite volumes for complex applications, IV (Marrakech, 2005), pages 483–492. Hermes Science, 2005.

A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P.-A. Raviart, and N. Seguin. The coupling of homogeneous models for two-phase flows. Int. J. Finite Volumes, 4(1):1–39, 2007.

A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P.-A. Raviart, and N. Seguin, Extension of interface coupling to general Lagrangian systems, Numerical mathematics and advanced applications, Springer, Berlin, 2006, pp. 852–860. MR 2303716, DOI https://doi.org/10.1007/978-3-540-34288-5_84

A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P.-A. Raviart, and N. Seguin, Coupling of general Lagrangian systems, Math. Comp. 77 (2008), no. 262, 909–941. MR 2373185, DOI https://doi.org/10.1090/S0025-5718-07-02064-9

Rémi Abgrall and Smadar Karni, Computations of compressible multifluids, J. Comput. Phys. 169 (2001), no. 2, 594–623. MR 1836526, DOI https://doi.org/10.1006/jcph.2000.6685

Mapundi K. Banda, Michael Herty, and Axel Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media 1 (2006), no. 2, 295–314. MR 2223073, DOI https://doi.org/10.3934/nhm.2006.1.295

Mapundi K. Banda, Michael Herty, and Axel Klar, Gas flow in pipeline networks, Netw. Heterog. Media 1 (2006), no. 1, 41–56. MR 2219276, DOI https://doi.org/10.3934/nhm.2006.1.41

B. Boutin. Couplage de lois de conservation scalaires par une régularisation à la Dafermos. Master’s thesis, Université Pierre et Marie Curie-Paris 6, 2006.

Florence Bachmann and Julien Vovelle, Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients, Comm. Partial Differential Equations 31 (2006), no. 1-3, 371–395. MR 2209759, DOI https://doi.org/10.1080/03605300500358095

F. Caetano. Sur certains problèmes de linéarisation et de couplage pour les systèmes hyperboliques non linéaires. Ph.D. thesis, Université Pierre et Marie Curie-Paris 6, France, 2006.

Rinaldo M. Colombo and Mauro Garavello, A well posed Riemann problem for the $p$-system at a junction, Netw. Heterog. Media 1 (2006), no. 3, 495–511. MR 2247787, DOI https://doi.org/10.3934/nhm.2006.1.495

J. M. Greenberg and A. Y. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal. 33 (1996), no. 1, 1–16. MR 1377240, DOI https://doi.org/10.1137/0733001

Paola Goatin and Philippe G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 6, 881–902 (English, with English and French summaries). MR 2097035, DOI https://doi.org/10.1016/j.anihpc.2004.02.002

Edwige Godlewski, Kim-Claire Le Thanh, and Pierre-Arnaud Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. II. The case of systems, M2AN Math. Model. Numer. Anal. 39 (2005), no. 4, 649–692. MR 2165674, DOI https://doi.org/10.1051/m2an%3A2005029

Mauro Garavello and Benedetto Piccoli, Traffic flow on networks, AIMS Series on Applied Mathematics, vol. 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. Conservation laws models. MR 2328174

Mauro Garavello and Benedetto Piccoli, Traffic flow on a road network using the Aw-Rascle model, Comm. Partial Differential Equations 31 (2006), no. 1-3, 243–275. MR 2209753, DOI https://doi.org/10.1080/03605300500358053

E. Godlewski and P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. I. The scalar case, Numer. Math. 97 (2004), no. 1, 81–130 (English, with English and French summaries). MR 2045460, DOI https://doi.org/10.1007/s00211-002-0438-5

M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM J. Math. Anal. 38 (2006), no. 2, 595–616. MR 2237163, DOI https://doi.org/10.1137/05062617X

O. Hurisse. Techniques de couplage de modèles hyperboliques en thermohydraulique diphasique. Ph.D. thesis, Université de Provence, France, 2006.

Eli Isaacson and Blake Temple, Nonlinear resonance in systems of conservation laws, SIAM J. Appl. Math. 52 (1992), no. 5, 1260–1278. MR 1182123, DOI https://doi.org/10.1137/0152073

Eli Isaacson and Blake Temple, Convergence of the $2\times 2$ Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math. 55 (1995), no. 3, 625–640. MR 1331577, DOI https://doi.org/10.1137/S0036139992240711

Christian Klingenberg and Nils Henrik Risebro, Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior, Comm. Partial Differential Equations 20 (1995), no. 11-12, 1959–1990. MR 1361727, DOI https://doi.org/10.1080/03605309508821159