Choice of measure source terms in interface coupling for a model problem in gas dynamics

Coquel, Frédéric; Godlewski, Edwige; Haddaoui, Khalil; Marmignon, Claude; Renac, Florent

doi:10.1090/mcom/3063

Choice of measure source terms in interface coupling for a model problem in gas dynamics
HTML articles powered by AMS MathViewer

by Frédéric Coquel, Edwige Godlewski, Khalil Haddaoui, Claude Marmignon and Florent Renac;

Math. Comp. 85 (2016), 2305-2339

DOI: https://doi.org/10.1090/mcom/3063

Published electronically: February 1, 2016

PDF | Request permission

Abstract:

This paper is devoted to the mathematical and numerical analysis of a coupling procedure for one-dimensional Euler systems. The two systems have different closure laws and are coupled through a thin fixed interface. Following the work of Ambroso, Chalons, Coquel and Galié (2004), we propose to couple these systems by a bounded vector-valued Dirac measure, concentrated at the coupling interface, which in the applications may have a physical meaning. We show that the proposed framework allows the control of the coupling conditions and we propose an approximate Riemann solver based on a relaxation approach preserving equilibrium solutions of the coupled problem. Numerical experiments in constrained optimization problems are then presented to assess the performances of the present method.

References

Rémi Abgrall and Smadar Karni, Computations of compressible multifluids, J. Comput. Phys. 169 (2001), no. 2, 594–623. MR 1836526, DOI 10.1006/jcph.2000.6685
Debora Amadori, Laurent Gosse, and Graziano Guerra, Godunov-type approximation for a general resonant balance law with large data, J. Differential Equations 198 (2004), no. 2, 233–274. MR 2038581, DOI 10.1016/j.jde.2003.10.004
Luigi Ambrosio, Alberto Bressan, Dirk Helbing, Axel Klar, and Enrique Zuazua, Modelling and optimisation of flows on networks, Lecture Notes in Mathematics, vol. 2062, Springer, Heidelberg; Fondazione C.I.M.E., Florence, 2013. Lectures from the Centro Internazionale Matematico Estivo (C.I.M.E.) Summer School held in Cetraro, June 15–19, 2009; Edited by Benedetto Piccoli and Michel Rascle; Fondazione CIME/CIME Foundation Subseries. MR 3075424, DOI 10.1007/978-3-642-32160-3
A. Ambroso, C. Chalons, F. Coquel, T. Galié, Interface model coupling via prescribed local flux balance, 18th AIAA Computational Fluid Dynamics Conference, AIAA paper 2007–3822, 2007.
Annalisa Ambroso, Christophe Chalons, Frédéric Coquel, and Thomas Galié, Interface model coupling via prescribed local flux balance, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 3, 895–918. MR 3264339, DOI 10.1051/m2an/2013125
A. Ambroso, C. Chalons, F. Coquel, T. Galié, E. Godlewski, F. Lagoutière, P-A. Raviart, N. Seguin, Numerical coupling of two-phase flows, Recent progress in scientific computing (SCPDE, Hong Kong), 168–178, 2005.
Annalisa Ambroso, Christophe Chalons, Frédéric Coquel, Edwige Godlewski, Frédéric Lagoutière, Pierre-Arnaud Raviart, and Nicolas Seguin, The coupling of homogeneous models for two-phase flows, Int. J. Finite Vol. 4 (2007), no. 1, 39. MR 2465468
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 10.1090/S0025-5718-07-02064-9
A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P.-A. Raviart, and N. Seguin, Relaxation methods and coupling procedures, Internat. J. Numer. Methods Fluids 56 (2008), no. 8, 1123–1129. MR 2393506, DOI 10.1002/fld.1680
A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P-A. Raviart, N. Seguin, J-M. Hérard, Coupling of multiphase flow models, Proceedings of the Eleventh International Meeting on Nuclear Thermal-Hydraulics (NURETH), 2005.
Boris Andreianov, Kenneth Hvistendahl Karlsen, and Nils Henrik Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal. 201 (2011), no. 1, 27–86. MR 2807133, DOI 10.1007/s00205-010-0389-4
Boris Andreianov, Frédéric Lagoutière, Nicolas Seguin, and Takéo Takahashi, Small solids in an inviscid fluid, Netw. Heterog. Media 5 (2010), no. 3, 385–404. MR 2670647, DOI 10.3934/nhm.2010.5.385
Boris Andreianov and Nicolas Seguin, Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes, Discrete Contin. Dyn. Syst. 32 (2012), no. 6, 1939–1964. MR 2885792, DOI 10.3934/dcds.2012.32.1939
M. Arienti, P. Hung, E. Morano, J. E. Shepherd, A level set approach to Eulerian–Lagrangian coupling, J. Comput. Phys., 185, 213–251, 2003.
F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws, J. Statist. Phys. 95 (1999), no. 1-2, 113–170. MR 1705583, DOI 10.1023/A:1004525427365
François Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004. MR 2128209, DOI 10.1007/b93802
François Bouchut, A reduced stability condition for nonlinear relaxation to conservation laws, J. Hyperbolic Differ. Equ. 1 (2004), no. 1, 149–170. MR 2052474, DOI 10.1142/S0219891604000020
B. Boutin, Étude mathématique et numérique d’équations hyperboliques non-linéaires: couplage de modèles et chocs non classiques, PhD thesis, Université Pierre et Marie Curie-Paris VI, France, 2009.
Carlos Castro, Francisco Palacios, and Enrique Zuazua, An alternating descent method for the optimal control of the inviscid Burgers equation in the presence of shocks, Math. Models Methods Appl. Sci. 18 (2008), no. 3, 369–416. MR 2397976, DOI 10.1142/S0218202508002723
C. Chalons and F. Coquel, Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes, Numer. Math. 101 (2005), no. 3, 451–478. MR 2194824, DOI 10.1007/s00211-005-0612-7
Christophe Chalons, Frédéric Coquel, Edwige Godlewski, Pierre-Arnaud Raviart, and Nicolas Seguin, Godunov-type schemes for hyperbolic systems with parameter-dependent source. The case of Euler system with friction, Math. Models Methods Appl. Sci. 20 (2010), no. 11, 2109–2166. MR 2740716, DOI 10.1142/S021820251000488X
Christophe Chalons and Jean-François Coulombel, Relaxation approximation of the Euler equations, J. Math. Anal. Appl. 348 (2008), no. 2, 872–893. MR 2446042, DOI 10.1016/j.jmaa.2008.07.034
Christophe Chalons, Pierre-Arnaud Raviart, and Nicolas Seguin, The interface coupling of the gas dynamics equations, Quart. Appl. Math. 66 (2008), no. 4, 659–705. MR 2465140, DOI 10.1090/S0033-569X-08-01087-X
Gui Qiang Chen, C. David Levermore, and Tai-Ping Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47 (1994), no. 6, 787–830. MR 1280989, DOI 10.1002/cpa.3160470602
F. Coquel, E. Godlewski, B. Perthame, A. In, and P. Rascle, Some new Godunov and relaxation methods for two-phase flow problems, Godunov methods (Oxford, 1999) Kluwer/Plenum, New York, 2001, pp. 179–188. MR 1963591
Frédéric Coquel, Edwige Godlewski, and Nicolas Seguin, Relaxation of fluid systems, Math. Models Methods Appl. Sci. 22 (2012), no. 8, 1250014, 52. MR 2928102, DOI 10.1142/S0218202512500145
Frédéric Coquel and Benoît Perthame, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics, SIAM J. Numer. Anal. 35 (1998), no. 6, 2223–2249. MR 1655844, DOI 10.1137/S0036142997318528
Stefan Diehl, A conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. Appl. Math. 56 (1996), no. 2, 388–419. MR 1381652, DOI 10.1137/S0036139994242425
François Dubois and Philippe LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations 71 (1988), no. 1, 93–122. MR 922200, DOI 10.1016/0022-0396(88)90040-X
E. Duret, Y. Peysson, Q. H. Tran, P. Rouchon, Active bypass to eliminate severe-slugging in multiphase production, Proceedings of the 4th North American Conference on Multiphase Technology, Ban, June 2004, J. Brill, ed., Cranfield, 2004, BHR Group Limited, 205–223.
M-P. Errera, S. Chemin, A fluid-solid thermal coupling applied to an effusion cooling system, 34th AIAA Computational Fluid Dynamics Conference, AIAA paper 2004–2140, 2004.
M. Gad-el-Hak, Flow Control, Appl. Mech. Rev., 42, 261–293, 1989.
M. Gad-el-Hak, D.M. Bushnell, Separation Control: Review, J. Fluids Eng., 113, 5–30, 1991.
T. Galié, Couplage interfacial de modèles en dynamique des fluides. Application aux écoulements diphasiques, PhD thesis, Université Pierre et Marie Curie-Paris VI, France, 2009.
Sergey Gavrilyuk and Richard Saurel, Mathematical and numerical modeling of two-phase compressible flows with micro-inertia, J. Comput. Phys. 175 (2002), no. 1, 326–360. MR 1877822, DOI 10.1006/jcph.2001.6951
James Glimm, The interaction of nonlinear hyperbolic waves, Comm. Pure Appl. Math. 41 (1988), no. 5, 569–590. MR 948072, DOI 10.1002/cpa.3160410505
J. Glimm and D. H. Sharp, An $S$ matrix theory for classical nonlinear physics, Found. Phys. 16 (1986), no. 2, 125–141. MR 836850, DOI 10.1007/BF01889377
Paola Goatin and Philippe G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincaré C Anal. Non Linéaire 21 (2004), no. 6, 881–902 (English, with English and French summaries). MR 2097035, DOI 10.1016/j.anihpc.2004.02.002
Edwige Godlewski, Coupling fluid models. Exploring some features of interfacial coupling, Finite volumes for complex applications V, ISTE, London, 2008, pp. 87–102. MR 2441448
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 10.1051/m2an:2005029
Edwige Godlewski and Pierre-Arnaud Raviart, Hyperbolic systems of conservation laws, Mathématiques & Applications (Paris) [Mathematics and Applications], vol. 3/4, Ellipses, Paris, 1991. MR 1304494
Edwige Godlewski and Pierre-Arnaud Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. MR 1410987, DOI 10.1007/978-1-4612-0713-9
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 10.1007/s00211-002-0438-5
Laurent Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms, Math. Models Methods Appl. Sci. 11 (2001), no. 2, 339–365. MR 1820677, DOI 10.1142/S021820250100088X
Laurent Gosse, Localization effects and measure source terms in numerical schemes for balance laws, Math. Comp. 71 (2002), no. 238, 553–582. MR 1885615, DOI 10.1090/S0025-5718-01-01354-0
Laurent Gosse, Time-splitting schemes and measure source terms for a quasilinear relaxing system, Math. Models Methods Appl. Sci. 13 (2003), no. 8, 1081–1101. MR 1998816, DOI 10.1142/S0218202503002829
Laurent Gosse, Computing qualitatively correct approximations of balance laws, SIMAI Springer Series, vol. 2, Springer, Milan, 2013. Exponential-fit, well-balanced and asymptotic-preserving. MR 3053000, DOI 10.1007/978-88-470-2892-0
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 10.1137/0733001
Graziano Guerra, Well-posedness for a scalar conservation law with singular nonconservative source, J. Differential Equations 206 (2004), no. 2, 438–469. MR 2095821, DOI 10.1016/j.jde.2004.04.008
Jean-Marc Hérard, A rough scheme to couple free and porous media, Int. J. Finite Vol. 3 (2006), no. 2, 28. MR 2465466
J-M. Hérard, O. Hurisse, Coupling two and one-dimensional unsteady Euler equations through a thin interface, Computers & Fluids, 36, 651–666, 2007.
Jean-Marc Hérard and Olivier Hurisse, The numerical coupling of a two-fluid model with an homogeneous relaxation model, Finite volumes for complex applications V, ISTE, London, 2008, pp. 503–510. MR 2451446
Shi Jin and Zhou Ping Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), no. 3, 235–276. MR 1322811, DOI 10.1002/cpa.3160480303
Young-Sam Kwon and Alexis Vasseur, Strong traces for solutions to scalar conservation laws with general flux, Arch. Ration. Mech. Anal. 185 (2007), no. 3, 495–513. MR 2322819, DOI 10.1007/s00205-007-0055-7
Tai-Ping Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), no. 1, 153–175. MR 872145
Hélène Mathis, Clément Cancès, Edwige Godlewski, and Nicolas Seguin, Dynamic model adaptation for multiscale simulation of hyperbolic systems with relaxation, J. Sci. Comput. 63 (2015), no. 3, 820–861. MR 3342726, DOI 10.1007/s10915-014-9915-0
A.M. Mitchell, J. Delery, Research into Vortex Breakdown Control, Progress in Aerospace Sciences, 37, 385–418, 2001.
A. Murrone, P. Villedieu, Numerical modeling of dispersed two-phase flows, Aerospace Lab, 2, 1–13, 2011.
K. Saleh, Analyse et simulation numérique par relaxation d’écoulements diphasiques compressibles. Contribution au traitement des phases évanescentes, PhD thesis, Université Pierre et Marie Curie-Paris 6, France, 2012.
I. Suliciu, On the thermodynamics of rate-type fluids and phase transitions. I. Rate-type fluids, Internat. J. Engrg. Sci. 36 (1998), no. 9, 921–947. MR 1636396, DOI 10.1016/S0020-7225(98)00005-6
Alexis Vasseur, Well-posedness of scalar conservation laws with singular sources, Methods Appl. Anal. 9 (2002), no. 2, 291–312. MR 1957491, DOI 10.4310/MAA.2002.v9.n2.a6
S. Tiwari and A. Klar, An adaptive domain decomposition procedure for Boltzmann and Euler equations, J. Comput. Appl. Math. 90 (1998), no. 2, 223–237. MR 1624342, DOI 10.1016/S0377-0427(98)00027-2