Entropy-satisfying relaxation method with large time-steps for Euler IBVPs

Coquel, Frédéric; Nguyen, Quang; Postel, Marie; Tran, Quang

doi:10.1090/S0025-5718-10-02339-2

Entropy-satisfying relaxation method with large time-steps for Euler IBVPs
HTML articles powered by AMS MathViewer

by Frédéric Coquel, Quang Long Nguyen, Marie Postel and Quang Huy Tran;

Math. Comp. 79 (2010), 1493-1533

DOI: https://doi.org/10.1090/S0025-5718-10-02339-2

Published electronically: February 23, 2010

PDF | Request permission

Abstract:

This paper could have been given the title: “How to positively and implicitly solve Euler equations using only linear scalar advections.” The new relaxation method we propose is able to solve Euler-like systems—as well as initial and boundary value problems—with real state laws at very low cost, using a hybrid explicit-implicit time integration associated with the Arbitrary Lagrangian-Eulerian formalism. Furthermore, it possesses many attractive properties, such as: (i) the preservation of positivity for densities; (ii) the guarantee of min-max principle for mass fractions; (iii) the satisfaction of entropy inequality, under an expressible bound on the CFL ratio. The main feature that will be emphasized is the design of this optimal time-step, which takes into account data not only from the inner domain but also from the boundary conditions.

References

A. A. Amsden, P. J. O’Rourke, and T. D. Butler, KIVA-II: A computer program for chemically reactive flows with sprays, Report LA-11560-MS, Los Alamos National Laboratory, 1989.
N. Andrianov, F. Coquel, M. Postel, and Q. H. Tran, A relaxation multiresolution scheme for accelerating realistic two-phase flows calculations in pipelines, Internat. J. Numer. Methods Fluids 54 (2007), no. 2, 207–236. MR 2313539, DOI 10.1002/fld.1397
Michaël Baudin, Christophe Berthon, Frédéric Coquel, Roland Masson, and Quang Huy Tran, A relaxation method for two-phase flow models with hydrodynamic closure law, Numer. Math. 99 (2005), no. 3, 411–440. MR 2117734, DOI 10.1007/s00211-004-0558-1
Michaël Baudin, Frédéric Coquel, and Quang-Huy Tran, A semi-implicit relaxation scheme for modeling two-phase flow in a pipeline, SIAM J. Sci. Comput. 27 (2005), no. 3, 914–936. MR 2199914, DOI 10.1137/030601624
F. Bouchut, Entropy satisfying flux vector splittings and kinetic BGK models, Numer. Math. 94 (2003), no. 4, 623–672. MR 1990588, DOI 10.1007/s00211-002-0426-9
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
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
Jean-Jacques Chattot and Sylvie Malet, A “box-scheme” for the Euler equations, Nonlinear hyperbolic problems (St. Etienne, 1986) Lecture Notes in Math., vol. 1270, Springer, Berlin, 1987, pp. 82–102. MR 910106, DOI 10.1007/BFb0078319
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
Frédéric Coquel, Edwige Godlewski, Benoît Perthame, Arun In, and P. Rascle, Some new Godunov and relaxation methods for two-phase flow problems, Godunov methods: Theory and Applications (New York) (E. Toro, ed.), Proceedings of the International Conference on Godunov methods in Oxford, 1999, Kluwer Academic/Plenum Publishers, 2001, pp. 179–188.
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
Constantine M. Dafermos, Hyperbolic conservation laws in continuum physics, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Springer-Verlag, Berlin, 2000. MR 1763936, DOI 10.1007/3-540-29089-3_{1}4
Bruno Després and Constant Mazeran, Lagrangian gas dynamics in two dimensions and Lagrangian systems, Arch. Ration. Mech. Anal. 178 (2005), no. 3, 327–372. MR 2196496, DOI 10.1007/s00205-005-0375-4
J. Donea, A. Huerta, J. P. Ponthot, and A. Rodríguez-Ferran, Arbitrary Lagragian-Eulerian methods, Encyclopedia of Computational Mechanics (E. Stein, R. de Borst, and T. Hughes, eds.), vol. 1, John Wiley & Sons, 2004, pp. 413–437.
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
Steinar Evje and Tore Flåtten, Weakly implicit numerical schemes for a two-fluid model, SIAM J. Sci. Comput. 26 (2005), no. 5, 1449–1484. MR 2142581, DOI 10.1137/030600631
I. Faille and E. Heintzé, A rough finite volume scheme for modeling two phase flow in a pipeline, Computers and Fluids 28 (1999), 213–241.
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
Charles Hirsch, Numerical computation of internal and external flows, vol. I & II, Wiley Series in Numerical Methods in Engineering, John Wiley and Sons, New York, 1990.
C. W. Hirt, A. A. Amsden, and J. L. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys. 14 (1974), 227–253.
M. J. Holst, Notes on the KIVA-II software and chemically reactive fluid mechanics, Report UCRL-ID-112019, Lawrence Livermore National Laboratory, California, 1992.
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
Claes Johnson and Anders Szepessy, On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp. 49 (1987), no. 180, 427–444. MR 906180, DOI 10.1090/S0025-5718-1987-0906180-5
Randall J. LeVeque, Numerical methods for conservation laws, 2nd ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. MR 1153252, DOI 10.1007/978-3-0348-8629-1
Tai-Ping Liu, Hyperbolic conservation laws with relaxation, Comm. Math. Phys. 108 (1987), no. 1, 153–175. MR 872145
Jean-Marie Masella, Isabelle Faille, and Thierry Gallouët, On an approximate Godunov scheme, Int. J. Comput. Fluid Dyn. 12 (1999), no. 2, 133–149. MR 1729206, DOI 10.1080/10618569908940819
J. M. Masella, Q. H. Tran, D. Ferré, and C. Pauchon, Transient simulation of two-phase flows in pipes, Int. J. Multiph. Flow 24 (1998), no. 5, 739–755.
Wim A. Mulder and Bram van Leer, Experiments with implicit upwind methods for the Euler equations, J. Comput. Phys. 59 (1985), no. 2, 232–246. MR 796608, DOI 10.1016/0021-9991(85)90144-5
Roberto Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49 (1996), no. 8, 795–823. MR 1391756, DOI 10.1002/(SICI)1097-0312(199608)49:8<795::AID-CPA2>3.0.CO;2-3
Christian Pauchon, Henri Dhulesia, G. Binh-Cirlot, and Jean Fabre, TACITE: A transient tool for multiphase pipeline and well simulation, SPE Annual Technical Conference and Exhibition, New Orleans, September 1994, 1994, SPE Paper 28545.
David L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev. 20 (1978), no. 4, 639–739. MR 508380, DOI 10.1137/1020095
Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779, DOI 10.1007/978-1-4612-0873-0
Eleuterio F. Toro, Riemann solvers and numerical methods for fluid dynamics, Springer-Verlag, Berlin, 1997. A practical introduction. MR 1474503, DOI 10.1007/978-3-662-03490-3
David H. Wagner, Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions, J. Differential Equations 68 (1987), no. 1, 118–136. MR 885816, DOI 10.1016/0022-0396(87)90188-4
Hermann Weyl, Shock waves in arbitrary fluids, Comm. Pure Appl. Math. 2 (1949), 103–122. MR 34677, DOI 10.1002/cpa.3160020201
H. C. Yee, R. F. Warming, and A. Harten, Implicit total variation diminishing (TVD) schemes for steady-state calculations, J. Comput. Phys. 57 (1985), no. 3, 327–360. MR 782986, DOI 10.1016/0021-9991(85)90183-4