Approximation of mono-dimensional hyperbolic systems: a lattice Boltzmann scheme as a relaxation method. (English) Zbl 1310.76145

Summary: We focus on mono-dimensional hyperbolic systems approximated by a particular lattice Boltzmann scheme. The scheme is described in the framework of the multiple relaxation times method and stability conditions are given. An analysis is done to link the scheme with an explicit finite differences approximation of the relaxation method proposed by S. Jin and Z. Xin [Commun. Pure Appl. Math. 48, No. 3, 235–276 (1995; Zbl 0826.65078)]. Several numerical illustrations are given for the transport equation, Burger’s equation, the p-system, and full compressible Euler’s system.


76M28 Particle methods and lattice-gas methods


Zbl 0826.65078
Full Text: DOI HAL


[1] Wang, J.; Wang, D.; Lallemand, P.; Luo, L.-S., Lattice Boltzmann simulations of thermal convective flows in two dimensions, Comput. Math. Appl., 65, 262-286, (2013) · Zbl 1268.76050
[2] Dellar, P. J., Lattice and discrete Boltzmann equations for fully compressible flow, (Bathe, K.-J., Computational Fluid and Solid Mechanics 2005, Proceedings of The Third MIT Conference on Computational Fluid and Solid Mechanics, (2005), Elsevier Amsterdam), 632-635
[3] Dellar, P. J., Two routes from the Boltzmann equation to compressible flow of polyatomic gases, Prog. Comput. Fluid Dyn., 8, 94-96, (2008) · Zbl 1187.76725
[4] Chen, F.; Xu, A.; Zhang, G.; Li, Y., Prandtl number effects in MRT lattice Boltzmann models for shocked and unshocked compressible fluids, Theor. Appl. Mech. Lett., 1, 052004, (2011), (4 pp.)
[5] Chen, F.; Xu, A.; Zhang, G.; Li, Y., Multiple-relaxation-time lattice Boltzmann model for compressible fluids, Phys. Lett. A, 375, 2129-2139, (2011)
[6] Xu, A.; Zhang, G.; Gan, Y.; Chen, F.; Yu, X., Lattice Boltzmann modeling and simulation of compressible flows, Front. Phys., 7, 582-600, (2012)
[7] Giovangigli, V., Multicomponent flow modeling, (1999), Birkhäuser Boston, Erratum at · Zbl 0956.76003
[8] Jin, S.; Xin, Z., The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Commun. Pure Appl. Math., 48, 235-277, (1995) · Zbl 0826.65078
[9] Aregba-Driollet, D.; Natalini, R., Convergence of relaxation schemes for conservation laws, Appl. Anal., 61, 163-193, (1996) · Zbl 0887.65100
[10] Natalini, R.; Hanouzet, B., Weakly coupled systems of quasilinear hyperbolic equations, Differ. Integral Equ., 9, 1279-1292, (1993) · Zbl 0879.35093
[11] Berthelin, F.; Bouchut, F., Relaxation to isentropic gas dynamics for a BGK system with single kinetic entropy, Methods Appl. Anal., 9, 313-327, (2002) · Zbl 1173.82351
[12] Berthelin, F.; Bouchut, F., Kinetic invariant domains and relaxation limit from a BGK model to isentropic gas dynamics, Asymptot. Anal., 31, 153-176, (2002) · Zbl 1032.76064
[13] Pelanti, M.; Bouchut, F., A relaxation method for modeling two-phase shallow granular flows, (Hyperbolic Problems. Theory, Numerics and Applications, Proceedings of the 12th International Conference on Hyperbolic Problems, Proc. Symp. Appl. Math., vol. 67(2), (2009), Univ. Maryland College Park), 835-844 · Zbl 1407.76166
[14] Junk, M., A finite difference interpretation of the lattice Boltzmann method, Numer. Methods Partial Differ. Equ., 17, 383-402, (2001) · Zbl 0987.76082
[15] d’Humière, D., Generalized lattice-Boltzmann equations, (Rarefied Gas Dynamics: Theory and Simulation, Prog. Astronaut. Aeronaut., vol. 159, (1992), AIAA), 450-458
[16] Dubois, F., Une introduction au schéma de Boltzmann sur réseau, ESAIM Proc., 18, 181-215, (2007) · Zbl 1359.76227
[17] Dubois, F., Equivalent partial differential equations of a lattice Boltzmann scheme, Comput. Math. Appl., 55, 1441-1449, (2008) · Zbl 1142.76449
[18] Kružkov, S. N., First order quasilinear equations in several independent variables, Math. USSR Sb., 10, (1970)
[19] Kuznetsov, N. N., Accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation, Ž. Vyčisl. Mat. Mat. Fiz., 16, 1489-1502, (1976) · Zbl 0354.35021
[20] Delarue, F.; Lagoutière, F., Probabilistic analysis of the upwind scheme for transport equations, Arch. Ration. Mech. Anal., 199, 229-268, (2011) · Zbl 1230.65008
[21] Sabac, F., The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws, SIAM J. Numer. Anal., 34, 2306-2318, (1997) · Zbl 0992.65099
[22] Brooks, A. N.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 32, 199-259, (1982) · Zbl 0497.76041
[23] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S. R., Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys., 71, 231-303, (1987) · Zbl 0652.65067
[24] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Eng., 45, 285-312, (1984) · Zbl 0526.76087
[25] Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, (Quarteroni, A., Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lect. Notes Math., vol. 1697, (1998), Springer Berlin, Heidelberg), 325-432 · Zbl 0927.65111
[26] Bouzidi, M.; Firdaouss, M.; Lallemand, P., Momentum transfer of a Boltzmann-lattice fluid with boundaries, Phys. Fluids, 13, 3452-3459, (2001) · Zbl 1184.76068
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