Lattice-Boltzmann type relaxation systems and high order relaxation schemes for the incompressible Navier-Stokes equations

Banda, Mapundi; Klar, Axel; Pareschi, Lorenzo; Seaïd, Mohammed

doi:10.1090/S0025-5718-07-02034-0

Lattice-Boltzmann type relaxation systems and high order relaxation schemes for the incompressible Navier-Stokes equations
HTML articles powered by AMS MathViewer

by Mapundi Banda, Axel Klar, Lorenzo Pareschi and Mohammed Seaïd;

Math. Comp. 77 (2008), 943-965

DOI: https://doi.org/10.1090/S0025-5718-07-02034-0

Published electronically: December 17, 2007

PDF | Request permission

Abstract:

A relaxation system based on a Lattice-Boltzmann type discrete velocity model is considered in the low Mach number limit. A third order relaxation scheme is developed working uniformly for all ranges of the mean free path and Mach number. In the incompressible Navier-Stokes limit the scheme reduces to an explicit high order finite difference scheme for the incompressible Navier-Stokes equations based on nonoscillatory upwind discretization. Numerical results and comparisons with other approaches are presented for several test cases in one and two space dimensions.

References

Alexander Kurganov and Doron Levy, A third-order semidiscrete central scheme for conservation laws and convection-diffusion equations, SIAM J. Sci. Comput. 22 (2000), no. 4, 1461–1488. MR 1797891, DOI 10.1137/S1064827599360236
Doron Levy, Gabriella Puppo, and Giovanni Russo, Central WENO schemes for hyperbolic systems of conservation laws, M2AN Math. Model. Numer. Anal. 33 (1999), no. 3, 547–571 (English, with English and French summaries). MR 1713238, DOI 10.1051/m2an:1999152
Doron Levy, Gabriella Puppo, and Giovanni Russo, Compact central WENO schemes for multidimensional conservation laws, SIAM J. Sci. Comput. 22 (2000), no. 2, 656–672. MR 1780619, DOI 10.1137/S1064827599359461
Uri M. Ascher, Steven J. Ruuth, and Raymond J. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math. 25 (1997), no. 2-3, 151–167. Special issue on time integration (Amsterdam, 1996). MR 1485812, DOI 10.1016/S0168-9274(97)00056-1
Claude Bardos, François Golse, and David Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Statist. Phys. 63 (1991), no. 1-2, 323–344. MR 1115587, DOI 10.1007/BF01026608
R. Benzi, S. Succi, and M. Vergassola. The Lattice-Boltzmann equation: Theory and applications. Physics Reports, 222:145–197, 1992.
Russel E. Caflisch, Shi Jin, and Giovanni Russo, Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J. Numer. Anal. 34 (1997), no. 1, 246–281. MR 1445737, DOI 10.1137/S0036142994268090
N. Cao, S. Chen, S. Jin, and D. Martinez. Physical symmetry and lattice symmetry in Lattice Boltzmann methods. Phys. Rev. E, 55:21, 1997.
H. Chen, S. Chen, and W. Matthaeus. Recovery of the Navier-Stokes equations using a Lattice-gas Boltzmann method. Physical Review A, 45:5339–5342, 1992.
Shiyi Chen and Gary D. Doolen, Lattice Boltzmann method for fluid flows, Annual review of fluid mechanics, Vol. 30, Annu. Rev. Fluid Mech., vol. 30, Annual Reviews, Palo Alto, CA, 1998, pp. 329–364. MR 1609606, DOI 10.1146/annurev.fluid.30.1.329
D. Aregba-Driollet, R. Natalini, and S.Q. Tang. Diffusive kinetic explicit schemes for nonlinear degenerate parabolic systems. Quaderno IAC 26, 2000.
A. De Masi, R. Esposito, and J. L. Lebowitz, Incompressible Navier-Stokes and Euler limits of the Boltzmann equation, Comm. Pure Appl. Math. 42 (1989), no. 8, 1189–1214. MR 1029125, DOI 10.1002/cpa.3160420810
D. d’Humières. Generalized Lattice-Boltzmann Equations in: AIAA Rarefied Gas Dynamics: Theory and Applications. Progress in Astronautics and Aeoronautics, 159:450–458, 1992.
Franca Bianco, Gabriella Puppo, and Giovanni Russo, High-order central schemes for hyperbolic systems of conservation laws, SIAM J. Sci. Comput. 21 (1999), no. 1, 294–322. MR 1722134, DOI 10.1137/S1064827597324998
Salvatore Fabio Liotta, Vittorio Romano, and Giovanni Russo, Central schemes for balance laws of relaxation type, SIAM J. Numer. Anal. 38 (2000), no. 4, 1337–1356. MR 1790036, DOI 10.1137/S0036142999363061
P. H. Gaskell and A. K. C. Lau, Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm, Internat. J. Numer. Methods Fluids 8 (1988), no. 6, 617–641. MR 944575, DOI 10.1002/fld.1650080602
L. Giraud, D. d’Humieres, and P. Lallemand. A lattice Boltzmann model for Jeffreys viscoelastic fluid. Europhys. Lett., 42:625–630, 1998.
Giovanni Naldi, Lorenzo Pareschi, and Giuseppe Toscani, Relaxation schemes for partial differential equations and applications to degenerate diffusion problems, Surveys Math. Indust. 10 (2002), no. 4, 315–343. MR 2012453
X. He and L.S. Luo. A-priori derivation of the lattice Boltzmann equation. Phys. Rev. E, 55:6333–6336, 1997.
Xiaoyi He and Li-Shi Luo, Lattice Boltzmann model for the incompressible Navier-Stokes equation, J. Statist. Phys. 88 (1997), no. 3-4, 927–944. MR 1467637, DOI 10.1023/B:JOSS.0000015179.12689.e4
Takaji Inamuro, Masato Yoshino, and Fumimaru Ogino, Accuracy of the lattice Boltzmann method for small Knudsen number with finite Reynolds number, Phys. Fluids 9 (1997), no. 11, 3535–3542. MR 1478126, DOI 10.1063/1.869426
Guang-Shan Jiang and Eitan Tadmor, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput. 19 (1998), no. 6, 1892–1917. MR 1638064, DOI 10.1137/S106482759631041X
Shi Jin and C. David Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys. 126 (1996), no. 2, 449–467. MR 1404381, DOI 10.1006/jcph.1996.0149
Shi Jin, Lorenzo Pareschi, and Giuseppe Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numer. Anal. 35 (1998), no. 6, 2405–2439. MR 1655853, DOI 10.1137/S0036142997315962
Shi Jin, Lorenzo Pareschi, and Giuseppe Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numer. Anal. 35 (1998), no. 6, 2405–2439. MR 1655853, DOI 10.1137/S0036142997315962
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
C. T. Kelley, Iterative methods for linear and nonlinear equations, Frontiers in Applied Mathematics, vol. 16, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. With separately available software. MR 1344684, DOI 10.1137/1.9781611970944
Axel Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit, SIAM J. Numer. Anal. 35 (1998), no. 3, 1073–1094. MR 1619859, DOI 10.1137/S0036142996305558
Axel Klar, Relaxation scheme for a lattice-Boltzmann-type discrete velocity model and numerical Navier-Stokes limit, J. Comput. Phys. 148 (1999), no. 2, 416–432. MR 1669711, DOI 10.1006/jcph.1998.6123
Raz Kupferman and Eitan Tadmor, A fast, high resolution, second-order central scheme for incompressible flows, Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 10, 4848–4852. MR 1453829, DOI 10.1073/pnas.94.10.4848
Alexander Kurganov, Sebastian Noelle, and Guergana Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput. 23 (2001), no. 3, 707–740. MR 1860961, DOI 10.1137/S1064827500373413
Ann S. Almgren, John B. Bell, and William G. Szymczak, A numerical method for the incompressible Navier-Stokes equations based on an approximate projection, SIAM J. Sci. Comput. 17 (1996), no. 2, 358–369. MR 1374285, DOI 10.1137/S1064827593244213
John B. Bell, Phillip Colella, and Harland M. Glaz, A second-order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys. 85 (1989), no. 2, 257–283. MR 1029192, DOI 10.1016/0021-9991(89)90151-4
Alexander Kurganov and Eitan Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys. 160 (2000), no. 1, 241–282. MR 1756766, DOI 10.1006/jcph.2000.6459
Lorenzo Pareschi, Central differencing based numerical schemes for hyperbolic conservation laws with relaxation terms, SIAM J. Numer. Anal. 39 (2001), no. 4, 1395–1417. MR 1870848, DOI 10.1137/S0036142900375906
Alexei A. Medovikov, High order explicit methods for parabolic equations, BIT 38 (1998), no. 2, 372–390. MR 1638136, DOI 10.1007/BF02512373
Haim Nessyahu and Eitan Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys. 87 (1990), no. 2, 408–463. MR 1047564, DOI 10.1016/0021-9991(90)90260-8
L. Pareschi and G. Russo. Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations. In L. Brugnano and D. Trigiante, editors, Recent Trends in Numerical Analysis, pages 269 – 289, 2000.
Roger Peyret and Thomas Darwin Taylor, Computational methods for fluid flow, Springer Series in Computational Physics, Springer-Verlag, New York-Berlin, 1983. MR 681481
Y.H. Qian, D. d’Humieres, and P. Lallemand. Lattice BGK models for the Navier Stokes equation. Europhys. Letters, 17:479–484, 1992.
Richard Sanders and Alan Weiser, A high order staggered grid method for hyperbolic systems of conservation laws in one space dimension, Proceedings of the Eighth International Conference on Computing Methods in Applied Sciences and Engineering (Versailles, 1987), 1989, pp. 91–107. MR 1035749, DOI 10.1016/0045-7825(89)90017-0
X. Shan and X. He. Discretization of the Velocity Space in the Solution of the Boltzmann Equation. Phys. Rev. Letter, 80:65–68, 1998.
Guang-Shan Jiang and Chi-Wang Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126 (1996), no. 1, 202–228. MR 1391627, DOI 10.1006/jcph.1996.0130
Y. Sone. Asymptotic theory of a steady flow of a rarefied gas past bodies for small Knudsen numbers. In R. Gatignol and Soubbaramayer, editors, Advances in Kinetic Theory and Continuum Mechanics, Proceedings of a Symposium Held in Honour of Henri Cabannes (Paris), Springer, pages 19–31, 1990.