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Analysis of the Lattice Boltzmann Equation

  • Timm Krüger
  • Halim Kusumaatmaja
  • Alexandr Kuzmin
  • Orest Shardt
  • Goncalo Silva
  • Erlend Magnus Viggen
Chapter
Part of the Graduate Texts in Physics book series (GTP)

Abstract

After reading this chapter, you will be familiar with many in-depth aspects of the lattice Boltzmann method. You will have a detailed understanding of how the Chapman-Enskog analysis can be used to determine how the lattice Boltzmann equation and its variations behave on the macroscopic Navier-Stokes level. You will know a number of such variations that result in different macroscopic behaviour from the standard lattice Boltzmann equation. Necessary and sufficient conditions that serve as stability guidelines for lattice Boltzmann simulations will be known to you, along with how to improve the stability of a given simulation. You will also have insight into the accuracy of both general simulations and lattice Boltzmann simulations. For the latter, you will understand what the sources of inaccuracy are, and how they may be reduced or nullified.

Keywords

Truncation Error Knudsen Number Discretisation Error Collision Operator Macroscopic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    S. Chapman, T.G. Cowling, The Mathematical Theory of Non-uniform Gases, 2nd edn. (Cambridge University Press, Cambridge, 1952)zbMATHGoogle Scholar
  2. 2.
    I. Ginzburg, F. Verhaeghe, D. d’Humières, Commun. Comput. Phys. 3, 427 (2008)MathSciNetGoogle Scholar
  3. 3.
    Y.Q. Zu, S. He, Phys. Rev. E 87, 043301 (2013)ADSCrossRefGoogle Scholar
  4. 4.
    G. Silva, V. Semiao, J. Comput. Phys. 269, 259 (2014)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    E.M. Viggen, The lattice Boltzmann method: Fundamentals and acoustics. Ph.D. thesis, Norwegian University of Science and Technology (NTNU), Trondheim (2014)Google Scholar
  6. 6.
    C.M. Bender, S.A. Orszag, Advanced mathematical methods for scientists and engineers (McGraw-Hill, New York, 1978)zbMATHGoogle Scholar
  7. 7.
    Y.H. Qian, S.A. Orszag, Europhys. Lett. 21 (3), 255 (1993)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    S. Geller, M. Krafczyk, J. Tölke, S. Turek, J. Hron, Comput. Fluids 35 (8-9), 888 (2006)CrossRefGoogle Scholar
  9. 9.
    J. Latt, B. Chopard, O. Malaspinas, M. Deville, A. Michler, Phys. Rev. E 77 (5), 056703 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    P.A. Thompson, Compressible-Fluid Dynamics (McGraw-Hill, New York, 1972)zbMATHGoogle Scholar
  11. 11.
    L.E. Kinsler, A.R. Frey, A.B. Coppens, J.V. Sanders, Fundamentals of Acoustics, 4th edn. (Wiley, New York, 2000)Google Scholar
  12. 12.
    P. Dellar, Phys. Rev. E 64 (3) (2001)Google Scholar
  13. 13.
    P. Lallemand, L.S. Luo, Phys. Rev. E 61 (6), 6546 (2000)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    N. Prasianakis, I. Karlin, Phys. Rev. E 76 (1) (2007)Google Scholar
  15. 15.
    I. Ginzburg, J. Stat. Phys. 126, 157 (2007)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    I. Ginzburg, Phys. Rev. E 77, 066704 (2008)ADSCrossRefGoogle Scholar
  17. 17.
    J. Latt, Hydrodynamic limit of lattice Boltzmann equations. Ph.D. thesis, University of Geneva (2007)Google Scholar
  18. 18.
    Y. Sone, Kinetic Theory and Fluid Dynamics (Birkhäuser, Boston, 2002)CrossRefzbMATHGoogle Scholar
  19. 19.
    T. Inamuro, M. Yoshino, F. Ogino, Phys. Fluids 9, 3535 (1997)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    M. Junk, A. Klar, L.S. Luo, J. Comput. Phys. 210, 676 (2005)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    M. Junk, Z. Yang, J. Stat. Phys. 121, 3 (2005)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    H. Grad, Commun. Pure Appl. Maths 2, 331 (1949)MathSciNetCrossRefGoogle Scholar
  23. 23.
    X. Shan, H. X., Phys. Rev. Lett. 80, 65 (1998)Google Scholar
  24. 24.
    X. Shan, X.F. Yuan, H. Chen, J. Fluid Mech. 550, 413 (2006)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    O. Malaspinas, P. Sagaut, J. Fluid Mech. 700, 514 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    E. Ikenberry, C. Truesdell, J. Ration. Mech. Anal. 5, 1 (1956)MathSciNetGoogle Scholar
  27. 27.
    P. Asinari, T. Ohwada, Comp. Math. Appl. 58, 841 (2009)MathSciNetCrossRefGoogle Scholar
  28. 28.
    S. Bennett, P. Asinari, P.J. Dellar, Int. J. Num. Meth. Fluids 69, 171 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    W.A. Yong, W. Zhao, L.S. Luo, Phys. Rev. E 93, 033310 (2016)ADSCrossRefGoogle Scholar
  30. 30.
    F. Dubois, Comp. Math. Appl. 55, 1441 (2008)CrossRefGoogle Scholar
  31. 31.
    D.J. Holdych, D.R. Noble, J.G. Georgiadis, R.O. Buckius, J. Comput. Phys. 193 (2), 595 (2004)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    A.J. Wagner, Phys. Rev. E 74, 056703 (2006)ADSCrossRefGoogle Scholar
  33. 33.
    D. Lycett-Brown, K.H. Luo, Phys. Rev. E 91, 023305 (2015)ADSCrossRefGoogle Scholar
  34. 34.
    A. Caiazzo, M. Junk, M. Rheinländer, Comp. Math. Appl. 58, 883 (2009)CrossRefGoogle Scholar
  35. 35.
    B. Chopard, A. Dupuis, A. Masselot, P. Luthi, Adv. Complex Syst. 05 (02n03), 103 (2002)Google Scholar
  36. 36.
    D.A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models (Springer, New York, 2005)zbMATHGoogle Scholar
  37. 37.
    B. Dünweg, A.J.C. Ladd, in Advances in Polymer Science (Springer, Berlin, Heidelberg, 2008), pp. 1–78Google Scholar
  38. 38.
    E.M. Viggen, Phys. Rev. E 87 (2) (2013)Google Scholar
  39. 39.
    Q. Zou, S. Hou, S. Chen, G.D. Doolen, J. Stat. Phys. 81 (1–2), 35 (1995)ADSCrossRefGoogle Scholar
  40. 40.
    X. He, L.S. Luo, J. Stat. Phys. 88 (3–4), 927 (1997)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    X. Shan, H. Chen, Phys. Rev. E 47 (3), 1815 (1993)ADSCrossRefGoogle Scholar
  42. 42.
    X. Shan, H. Chen, Phys. Rev. E 49 (4), 2941 (1994)ADSCrossRefGoogle Scholar
  43. 43.
    H. Yu, K. Zhao, Phys. Rev. E 61 (4), 3867 (2000)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    J.M. Buick, J.A. Cosgrove, J. Phys. A 39 (44), 13807 (2006)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    A. Kupershtokh, D. Medvedev, D. Karpov, Comput. Math. Appl. 58 (5), 965 (2009)MathSciNetCrossRefGoogle Scholar
  46. 46.
    E.M. Viggen, Phys. Rev. E 90, 013310 (2014)ADSCrossRefGoogle Scholar
  47. 47.
    S. Bennett, A lattice Boltzmann model for diffusion of binary gas mixtures. Ph.D. thesis, University of Cambridge (2010)Google Scholar
  48. 48.
    F. Alexander, H. Chen, S. Chen, G. Doolen, Phys. Rev. A 46 (4), 1967 (1992)ADSCrossRefGoogle Scholar
  49. 49.
    B.J. Palmer, D.R. Rector, J. Comput. Phys. 161 (1), 1 (2000)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    P.J. Dellar, Phys. Rev. E 65 (3) (2002)Google Scholar
  51. 51.
    R. Salmon, J. Mar. Res. 57 (3), 503 (1999)MathSciNetCrossRefGoogle Scholar
  52. 52.
    J.G. Zhou, Comput. Method. Appl. M. 191 (32), 3527 (2002)CrossRefGoogle Scholar
  53. 53.
    S. Li, P. Huang, J. Li, Int. J. Numer. Meth. Fl. 77 (8), 441 (2015)CrossRefGoogle Scholar
  54. 54.
    R. Mei, L.S. Luo, P. Lallemand, D. d’Humières, Comput. Fluids 35 (8-9), 855 (2006)MathSciNetCrossRefGoogle Scholar
  55. 55.
    S. Chen, J. Tölke, M. Krafczyk, Comput. Method. Appl. M. 198 (3-4), 367 (2008)CrossRefGoogle Scholar
  56. 56.
    M. Mendoza, B.M. Boghosian, H.J. Herrmann, S. Succi, Phys. Rev. Lett. 105 (1), 014502 (2010)ADSCrossRefGoogle Scholar
  57. 57.
    M. Mendoza, J.D. Muñoz, Phys. Rev. E 82 (5), 056708 (2010)ADSCrossRefGoogle Scholar
  58. 58.
    J. Chen, Z. Chai, B. Shi, W. Zhang, Comput. Math. Appl. 68 (3), 257 (2014)CrossRefGoogle Scholar
  59. 59.
    D.V. Patil, K.N. Premnath, S. Banerjee, J. Comput. Phys. 265, 172 (2014)ADSMathSciNetCrossRefGoogle Scholar
  60. 60.
    K. Li, C. Zhong, Int. J. Numer. Meth. Fl. 77 (6), 334 (2015)MathSciNetCrossRefGoogle Scholar
  61. 61.
    D.N. Siebert, L.A. Hegele, P.C. Philippi, Phys. Rev. E 77 (2), 026707 (2008)ADSCrossRefGoogle Scholar
  62. 62.
    X.D. Niu, C. Shu, Y.T. Chew, T.G. Wang, J. Stat. Phys. 117 (3–4), 665 (2004)ADSMathSciNetCrossRefGoogle Scholar
  63. 63.
    A. Kuzmin, I. Ginzburg, A. Mohamad, Comp. Math. Appl. 61, 1090 (2011)MathSciNetCrossRefGoogle Scholar
  64. 64.
    J. Hoffmann, Numerical Methods for Engineers and Scientists (McGraw-Hill, New York, 1992)Google Scholar
  65. 65.
    R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady State and Time Dependent Problems (SIAM, Philadelphia, 2007)CrossRefzbMATHGoogle Scholar
  66. 66.
    J.D. Sterling, S. Chen, J. Comput. Phys. 123 (1), 196 (1996)ADSCrossRefGoogle Scholar
  67. 67.
    I. Ginzburg, D. d’Humières, A. Kuzmin, J. Stat. Phys. 139, 1090 (2010)ADSMathSciNetCrossRefGoogle Scholar
  68. 68.
    S. Suga, Int. J. Mod. Phys. C 20, 633 (2009)ADSCrossRefGoogle Scholar
  69. 69.
    X. Aokui, Acta Mech. Sinica 18 (6), 603 (2002)ADSCrossRefGoogle Scholar
  70. 70.
    R. Brownlee, A. Gorban, J. Levesley, Physica A 387, 385 (2008)ADSMathSciNetCrossRefGoogle Scholar
  71. 71.
    F. Tosi, S. Ubertini, S. Succi, H. Chen, I. Karlin, Math. Comp. Simul. 72 (2–6), 227 (2006)MathSciNetCrossRefGoogle Scholar
  72. 72.
    D. d’Humières, I. Ginzburg, Comput. Math. Appl. 58, 823 (2009)MathSciNetCrossRefGoogle Scholar
  73. 73.
    P.A. Skordos, Phys. Rev. E 48 (6), 4823 (1993)ADSMathSciNetCrossRefGoogle Scholar
  74. 74.
    M. Reider, J. Sterling, Comput. Fluids 118, 459 (1995)ADSMathSciNetCrossRefGoogle Scholar
  75. 75.
    R.S. Maier, Int. J. Mod. Phys. C 8, 747 (1997)ADSCrossRefGoogle Scholar
  76. 76.
    I. Ginzburg, Commun. Comput. Phys. 11, 1439 (2012)CrossRefGoogle Scholar
  77. 77.
    P.J. Roache, Verification and Validation in Computational Science and Engineering, 1998, 1st edn. (Hermosa Publishers, New Mexico, 1998)Google Scholar
  78. 78.
    C.J. Roy, J. Comp. Phys. 205, 131 (2005)ADSCrossRefGoogle Scholar
  79. 79.
    J.H. Ferziger, M. Peric, A. Leonard, Computational Methods for Fluid Dynamics, vol. 50, 3rd edn. (Springer, New York, 2002)Google Scholar
  80. 80.
    L.S. Luo, W. Lia, X. Chen, Y. Peng, W. Zhang, Phys. Rev. E 83, 056710 (2011)ADSCrossRefGoogle Scholar
  81. 81.
    R. Verberg, A.J.C. Ladd, Phys. Rev. E 60, 3366 (1999)ADSCrossRefGoogle Scholar
  82. 82.
    M. Bernaschi, S. Succi, H. Chen, J. Stat. Phys. 16, 135 (2001)MathSciNetGoogle Scholar
  83. 83.
    J. Tölke, M. Krafczyk, E. Rank, J. Stat. Phys. 107, 573 (2002)CrossRefGoogle Scholar
  84. 84.
    O. Filippova, D. Hänel, J. Comp. Phys. 165, 407 (2000)ADSCrossRefGoogle Scholar
  85. 85.
    Z. Guo, T.S. Zhao, Y. Shi, Phys. Rev. E 70 (6), 066706 (2004)ADSCrossRefGoogle Scholar
  86. 86.
    S. Izquierdo, N. Fueyo, J. Comput. Phys. 228 (17), 6479 (2009)ADSMathSciNetCrossRefGoogle Scholar
  87. 87.
    S. Khirevich, I. Ginzburg, U. Tallarek, J. Comp. Phys. 281, 708 (2015)ADSMathSciNetCrossRefGoogle Scholar
  88. 88.
    L. Talon, D. Bauer, D. Gland, H. Auradou, I. Ginzburg, Water Resour. Res. 48, W04526 (2012)ADSCrossRefGoogle Scholar
  89. 89.
    B. Servan-Camas, F. Tsai, Adv. Water Res. 31, 1113 (2008)CrossRefGoogle Scholar
  90. 90.
    R.G.M. van der Sman, Comput. Fluids 35, 849 (2006)CrossRefGoogle Scholar
  91. 91.
    I. Ginzburg, G. Silva, L. Talon, Phys. Rev. E 91, 023307 (2015)ADSMathSciNetCrossRefGoogle Scholar
  92. 92.
    P.C. Philippi, L.A. Hegele, L.O.E. Santos, R. Surmas, Phys. Rev. E 73, 056702 (2006)ADSMathSciNetCrossRefGoogle Scholar
  93. 93.
    P. Lallemand, L.S. Luo, Phys. Rev. E 68, 1 (2003)MathSciNetCrossRefGoogle Scholar
  94. 94.
    X. He, G.D. Doolen, T. Clark, J. Comp. Phys. 179, 439 (2002)ADSMathSciNetCrossRefGoogle Scholar
  95. 95.
    A.J. Chorin, J. Comput. Phys 2, 12 (1967)ADSCrossRefGoogle Scholar
  96. 96.
    P.J. Dellar, J. Comp. Phys. 190, 351 (2003)ADSMathSciNetCrossRefGoogle Scholar
  97. 97.
    G. Hazi, C. Jimenez, Comput. Fluids 35, 280–303 (2006)CrossRefGoogle Scholar
  98. 98.
    G. Silva, V. Semiao, J. Fluid Mech. 698, 282 (2012)ADSMathSciNetCrossRefGoogle Scholar
  99. 99.
    M.G. Ancona, J. Comp. Phys. 115, 107 (1994)ADSMathSciNetCrossRefGoogle Scholar
  100. 100.
    S. Ubertini, P. Asinari, S. Succi, Phys. Rev. E 81 (1), 016311 (2010)ADSCrossRefGoogle Scholar
  101. 101.
    S. Marié, D. Ricot, P. Sagaut, J. Comput. Phys. 228 (4), 1056 (2009)ADSMathSciNetCrossRefGoogle Scholar
  102. 102.
    Y. Peng, W. Liao, L.S. Luo, L.P. Wang, Comput. Fluids 39, 568 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Timm Krüger
    • 1
  • Halim Kusumaatmaja
    • 2
  • Alexandr Kuzmin
    • 3
  • Orest Shardt
    • 4
  • Goncalo Silva
    • 5
  • Erlend Magnus Viggen
    • 6
  1. 1.School of Engineering University of EdinburghEdinburghUK
  2. 2.Department of PhysicsDurham UniversityDurhamUK
  3. 3.Maya Heat Transfer TechnologiesWestmountCanada
  4. 4.Department of Mechanical and Aerospace EngineeringPrinceton UniversityPrincetonUSA
  5. 5.IDMEC/IST, University of LisbonLisbonPortugal
  6. 6.Acoustics Research Centre, SINTEF ICTTrondheimNorway

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