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MRT and TRT Collision Operators

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The Lattice Boltzmann Method

Part of the book series: Graduate Texts in Physics ((GTP))

Abstract

After reading this chapter, you will have a solid understanding of the general principles of multiple-relaxation-time (MRT) and two-relaxation-time (TRT) collision operators. You will know how to implement these and how to choose the various relaxation times in order to increase the stability, the accuracy, and the possibilities of lattice Boltzmann simulations.

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Notes

  1. 1.

    In the following we will only explain the force-free algorithm; the inclusion of forces is discussed in Sect. 10.5.

  2. 2.

    Alternatively, because of the orthogonality condition (10.12) one can represent each column of \({\boldsymbol M}^{-1}\) through the corresponding row vector of \({\boldsymbol M}\). We leave this as an exercise for an interested reader.

References

  1. I. Ginzburg, F. Verhaeghe, D. d’Humières, Commun. Comput. Phys. 3, 519 (2008)

    MathSciNet  Google Scholar 

  2. A. Kuzmin, Multiphase simulations with lattice Boltzmann scheme. Ph.D. thesis, University of Calgary (2010)

    Google Scholar 

  3. X. He, Q. Zou, L.S. Luo, M. Dembo, J. Stat. Phys. 87 (1–2), 115 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  4. I. Ginzbourg, P.M. Adler, J. Phys. II France 4 (2), 191 (1994)

    Article  Google Scholar 

  5. I. Ginzburg, D. d’Humières, Phys. Rev. E 68, 066614 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  6. S. Khirevich, I. Ginzburg, U. Tallarek, J. Comp. Phys. 281, 708 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  7. D. d’Humières, I. Ginzburg, Comput. Math. Appl. 58, 823 (2009)

    Article  MathSciNet  Google Scholar 

  8. Y.H. Qian, Y. Zhou, Europhys. Lett. 42 (4), 359 (1998)

    Article  ADS  Google Scholar 

  9. P. Lallemand, L.S. Luo, Phys. Rev. E 61 (6), 6546 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  10. P.J. Dellar, J. Comput. Phys. 259, 270 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  11. G. Silva, V. Semiao, J. Comput. Phys. 269, 259 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  12. B. Servan-Camas, F. Tsai, Adv. Water Resour. 31, 1113 (2008)

    Article  ADS  Google Scholar 

  13. I. Ginzburg, Phys. Rev. E 77, 066704 (2008)

    Article  ADS  Google Scholar 

  14. I. Ginzburg, Adv. Water Resour. 28 (11), 1171 (2005)

    Article  ADS  Google Scholar 

  15. J. Latt, Hydrodynamic limit of lattice Boltzmann equations. Ph.D. thesis, University of Geneva (2007)

    Google Scholar 

  16. J. Latt, B. Chopard, Math. Comput. Simul. 72 (2–6), 165 (2006)

    Article  MathSciNet  Google Scholar 

  17. R. Zhang, X. Shan, H. Chen, Phys. Rev. E 74, 046703 (2006)

    Article  ADS  Google Scholar 

  18. A. Montessori, G. Facucci, P. Prestininzi, A. La Rocca, S. Succi, Phys. Rev. E 89, 053317 (2014)

    Article  ADS  Google Scholar 

  19. B.M. Boghosian, J. Yepez, P.V. Coveney, A. Wagner, Proc. R. Soc. A 457 (2007), 717 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  20. S. Ansumali, I.V. Karlin, H.C. Öttinger, Europhys. Lett. 63 (6), 798 (2003)

    Article  ADS  Google Scholar 

  21. M. Geier, A. Greiner, J. Korvink, Phys. Rev. E 73 (066705), 1 (2006)

    Google Scholar 

  22. Y. Ning, K.N. Premnath, D.V. Patil, Int. J. Num. Meth. Fluids 82 (2), 59 (2015)

    Article  MathSciNet  Google Scholar 

  23. M. Geier, M. Schönherr, A. Pasquali, M. Krafczyk, Comput. Math. Appl. 70 (4), 507 (2015)

    Article  MathSciNet  Google Scholar 

  24. I. Karlin, P. Asinari, Physica A 389 (8), 1530 (2010)

    Article  ADS  Google Scholar 

  25. R. Adhikari, S. Succi, Phys. Rev. E 78 (066701), 1 (2008)

    Google Scholar 

  26. P.J. Dellar, J. Comp. Phys. 190, 351 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  27. A. Kuzmin, A. Mohamad, S. Succi, Int. J. Mod. Phys. C 19 (6), 875 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  28. F. Higuera, S. Succi, R. Benzi, Europhys. Lett. 9 (4), 345 (1989)

    Article  ADS  Google Scholar 

  29. D. d’Humières, I. Ginzburg, M. Krafczyk, P. Lallemand, L.S. Luo, Phil. Trans. R. Soc. Lond. A 360, 437 (2002)

    Article  ADS  Google Scholar 

  30. P.J. Dellar, Phys. Rev. E 65 (3) (2002)

    Google Scholar 

  31. R. Benzi, S. Succi, M. Vergassola, Phys. Rep. 222 (3), 145 (1992)

    Article  ADS  Google Scholar 

  32. P. Asinari, Phys. Rev. E 77 (056706), 1 (2008)

    Google Scholar 

  33. R. Rubinstein, L.S. Luo, Phys. Rev. E 77 (036709), 1 (2008)

    MathSciNet  Google Scholar 

  34. D.N. Siebert, L.A. Hegele Jr., P.C. Philippi, Phys. Rev. E 77, 026707 (2008)

    Article  ADS  Google Scholar 

  35. P. Asinari, I. Karlin, Phys. Rev. E 81 (016702), 1 (2010)

    Google Scholar 

  36. P. Dellar, Phys. Rev. E 64 (3) (2001)

    Google Scholar 

  37. Z. Guo, C. Zheng, B. Shi, Phys. Rev. E 65, 46308 (2002)

    Article  ADS  Google Scholar 

  38. A. Kuzmin, Z. Guo, A. Mohamad, Phil. Trans. Royal Soc. A 369, 2219 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  39. G. Silva, V. Semiao, J. Fluid Mech. 698, 282 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  40. S. Mukherjee, J. Abraham, Comput. Fluids 36, 1149 (2007)

    Article  Google Scholar 

  41. K. Premnath, J. Abraham, J. Comput. Phys. 224, 539 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  42. P. Lallemand, L.S. Luo, Phys. Rev. E 68, 1 (2003)

    Article  MathSciNet  Google Scholar 

  43. I. Ginzburg, F. Verhaeghe, D. d’Humières, Commun. Comput. Phys. 3, 427 (2008)

    MathSciNet  Google Scholar 

  44. I. Ginzburg, Commun. Comput. Phys. 11, 1439 (2012)

    Article  Google Scholar 

  45. I. Ginzburg, Adv. Water Resour. 28 (11), 1196 (2005)

    Article  ADS  Google Scholar 

  46. I. Ginzburg, D. d’Humières, A. Kuzmin, J. Stat. Phys. 139, 1090 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  47. A. Kuzmin, I. Ginzburg, A. Mohamad, Comp. Math. Appl. 61, 1090 (2011)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Engineering University of Edinburgh, Edinburgh, UK

    Timm Krüger

  2. Department of Physics, Durham University, Durham, UK

    Halim Kusumaatmaja

  3. Maya Heat Transfer Technologies, Westmount, Québec, Canada

    Alexandr Kuzmin

  4. Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, USA

    Orest Shardt

  5. IDMEC/IST, University of Lisbon, Lisbon, Portugal

    Goncalo Silva

  6. Acoustics Research Centre, SINTEF ICT, Trondheim, Norway

    Erlend Magnus Viggen

Authors
  1. Timm Krüger
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  2. Halim Kusumaatmaja
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  3. Alexandr Kuzmin
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  4. Orest Shardt
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  5. Goncalo Silva
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  6. Erlend Magnus Viggen
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Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., Viggen, E.M. (2017). MRT and TRT Collision Operators. In: The Lattice Boltzmann Method. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-44649-3_10

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