Non-negativity and stability analyses of lattice Boltzmann method for advection-diffusion equation.

*(English)*Zbl 1245.76119Summary: Stability is one of the main concerns in lattice Boltzmann method (LBM). The objectives of this study are to investigate the linear stability of lattice Boltzmann equation with the Bhatnagar-Gross-Krook collision operator (LBGK) for the advection-diffusion equation, and to understand the relationship between the stability of LBGK and non-negativity of equilibrium distribution functions (EDFs). This study conducted linear stability analysis on the LBGK, whose stability depends on the lattice Peclet number, the Courant number, the single relaxation time, and the flow direction. The von Neumann analysis was applied to delineate the stability domains by systematically varying these parameters. Moreover, the dimensionless EDFs were analyzed to identify non-negative domains of dimensionless EDFs. As a result, this study obtained linear stability and non-negativity domains for three different lattices with linear and second-order EDFs. It was found that the second-order EDFs have larger stability and non-negativity domains than the linear EDFs, and outperform linear EDFs in terms of stability and numerical dispersion. Furthermore, the non-negativity of the EDFs is a sufficient condition for linear stability and becomes a necessary condition when the relaxation time is very close to 0.5. The stability and non-negativity domains provide useful information to guide the selection of dimensionless parameters to obtain stable LBM solutions. We use mass transport problems to demonstrate the consistence between theoretical findings and LBM solutions.

##### MSC:

76M28 | Particle methods and lattice-gas methods |

76R99 | Diffusion and convection |

80A20 | Heat and mass transfer, heat flow (MSC2010) |

##### Keywords:

mass transport; Bhatnagar-Gross-Krook collision operator; equilibrium distribution functions; von Neumann analysis
PDF
BibTeX
XML
Cite

\textit{B. Servan-Camas} and \textit{F. T. C. Tsai}, J. Comput. Phys. 228, No. 1, 236--256 (2009; Zbl 1245.76119)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Chen, S.; Doolen, G.D., Lattice Boltzmann method for fluid flows, Annual review of fluid mechanics, 30, 1, 329-364, (1998) · Zbl 1398.76180 |

[2] | Rothman, D.H.; Zaleski, S., Lattice-gas cellular automata: simple models of complex hydrodynamics, (1997), Cambridge University Press UK · Zbl 0931.76004 |

[3] | Dawson, S.P.; Chen, S.; Doolen, G.D., Lattice Boltzmann computations for reaction – diffusion equations, Journal of chemical physics, 98, 2, 1514-1523, (1993) |

[4] | Deng, J.Q.; Ghidaoui, M.S.; Gray, W.G.; Xu, K., A Boltzmann-based mesoscopic model for contaminant transport in flow systems, Advances in water resources, 24, 5, 531-550, (2001) |

[5] | Yoshino, M.; Inamuro, T., Lattice Boltzmann simulations for flow and heat/mass transfer problems in a three-dimensional porous structure, International journal of numerical methods in fluids, 43, 2, 183-198, (2003) · Zbl 1032.76661 |

[6] | Wang, J.; Wang, M.; Li, Z., A lattice Boltzmann algorithm for fluid – solid conjugate heat transfer, International journal of thermal sciences, 46, 3, 228-234, (2007) |

[7] | Bhatnagar, P.; P Gross, E.; Krook, M.K., A model for collision processes in gases: small amplitude processes in charged and neutral one component systems, Physical review, 94, 3, 511-525, (1954) · Zbl 0055.23609 |

[8] | Sterling, J.D.; Chen, S., Stability analysis of lattice Boltzmann methods, Journal of computational physics, 123, 196-206, (1996) · Zbl 0840.76078 |

[9] | Worthing, R.A.; Mozer, J.; Seeley, G., Stability of lattice Boltzmann methods in hydrodynamics regimes, Physical review E, 56, 2, 2243-2253, (1997) |

[10] | Karlin, I.V.; Gorban, A.N.; Succi, S.; Boffi, V., Maximum entropy principle for lattice kinetic equations, Physical review letters, 81, 1, 4, (1998) |

[11] | Karlin, I.V.; Ferrante, A.; Otttinger, H.C., Perfect entropy functions of the lattice Boltzmann method, Europhysics letters, 47, 2, 182-188, (1999) |

[12] | Ansumali, S.; Karlin, I.V., Single relaxation time model for entropic lattice Boltzmann methods, Physical review E, 65, 056312, (2002) |

[13] | Boghosian, B.M.; Yepez, J.; Coveney, P.V.; Wagner, A., Entropic lattice Boltzmann methods, Proceedings of the royal society: mathematical, physical and engineering sciences, 457, 2007, 707-766, (2000) · Zbl 0984.76069 |

[14] | Yong, W.-A.; Luo, L.-S., Nonexistence of H theorem for the athermal lattice Boltzmann models with polynomial equilibria, Physical review E, 67, 051105, 1-4, (2003) |

[15] | Chikatamarla, S.S.; Ansumali, S.; Karlin, I.V., Entropic lattice Boltzmann models for hydrodynamics in three dimensions, Physical review letters, 97, 4, (2006) · Zbl 1228.82078 |

[16] | Ansumali, S.; Karlin, I.V., Stabilization of the lattice Boltzmann method by the H theorem: a numerical test, Physical review E, 62, 6, 7999-8003, (2000) |

[17] | Boghosian, B.M.; Love, P.; Yepez, J., Entropic lattice-Boltzmann model for burger’s equation, Philosophical transactions royal society of London A, 362, 1691-1701, (2004) · Zbl 1205.76212 |

[18] | Chikatamarla, S.S.; Karlin, I.V., Entropy and Galilean invariance of lattice Boltzmann theories, Physical review letters, 97, 190601, 1-4, (2006) · Zbl 1228.82079 |

[19] | Li, Y.; Shock, R.; Zhang, R.; Chen, H., Numerical study of flow past an impulsively started cylinder by the lattice-Boltzmann method, Journal of fluid mechanics, 519, 273-300, (2004) · Zbl 1065.76166 |

[20] | Tosi, F.; Ubertini, S.; Succi, S.; Chen, H.; Karlin, I.V., Numerical stability of entropic versus positivity-enforcing lattice Boltzmann schemes, Mathematics and computers in simulation, 72, 227-231, (2006) · Zbl 1116.76068 |

[21] | Brownlee, R.A.; Gorban, A.N.; Levesley, J., Stabilization of the lattice Boltzmann method using the ehrenfests’ coarse-graining idea, Physical review E, 74, 037703, 1-4, (2006) |

[22] | R.A. Brownlee, A.N. Gorban, J. Levesley, Nonequilibrium entropy limiters in lattice Boltzmann methods, Physica A 387, 385-406. |

[23] | Lallemand, P.; Luo, L.-S., Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance and stability, Physical review E, 61, 6, 6546-6562, (2000) |

[24] | McCracken, M.E.; Abraham, J., Multiple-relaxation-time lattice-Boltzmann model for multiphase flow, Physical review E, 71, 036701, (2005) |

[25] | Ginzburgh, I., Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation, Advances in water resources, 28, 11, 1171-1195, (2005) |

[26] | Wolf-Gladrow, D.A., Lattice-gas cellular automata and lattice Boltzmann models: an introduction, (2000), Springer Berlin/Heidelberg · Zbl 0999.82054 |

[27] | Yu, H.; Zhao, K., Lattice Boltzmann method for compressible flows with high Mach numbers, Physical review E, 61, 4, 3867-3870, (2000) |

[28] | Suga, S., Numerical schemes obtained from lattice Boltzmann equations for advection diffusion equations, International journal of modern physics C, 17, 11, 1563-1577, (2006) · Zbl 1121.82302 |

[29] | He, X.; Shan, X.; Doolen, G.D., Discrete Boltzmann equation model for nonideal gases, Physical review E, 57, 1, R13-R16, (1998) |

[30] | Flekkøy, E.G., Lattice bathnagar – gross – krook models for miscible fluids, Physical review E, 47, 6, 4247-4257, (1993) |

[31] | Inamuro, T.; Yoshino, M.; Inoue, H.; Mizuno, R.; Ogino, F., A lattice Boltzmann method for a binary miscible fluid mixture and its application to a heat-transfer problem, Journal of computational physics, 179, 1, 201-215, (2002) · Zbl 1065.76164 |

[32] | Karlin, I.V.; Ansumali, S.; Frouzakis, C.E.; Chikatamarla, S.S., Elements of the lattice Boltzmann method: linear advection equation, Communications in computational physics, 1, 616-655, (2006) · Zbl 1115.76396 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.