An upwind discretization scheme for the finite volume lattice Boltzmann method.

*(English)*Zbl 1177.76329Summary: The fact that the classic lattice Boltzmann method is restricted to Cartesian Grids has inspired several researchers to apply Finite Volume [F. Nannelli and S. Succi [J. Stat. Phys. 68, No. 3–4, 401–407 (1992; Zbl 0925.82036); G. Peng, H. Xi, C. Duncan, S. H. Chou, Finite volume scheme for the lattice Boltzmann method on unstructured meshes. Phys. Rev. E 59, 4675–4682 (1999); H. Chen, Volumetric formulation of the lattice Boltzmann method for fluid dynamics: basic concept. Phys. Rev. E 58, 3955–3963 (1998)] or Finite Element T. Lee, Ch.-L. Lin, J. Comput. Phys. 171, No. 1, 336–356 (2001; Zbl 1017.76043); X. Shi, J. Lin, Zh. Yu, Int. J. Numer. Methods Fluids 42, No. 11, 1249–1261 (2003; Zbl 1033.76046)] methods to the Discrete Boltzmann equation. The finite volume method proposed by Peng et al. works on unstructured grids, thus allowing an increased geometrical flexibility. However, the method suffers from substantial numerical instability compared to the standard LBE models. The computational efficiency of the scheme is not competitive with standard methods.

We propose an alternative way of discretizing the convection operator using an upwind scheme, as opposed to the central scheme described by Peng et al. We apply our method to some test problems in two spatial dimensions to demonstrate the improved stability of the new scheme and the significant improvement in computational efficiency. Comparisons with a lattice Boltzmann solver working on a hierarchical grid were done and we found that currently finite volume methods for the discrete Boltzmann equation are not yet competitive as stand alone fluid solvers.

We propose an alternative way of discretizing the convection operator using an upwind scheme, as opposed to the central scheme described by Peng et al. We apply our method to some test problems in two spatial dimensions to demonstrate the improved stability of the new scheme and the significant improvement in computational efficiency. Comparisons with a lattice Boltzmann solver working on a hierarchical grid were done and we found that currently finite volume methods for the discrete Boltzmann equation are not yet competitive as stand alone fluid solvers.

##### MSC:

76M28 | Particle methods and lattice-gas methods |

76M12 | Finite volume methods applied to problems in fluid mechanics |

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\textit{M. Stiebler} et al., Comput. Fluids 35, No. 8--9, 814--819 (2006; Zbl 1177.76329)

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##### References:

[1] | Nannelli, F.; Succi, S., The lattice Boltzmann equation on irregular lattices, J stat phys, 68, 401-407, (1992) · Zbl 0925.82036 |

[2] | Peng, G.; Xi, H.; Duncan, C.; Chou, S.-H., Finite volume scheme for the lattice Boltzmann method on unstructured meshes, Phys rev E, 59, 4675-4682, (1999) |

[3] | Yu, D.; Mei, R.; Shyy, W., A multi-block lattice Boltzmann method for viscous fluid flows, Int J numer methods fluids, 39, 99-120, (2002) · Zbl 1036.76051 |

[4] | Filippova, O.; Hänel, D., Grid refinement for lattice-BGK models, J comput phys, 147, 219-228, (1998) · Zbl 0917.76061 |

[5] | He, X.; Luo, L.-S.; Dembo, M., Some progress in lattice Boltzmann method: part I. nonuniform mesh grids, J comput phys, 129, 357-363, (1996) · Zbl 0868.76068 |

[6] | Geller S, Krafczyk M, Tölke J, Turek S, Hron J. Benchmark computations based on lattice Boltzmann, finite element and finite volume methods for laminar flows. Comput. Fluids 2005, accepted for publication. · Zbl 1177.76313 |

[7] | Hirsch, C., Numerical computation of internal and external flows, Fundamentals of numerical discretization, vol. 1, (1988), Wiley Chichester |

[8] | Bhatnagar, P.; Gross, E.; Krook, M., A model for collision processes in gases, Phys rev lett, 94, 511-525, (1954) · Zbl 0055.23609 |

[9] | Schäfer, M.; Turek, S., Benchmark computations of laminar flow around cylinder, Notes on numerical fluid mechanics, 52, (1996), Vieweg Verlag Braunschweig, p. 547-66 · Zbl 0874.76070 |

[10] | W. Coirier, An adaptively-refined, cartesian, cell-based scheme for the Euler and Navier-Stokes equations. NASA TM-106754; 1994. |

[11] | Qian, Y.H.; d’Humières, D.; Lallemand, P., Lattice BGK models for Navier-Stokes equation, Europhys lett, 17, 479-484, (1992) · Zbl 1116.76419 |

[12] | He, X.; Luo, L.-S., Lattice Boltzmann model for the incompressible Navier-Stokes equation, J stat phys, 88, 927-944, (1997) · Zbl 0939.82042 |

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