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Incompressible limits of lattice Boltzmann equations using multiple relaxation
times

P. J. Dellar (2003) *Incompressible limits of lattice Boltzmann
equations using multiple relaxation times* J.
Comput. Phys. **190** 351--370
Preprint available as Adobe PDF ( incompLB.pdf
1106K)

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Abstract

Lattice Boltzmann equations using multiple relaxation times are intended
to be more stable than those using a single relaxation time. The additional
relaxation times may be adjusted to suppress non-hydrodynamic modes that
do not appear directly in the continuum equations, but may contribute to
instabilities on the grid scale. If these relaxation times are fixed in
lattice units, as in previous work, solutions computed on a given lattice
are found to diverge in the incompressible (small Mach number) limit. This
non-existence of an incompressible limit is analysed for an inclined one
dimensional jet. An incompressible limit does exist if the non-hydrodynamic
relaxation times are not fixed, but scaled by the Mach number in the same
way as the hydrodynamic relaxation time that determines the viscosity.
This paper extends the decomposition of the 9 velocity lattice Boltzmann
equation introduced in my earlier paper

P. J. Dellar (2002) *Non-hydrodynamic modes
and *a priori *construction of shallow water lattice Boltzmann equations*
Phys. Rev. E
**65**
036309 (12 pages)

For recent work on multiple relaxation time collision operators see

P. Lallemand and L.-S. Luo (2000) *Theory of the lattice Boltzmann
method: Dispersion, dissipation, isotropy, Galilean invariance, and stability*
Phys. Rev. E **61**
6546--6562 DOI: 10.1103/PhysRevE.61.6546
Also ICASE
Report 2000-17 from NASA Langley Research Center

D. d'Humieres, I. Ginzburg, M. Krafczyk, P. Lallemand, and L.-S. Luo
(2002) *Multiple-relaxation-time lattice Boltzmann models in three dimensions*Phil.
Trans. R. Soc. Lond. A **360 **437-451 DOI: 10.1098/rsta.2001.0955
Also ICASE
Report 2002-20 from NASA Langley Research Center

For a general overview of collision operators in lattice Boltzmann equations
see

S. Succi, I. V. Karlin, and H. Chen (2002) *Role of the H theorem
in lattice Boltzmann hydrodynamic simulations* Rev.
Mod. Phys. **74** 1203-1220 ( cond-mat/0205639
) DOI: 10.1103/RevModPhys.74.1203

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BibTeX citation information:

@article{Dellar03incompLB,

author = "P. J. Dellar",

title = "Incompressible limits of lattice Boltzmann equations using
multiple relaxation times",

year = "2003",

journal = "J. Comput. Phys.",

volume = "190",

pages = "351--370"

}

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