Abstract
Stability and hydrodynamic behaviors of different lattice Boltzmann models including the lattice Boltzmann equation (LBE), the differential lattice Boltzmann equation (DLBE), the interpolation-supplemented lattice Boltzmann method (ISLBM) and the Taylor series expansion- and least square-based lattice Boltzmann method (TLLBM) are studied in detail. Our work is based on the von Neumann linearized stability analysis under a uniform flow condition. The local stability and hydrodynamic (dissipation) behaviors are studied by solving the evolution operator of the linearized lattice Boltzmann equations numerically. Our investigation shows that the LBE schemes with interpolations, such as DLBE, ISLBM and TLLBM, improve the numerical stability by increasing hyper-viscosities at large wave numbers (small scales). It was found that these interpolated LBE schemes with the upwind interpolations are more stable than those with central interpolations because of much larger hyper-viscosities.
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REFERENCES
U. Frisch, B. Hasslacher, and Y. Pomeau, Lattice-gas automata for the Navier-Stokes equations, Phys.Rev.Lett. 56:1505 (1986).
G. D. Doolen. Lattice Gas Methods for Partial Differential Equations, (Addison-Wesley, MA, 1989).
R. Mei and W. Shyy, On the finite difference-based lattice Boltzmann method in curvilinear coordinates, J.Comp.Phys. 134:306 (1997).
X. He, L-S. Luo, and M. Dembo, Some progress in lattice Boltzmann method. Part I: non-uniform mesh grids, J.Comp.Phys. 129:357 (1996).
C. Shu, Y. T. Chew, and X. D. Niu, Least-square-based lattice Boltzmann method: a meshless approach for simulation of flows with complex geometry, Phys.Rev.E 64:045701 (R) (2001).
Y. T. Chew, C. Shu, and X. D. Niu, A new differential lattice Boltzmann equation and its application to simulate incompressible flows on non-uniform grids, J.Stat.Phys. 107 (1/2):329 (2002).
X. He and L-S. Luo, Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation, Phys.Rev.E 56:6811 (1997).
T. Abe, Derivation of the lattice Boltzmann method by means of the discrete ordinate method for the Boltzmann equation, J.Comp.Phys. 131:241 (1997).
S. Chen and G. D. Doolen, Lattice Boltzmann method for fluid flows, Ann.Rev.Fluid Mech. 30:329 (1998).
J. D. Sterling and S. Chen, “Stability analysis of the lattice Boltzmann methods”, J.Comp.Phys. 123:196 (1996).
W.-A. Yong and L-S. Luo, “Nonexistence of Htheorems for the athermal lattice Boltzmann models with polynomial equilibria”, Phys.Rev.E 67:051105 (2003).
G. R. McNamara and G. Zanetti, Use of the Boltzmann equation to simulate latticegas automata, Phys.Rev.Lett. 61:2332 (1988).
H. Chen, S. Chen, and W. H. Matthaeus, Recovery of the Navier-Stokes equations using lattice-gas Boltzmann method, Phys.Rev.A 45:5339 (1992).
S. Chen, Z. Wang, X. Shan, and G. D. Doolen, Lattice Boltzmann computational fluid dynamics in three dimensions, J.Stat.Phys. 68:379 (1992).
R. A. Worthing, J. Mozer, and G. Seeley, Stability of lattice Boltzmann methods in hydrodynamic regimes, Phys.Rev.E 56:2243 (1997).
O. Behrend, R. Harris, and P. B. Warren, Hydrodynamic behavior of lattice Boltzmann and lattice Bhatnagar-Gross-Krook models, Phys.Rev.E 50:4586 (1994).
P. Lallemand and L.-S. Luo, Theory of the lattice Boltzmann method: Dissipation, Isotropy, Galilean invariance, and stability, Phys.Rev.E 61:6546 (2000).
P. Lallemand and L.-S. Luo, Theory of the lattice Boltzmann method: Acoustic and thermal properties in two and three dimensions, Phys.Rev.E 68:036706 (2003).
S. Succi, G. Amati, and R. Benzi, Challenges in lattice Boltzmann computating, J.Stat.Phys. 81(1/2):5 (1995).
X. He and L.-S. Luo, A priori derivation of the lattice Boltzmann equation, Phys.Rev.E 55:6333 (1997).
L. F. Shampine, Numerical Solution of Ordinary Differential Equations (Chapman & Hall, Newyork, 1994).
J. L. Buchanan and P. R Turner, Numerical Methods and Analysis, (McGraw-Hill, New York, 1992).
K. A. Hoffman and S. T. Chiang, Computational Fluid Dynamics, 3rd ed, (Engineering Education System, Wichita, Kansas, 1998).
S. P. Das, H. J. Bussemaker and M. H. Ernst, “Generalized hydrodynamics and dispersion relations in lattice gases”, Phys.Rev.E 48:245 (1993).
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Niu, X.D., Shu, C., Chew, Y.T. et al. Investigation of Stability and Hydrodynamics of Different Lattice Boltzmann Models. Journal of Statistical Physics 117, 665–680 (2004). https://doi.org/10.1007/s10955-004-2264-x
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DOI: https://doi.org/10.1007/s10955-004-2264-x