×

zbMATH — the first resource for mathematics

A novel lattice Boltzmann model for the Poisson equation. (English) Zbl 1145.82344
Summary: In this paper, a novel lattice Boltzmann model is proposed to solve the Poisson equation through modifying equilibrium distribution function. Compared with previous models, which can be viewed as the solvers to diffusion equation, the present model is a genuine solver to the Poisson equation, and the transient term derived by previous models is eliminated. Numerical solutions agree well with analytical solutions, which indicates the potential of the present model for solving the Poisson equation.

MSC:
82C70 Transport processes in time-dependent statistical mechanics
65N99 Numerical methods for partial differential equations, boundary value problems
PDF BibTeX Cite
Full Text: DOI
References:
[1] He, X.; Ling, N., Lattice Boltzmann simulation of electrochemical systems, Comput. phys. commun., 129, 158-166, (2000) · Zbl 0976.76066
[2] Guo, Z.; Zhao, T.S.; Shi, Y., A lattice Boltzmann algorithm for electro-osmotic flows in microfluidic devices, J. chem. phys., 122, 144907, (2005)
[3] Wang, J.; Wang, M.; Li, Z., Lattice poisson – boltzmann simulations of electro-osmotic flows in microchannels, J. colloid. interface sci., 296, 729-736, (2006)
[4] Chai, Z.; Shi, B., Simulation of electro-osmotic flow in microchannel with lattice Boltzmann method, Phys. lett. A, 364, 183-188, (2007) · Zbl 1203.76125
[5] Siyyam, H.I., An accurate solution of the Poisson equation by the finite difference-Chebyshev-tau method, Appl. math. mech., 22, 8, 0935-0939, (2001)
[6] Boschitsch, A.H.; Fenley, M.O., Hybrid boundary element and finite difference method for solving the nonlinear poisson – boltzmann equation, J. comput. chem., 25, 7, 935-955, (2004)
[7] Cai, Z.; Kim, S.; Kim, S.; Kong, S., A finite element method using singular functions for Poisson equations: mixed boundary conditions, Comput. methods appl. mech. engrg., 195, 2635-2648, (2006) · Zbl 1124.65108
[8] Kikuchi, F.; Saito, H., Remarks on a posteriori error estimation for finite element solutions, J. comput. appl. math., 199, 2, 329-336, (2007) · Zbl 1109.65094
[9] Liao, S.J., General boundary element method for Poisson equation with spatially varying conductivity, Eng. anal. bound. elem., 21, 23-38, (1998) · Zbl 0940.65134
[10] Benzi, R.; Succi, S.; Vergassola, M., The lattice Boltzmann equation: theory and applications, Phys. rep., 222, 147-197, (1992)
[11] Chen, S.; Doolen, G.D., Lattice Boltzmann method for fluid flows, Annu. rev. fluid mech., 30, 329-364, (1998) · Zbl 1398.76180
[12] Chen, H.; Kandasamy, S.; Orszag, S.; Shock, R.; Succi, S.; Yakhot, V., Extended Boltzmann kinetic equation for turbulent flows, Science, 301, 633-636, (2003)
[13] Frisch, U.; Hasslacher, B.; Pomeau, Y., Lattice-gas automata for the navier – stokes equation, Phys. rev. lett., 56, 1505-1508, (1986)
[14] He, X.; Luo, L.S., A priori derivation of the lattice Boltzmann equation, Phys. rev. E, 55, R6333-R6336, (1997)
[15] Shan, X.; He, X., Discretization of the velocity space in the solution of Boltzmann equation, Phys. rev. lett., 80, 65-68, (1998)
[16] Guo, Z.L.; Shi, B.C.; Wang, N.C., Fully Lagrangian and lattice Boltzmann methods for the advection – diffusion equation, J. sci. comput., 14, 3, 291-300, (1999) · Zbl 0971.76073
[17] Deng, B.; Shi, B.C.; Wang, G.C., A new lattice bhatnagar – gross – krook model for convection – diffusion equation with a source term, Chin. phys. lett., 22, 2, 267-270, (2005)
[18] Chen, S.; Dawson, S.P.; Doolen, G.D.; Janecky, D.R.; Lawniczak, A., Lattice methods and their applications to reacting systems, Comput. chem. engrg., 19, 6-7, 617-646, (1995)
[19] Zhang, C.Y.; Tan, H.L.; Liu, M.R.; Kong, L.J., A lattice Boltzmann model and simulation of kdv – burgers equation, Commun. theor. phys., 42, 2, 281-284, (2004) · Zbl 1167.37371
[20] Ma, C., A new lattice Boltzmann model for kdv – burgers equation, Chin. phys. lett., 29, 9, 2313-2315, (2005)
[21] Z. Chai, B. Shi, L. Zheng, A unified lattice Boltzmann model for some nonlinear partial differential equations, Chaos Solitons Fractals, in press, doi:10.1016/j.chaos.2006.07.023. · Zbl 1139.35333
[22] Yan, G., A lattice Boltzmann equation for waves, J. comput. phys., 161, 61-69, (2000) · Zbl 0969.76076
[23] Hirabayashi, M.; Chen, Y.; Ohashi, H., The lattice BGK model for the Poisson equation, JSME int. J. ser. B, 44, 1, 45-52, (2001)
[24] Guo, Z.; Shi, B.; Wang, N., Lattice BGK model for incompressible navier – stokes equation, J. comput. phys., 165, 288-306, (2000) · Zbl 0979.76069
[25] Dawson, S.P.; Chen, S.; Doolen, D.G., Lattice Boltzmann computations for reaction – diffusion equations, J. chem. phys., 98, 2, 1514-1523, (1993)
[26] Averbuch, A.; Vozovoi, L.; Israeli, M., On a fast direct elliptic solver by a modified Fourier method, Numer. algorithms, 15, 287-313, (1997) · Zbl 0892.65069
[27] Guo, Z.L.; Zheng, C.G.; Shi, B.C., Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method, Chin. phys., 11, 4, 366-374, (2002)
[28] Banda, M.K.; Yong, W.A.; Klar, A., A stability notion for lattice Boltzmann equations, SIAM J. sci. comput., 27, 2098-2111, (2006) · Zbl 1100.76051
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.