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A Lattice Boltzmann Method for the Advection-Diffusion Equation with Neumann Boundary Conditions

Published online by Cambridge University Press:  03 June 2015

Tobias Gebäck  and
Alexei Heintz
Tobias Gebäck*
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
Alexei Heintz*
Affiliation:
Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
*
Corresponding author.Email:tobias.geback@chalmers.se
Email:heintz@chalmers.se
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Abstract

In this paper, we study a lattice Boltzmann method for the advection-diffusion equation with Neumann boundary conditions on general boundaries. A novel mass conservative scheme is introduced for implementing such boundary conditions, and is analyzed both theoretically and numerically.

Second order convergence is predicted by the theoretical analysis, and numerical investigations show that the convergence is at or close to the predicted rate. The numerical investigations include time-dependent problems and a steady-state diffusion problem for computation of effective diffusion coefficients.

Keywords

76R5076M2865N75Lattice Boltzmanndiffusionadvection-diffusionNeumann boundary condition
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 2 , February 2014 , pp. 487 - 505
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Caiazzo, A.Analysis of lattice boltzmann initialization routines. J. Stat. Phys., 121(1-2):37–48, 2005.CrossRefGoogle Scholar
[2]Chen, H., Chen, S., and Matthaeus, W. H.Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. Phys. Rev. A, 45:R5339–R5342, 1992.Google ScholarPubMed
[3]Chen, S. and Doolen, G. D.Lattice Boltzmann method for fluid flows. Ann. Rev. Fluid Mech., 30:329–364, 1998.Google Scholar
[4]Chen, S. Y., Wang, Z., Shan, X., and Doolen, G. D.Lattice Boltzmann computational fluid dynamics in three dimensions. J. Stat. Phys., 68(3-4):379–400, 1992. Advanced Research Workshop on Lattice Gas Automata (Nice, 1991).CrossRefGoogle Scholar
[5]Geier, M., Greiner, A., and Korvink, J. G.Bubble functions for the lattice Boltzmann method and their application to grid refinement. Eur. Phys. J.–Special Topics, 171:173–179, 2009.CrossRefGoogle Scholar
[6]Ginzburg, I.Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation. Adv. Water Resour., 28(11):1171–1195, 2005.Google Scholar
[7]Ginzburg, I.Generic boundary conditions for lattice Boltzmann models and their application to advection and anisotropic dispersion equations. Adv. Water Resour., 28(11):1196– 1216, 2005.Google Scholar
[8]Ginzburg, I., Verhaeghe, F., and d’Humieres, D.Two-relaxation-time lattice Boltzmann scheme: About parametrization, velocity, pressure and mixed boundary conditions. Commun. Comp. Phys., 3(2):427–478, 2008.Google Scholar
[9]Higuera, F. J., Succi, S., and Benzi, R.Lattice gas dynamics with enhanced collisions. Euro-phys. Lett., 9(4):3451, 1989.Google Scholar
[10]Hiorth, A., Lad, U. H. A., Evje, S., and Skjaeveland, S. M.A lattice Boltzmann-BGK algorithm for a diffusion equation with Robin boundary condition-application to NMR relaxation. Int. J. Numer. Meth. Fluids, 59(4):405–421, 2009.CrossRefGoogle Scholar
[11]Huang, H.-B., Lu, X.-Y., and Sukop, M. C.Numerical study of lattice Boltzmann methods for a convection-diffusion equation coupled with Navier-Stokes equations. Phys, J. A, 44(5):055001, 2011.Google Scholar
[12]Junk, M., Klar, A., and Luo, L.-S.Asymptotic analysis of the lattice Boltzmann equation. J. Comp. Phys., 210:676–704, 2005.Google Scholar
[13]Kang, Q.Simulation of dissolution and precipitation in porous media. J. Geophys. Res., 108(B10):1–10, 2003.Google Scholar
[14]Moriyama, K. and Inamuro, T.Lattice boltzmann simulations of water transport from the gas diffusion layer to the gas channel in pefc. Commun. Comp. Phys., 9:1206–1218, 2011.Google Scholar
[15]Sangani, A. S. and Acrivos, A.The effective conductivity of a periodic array of spheres. Proc. Soc, R. A, 386(1791):263–275, 1983.Google Scholar
[16]Stiebler, M., Tölke, J., and Krafczyk, M.Advection-diffusion lattice Boltzmann scheme for hierarchical grids. Comput. Math. Appl., 55(7):1576–1584, 2008.CrossRefGoogle Scholar
[17]Verhaeghe, F., Arnout, S., Blanpain, B., and Wollants, P.Lattice-Boltzmann modeling of dissolution phenomena. Phys. Rev. E, 73(3):036316, 2006.Google Scholar
[18]Yoshida, H. and Nagaoka, M.Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation. J. Comp. Phys., 229(20):7774–7795, 2010.CrossRefGoogle Scholar
[19]Zhang, X. X., Crawford, J. W., Bengough, A. G., and Young, I. M.On boundary conditions in the lattice Boltzmann model for advection and anisotropic dispersion equation. Adv. Water Resour., 25(6):601–609, 2002.Google Scholar