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L2 Convergence of the Lattice Boltzmann Method for One Dimensional Convection-Diffusion-Reaction Equations

Published online by Cambridge University Press:  03 June 2015

Michael Junk  and
Zhaoxia Yang
Michael Junk
Affiliation:
FB Mathematik und Statistik, Universität Konstanz, Postfach D194, 78457 Konstanz, Germany
Zhaoxia Yang*
Affiliation:
FB Mathematik und Statistik, Universität Konstanz, Postfach D194, 78457 Konstanz, Germany
*
*Corresponding author. Email addresses: [email protected] (M. Junk), [email protected] (Z. Yang)
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Abstract

Combining asymptotic analysis and weighted L2 stability estimates, the convergence of lattice Boltzmann methods for the approximation of 1D convection-diffusion-reaction equations is proved. Unlike previous approaches, the proof does not require transformations to equivalent macroscopic equations.

Keywords

65417635Lattice Boltzmann methodconvection-diffusion-reaction equationL2 stabilityL2 convergence
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 5 , May 2015 , pp. 1225 - 1245
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Banda, M. K.Yong, W.-A., and Klar, A.. A stability notion for lattice Boltzmann equations. SIAMJ. Sci. Comput, 27:2098–2111,2006.Google Scholar
[2]Bhatnagar, P.Gross, E., and Krook, M.. A model for collision processes in gases i: small amplitude processes in charged and neutral one-component system. Phys. Rev., 94:511–525, 1954.Google Scholar
[3]Blaak, R. and Sloot, P. M. A.. Lattice dependence of reaction-diffusion in lattice Boltzmann modeling. Comput. Phys. Commun., 129(1–3):256–266,2000.Google Scholar
[4]Caflisch, R.. The fluid dynamic limit of the nonlinear Boltzmann equation. Comm. Pure Appl. Math., 30:651–666,1980.CrossRefGoogle Scholar
[5]Caiazzo, A.Junk, M., and Rheinländer, M.. Comparison of analysis techniques for the lattice Boltzmann method. Comput. Math. Appl., 58:883–897,2009.CrossRefGoogle Scholar
[6]Chen, S. and Doolen, G.D.. Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech., 30:329–364,1998.Google Scholar
[7]Dellacherie, S.. Construction and analysis of lattice Boltzmann methods applied to a 1d convection-diffusion equation. Acta. Appl. Math., 2013.Google Scholar
[8]Ginzburg, I.. Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation. Advances in Water Resources, 28:1171–1195, 2005.CrossRefGoogle Scholar
[9]Junk, M.Klar, A., and Luo, L.-S.. Asymptotic analysis of the lattice Boltzmann equation. J. Comput. Phys., 210:676–704, 2005.Google Scholar
[10]Junk, M. and Yang, Z.. Asymptotic analysis of lattice Boltzmann boundary conditions. J. Stat. Phys., 121:3–35, 2005.Google Scholar
[11]Junk, M. and Yang, Z.. Convergence of lattice Boltzmann methods for Stokes flows in periodic and bounded domains. Comput. Math. Appl., 55(7):1481–1491, 2008.Google Scholar
[12]Junk, M. and Yang, Z.. Convergence of lattice Boltzmann methods for Navier-Stokes flows in periodic and bounded domains. Numerische Mathematik, 112(1):65–87, 2009.CrossRefGoogle Scholar
[13]Junk, M. and Yong, W.-A.. Rigorous Navier-Stokes limit of the lattice Boltzmann equation. Asymp. Anal., 35:165–185, 2003.Google Scholar
[14]Junk, M. and Yong, W.-A.. Weighted L 2 stability of the lattice Boltzmann equation. SIAM J. Numer. Anal., 47:1651–1665, 2009.Google Scholar
[15]Masi, A. D.Esposito, R., and Lebowitz, J. L.. Incompressible Navier-Stokes and Euler limits of the Boltzmann equation. Comm. Pure Appl. Math., 42(8):1189–1214, 1989.Google Scholar
[16]Dawson, S. P., Chen, S., and Doolen, G. D.. Lattice Boltzmann computations for reactiondiffusion equations. The Journal of Chemical Physics, 98(2):1514–1523, 1993.Google Scholar
[17]Rheinländer, M.. Analysis of Lattice-Boltzmann Methods: Asymptotic and Numeric Investigation of a Singularly Perturbed System. Dissertation, 2007.Google Scholar
[18]Shi, B. and Guo, Z.. Lattice Boltzmann model for nonlinear convection-diffusion equations. Phys. Rev. E, 79:016701, 2009.Google Scholar
[19]Stiebler, M.Tölke, J., and Krafczyk, M.. Advection-diffusion lattice Boltzmann scheme for hierarchical grids. Comput. Math. Appl., 55(7):1576–1584, 2008.Google Scholar
[20]Strang, G.. Accurate partial difference methods ii. non-linear problems. Numeric Mathematik, 6:37–64, 1964.Google Scholar
[21]Wang, J.Wang, D.Lallemand, P., and Luo, L.-S.. Lattice Boltzmann simulations of thermal convective flows in two dimensions. Comput. Math. Appl., 65(2):262–286, 2013.Google Scholar
[22]Weiß, J.-P.. Numerical Analysis of Lattice Boltzmann Methods for the Heat Equation on a Bounded Interval. Dissertation, 2006.Google Scholar
[23]Yoshida, H. and Nagaoka, M.. Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation. J. Comput. Phys., 229(20):7774–7795, 2010.Google Scholar