Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T19:00:39.107Z Has data issue: false hasContentIssue false

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
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)
Get access

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.

Type
Research Article
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Banda, M. K.Yong, W.-A., and Klar, A.. A stability notion for lattice Boltzmann equations. SIAMJ. Sci. Comput, 27:20982111,2006.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., 94:511525, 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):256266,2000.Google Scholar
[4]Caflisch, R.. The fluid dynamic limit of the nonlinear Boltzmann equation. Comm. Pure Appl. Math., 30:651666,1980.CrossRefGoogle Scholar
[5]Caiazzo, A.Junk, M., and Rheinländer, M.. Comparison of analysis techniques for the lattice Boltzmann method. Comput. Math. Appl., 58:883897,2009.CrossRefGoogle Scholar
[6]Chen, S. and Doolen, G.D.. Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech., 30:329364,1998.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, 28:11711195, 2005.CrossRefGoogle Scholar
[9]Junk, M.Klar, A., and Luo, L.-S.. Asymptotic analysis of the lattice Boltzmann equation. J. Comput. Phys., 210:676704, 2005.Google Scholar
[10]Junk, M. and Yang, Z.. Asymptotic analysis of lattice Boltzmann boundary conditions. J. Stat. Phys., 121:335, 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):14811491, 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):6587, 2009.CrossRefGoogle Scholar
[13]Junk, M. and Yong, W.-A.. Rigorous Navier-Stokes limit of the lattice Boltzmann equation. Asymp. Anal., 35:165185, 2003.Google Scholar
[14]Junk, M. and Yong, W.-A.. Weighted L 2 stability of the lattice Boltzmann equation. SIAM J. Numer. Anal., 47:16511665, 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):11891214, 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):15141523, 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):15761584, 2008.Google Scholar
[20]Strang, G.. Accurate partial difference methods ii. non-linear problems. Numeric Mathematik, 6:3764, 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):262286, 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):77747795, 2010.Google Scholar