Hostname: page-component-7dd5485656-6kn8j Total loading time: 0 Render date: 2025-10-27T23:28:05.617Z Has data issue: false hasContentIssue false
  • English
  • Français

Truncation Errors, Exact And Heuristic Stability Analysis Of Two-Relaxation-Times Lattice Boltzmann Schemes For Anisotropic Advection-Diffusion Equation

Published online by Cambridge University Press:  20 August 2015

Irina Ginzburg
Irina Ginzburg*
Affiliation:
IRSTEA, Antony Regional Centre, HBAN, 1 rue Pierre-Gilles de Gennes CS 10030, 92761 Antony cedex, France
*
*Corresponding author.Email:irina.ginzburg@cemagref.fr
Get access

Abstract

This paper establishes relations between the stability and the high-order truncated corrections for modeling of the mass conservation equation with the two-relaxation-times (TRT) collision operator. First we propose a simple method to derive the truncation errors from the exact, central-difference type, recurrence equations of the TRT scheme. They also supply its equivalent three-time-level discretization form. Two different relationships of the two relaxation rates nullify the third (advection) and fourth (pure diffusion) truncation errors, for any linear equilibrium and any velocity set. However, the two relaxation times alone cannot remove the leading-order advection-d if fusion error, because of the intrinsic fourth-order numerical diffusion. The truncation analysis is carefully verified for the evolution of concentration waves with the anisotropic diffusion tensors. The anisotropic equilibrium functions are presented in a simple but general form, suitable for the minimal velocity sets and the d2Q9, d3Q13, d3Q15 and d3Q19 velocity sets. All anisotropic schemes are complemented by their exact necessary von Neumann stability conditions and equivalent finite-difference stencils. The sufficient stability conditions are proposed for the most stable (OTRT) family, which enables modeling at any Peclet numbers with the same velocity amplitude. The heuristic stability analysis of the fourth-order truncated corrections extends the optimal stability to larger relationships of the two relaxation rates, in agreement with the exact (one-dimensional) and numerical (multi-dimensional) stability analysis. A special attention is put on the choice of the equilibrium weights. By combining accuracy and stability predictions, several strategies for selecting the relaxation and free-tunable equilibrium parameters are suggested and applied to the evolution of the Gaussian hill.

Keywords

02.60.Lj02.30.Jr66.10.cTwo-relaxation-times Lattice Boltzmann schemeAADEtruncation errorsvon Neumann stability analysisnumerical diffusionheuristic stability analysis

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 5 , May 2012 , pp. 1439 - 1502
Copyright
Copyright © Global Science Press Limited 2012

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.)

Article purchase

Temporarily unavailable

References

[1]Adhikari, R. and Succi, S., Duality in matrix lattice Boltzmann methods, Phys. Rev. E, 78:066701–1-9, 2008.CrossRefGoogle Scholar
[2]Ahrenholz, B., Toölke, J., and Krafczyk, M., Lattice-Boltzmann simulations in reconstructed parametrized porous media. Int. J. Comput. Fluid Dyn., 20(6):369–377, 2006.Google Scholar
[3]Ancona, M. G., Fully-Lagrangian and Lattice-Boltzmann Methods for solving systems of conservation equations. J. Comput. Phys., 115:107–120, 1994.Google Scholar
[4]Benzi, R., Succi, S., and Vergassola, M.The lattice Boltzmann equation: theory and applications. Physics Reports, 222:145–197, 1992.Google Scholar
[5]Chun, B. and Ladd, A. J. C., Interpolated boundary conditions for Lattice Boltzmann simulations of flows in narrow gaps. Phys. Rev. E, 75:066705–1-11, 2007.Google Scholar
[6]Dong, Y., Zhang, J., and Yan, G., A higher-order moment method of the lattice Boltzmann model for the conservation law equation, Appl. Math. Model., 34:481–494, 2010.Google Scholar
[7]Dubois, F., Equivalent partial differential equations of a lattice Boltzmann scheme. Comp. Math. Appl., 55:1441–1449,2008.Google Scholar
[8]Dubois, F. and Lallemand, P., Towards higher order lattice Boltzmann schemes. J. Stat. Mech.: Theory and Experiment, PO6:P06006, 2009. doi: 10.1088/1742-5468/2009/06/006006.Google Scholar
[9]Dubois, F., Lallemand, P., and Tekitek, M., On a superconvergent lattice Boltzmann boundary scheme. Comp. Math. Appl., 59(7):2141–2149, 2010.Google Scholar
[10]Du Fort, E. C. and Frankel, S. P., Stability conditions in the numerical treatment of parabolic differential equations. M.T.A.C., 7:135, 1953.Google Scholar
[11]Ginzburg, I. and d’Humières, D., Multi-reflection boundary conditions for lattice Boltzmann models. Phys. Rev. E, 68:066614–1-30,2003.Google Scholar
[12]Ginzburg, I., Equilibrium-type and Link-type Lattice Boltzmann models for generic advection and anisotropic-dispersion equation. Adv. Wat. Res., 28:1171–1195, 2005.Google Scholar
[13]Ginzburg, I., Generic boundary conditions for Lattice Boltzmann models and their application to advection and anisotropic-dispersion equations. Adv. Wat. Res., 28:1196–1216, 2005.Google Scholar
[14]Ginzburg, I., Variably saturated flow described with the anisotropic Lattice Boltzmann methods. J. Comp. Fluids, 25:831–848, 2006.Google Scholar
[15]Ginzburg, I. and d’Humières, D., Lattice Boltzmann and analytical modeling of flow processes in anisotropic and heterogeneous stratified aquifers. Adv. Wat. Res., 30:2202–2234, 2007.CrossRefGoogle Scholar
[16]Ginzburg, I., Lattice Boltzmann modeling with discontinuous collision components. Hydro-dynamic and advection-diffusion equations. J. Stat. Phys., 126:157–203, 2007.CrossRefGoogle Scholar
[17]Ginzburg, I., Consistent Lattice Boltzmann schemes for the Brinkman model of porous flow and infinite Chapman-Enskog expansion. Phys. Rev. E, 77:0666704:1-12, 2008.Google Scholar
[18]Ginzburg, I., Verhaeghe, F., and d’Humières, D., Two-relaxation-time Lattice Boltzmann scheme: about parametrization, velocity, pressure and mixed boundary conditions. Commun. Comput. Phys, 3:427–478, 2008.Google Scholar
[19]Ginzburg, I., Verhaeghe, F., and d’Humières, D., Study of simple hydrodynamic solutions with the two-relaxation-times Lattice Boltzmann scheme. Commun. Comput. Phys, 3:519–581, 2008.Google Scholar
[20]Ginzburg, I., d’Humières, D., and Kuzmin, A., Optimal stability of advection-diffusion Lattice Boltzmann models with two relaxation times for positive/negative equilibrium. J. Stat. Phys., 139(6):1090–1143, 2010. DOI 10.1007/s10955-010-9969-9.Google Scholar
[21]Hammou, H., Ginzburg, I., and Boulerhcha, M., Two-relaxation-times Lattice Boltz-mann schemes for solute transport in unsaturated water flow, with a focus on stability. Adv. Wat. Res., 34(6):779–793, 2011.Google Scholar
[22]Higuera, F. J., Succi, S., and Benzi, R., Lattice gas dynamics with enhanced collisions. Euro-phys. Lett., 9:345–349, 1989.CrossRefGoogle Scholar
[23]Higuera, F. J. and Jiménez, J., Boltzmann approach to lattice gas simulations. Europhys. Lett., 9:663–668, 1989.CrossRefGoogle Scholar
[24]Hindmarsch, A. C., Grescho, P. M., and Griffiths, D. F., The stability of explicit time-integration for certain finite difference approximation of the multi-dimensional advection-diffusion equation. Int. J. Num. Methods Fluids, 4:853–897, 1984.CrossRefGoogle Scholar
[25]Hirt, C. W., Heuristic stability theory for finite-difference equations. J. Comput. Phys., 2:339–355, 1968.Google Scholar
[26]Holdych, D. J., Noble, D. R., Georgiadis, J. G., and Buckius, R. O., Truncation error analysis of Lattice Boltzmann methods. J. Comput. Phys., 193(2):595–619, 2004.CrossRefGoogle Scholar
[27]d’Humières, D., Generalized Lattice-Boltzmann equations. AIAA Rarefied Gas Dynamics: Theory and Simulations. Progress in Astronautics and Aeronautics, 59:450–548, 1992.Google Scholar
[28]d’Humières, D., Bouzidi, M., and Lallemand, P., Thirteen-velocity three-dimensional Lattice Boltzmann model. Phys. Rev. E, 63:066702–1-7, 2001.Google Scholar
[29]d’Humières, D., Ginzburg, I., Krafczyk, M., Lallemand, P., and Luo, L.-S., Multiple-relaxation-time Lattice Boltzmann models in three dimensions. Phil. Trans. R. Soc. Lond. A, 360:437–451, 2002.CrossRefGoogle ScholarPubMed
[30]d’Humières, D. and Ginzburg, I., Viscosity independent numerical errors for Lattice Boltz-mann models: from recurrence equations to “magic” collision numbers. Comp. Math. Appl., 58(5):823–840, 2009.Google Scholar
[31]d’Humières, D., Some stability results, in preparation, 2010.Google Scholar
[32]Junk, M. and Yang, Z., Analysis and invariant properties of a Lattice Boltzmann method. Adv. Appl. Math. Mech., 2/5:640–669, 2010.Google Scholar
[33]Koivu, V., Decain, M., Geindreau, C., Mattila, K., Bloch, J.-F., and Kataja, M., Transport properties of heterogeneous materials. Combining computerised X-ray micro-tomography and direct numerical simulations. Int. J. Comput. Fluid Dyn., 23(10):713–721, 2009.Google Scholar
[34]Kuzmin, A., Ginzburg, I., and Mohamad, A. A., A role of the kinetic parameter on the stability of two-relaxation-times advection-diffusion Lattice Boltzmann scheme. Comp. Math. Appl., 61:3417–3442, 2011.Google Scholar
[35]Kwok, Y.-K. and Tam, K.-K., Stability analysis of three-level difference schemes for initial boundary problems for multi-dimensional convective-diffusion equations. Commun. Numer. Methods Engr., 9:595–605, 1993.CrossRefGoogle Scholar
[36]Lallemand, P. and Luo, L.-S., Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E, 61:6546–6562, 2000.Google Scholar
[37]McCracken, M. E. and Abraham, J., Multiple-relaxation-time lattice-Boltzmann model for multiphase flow. Phys. Rev. E, 71:036701–1-9, 2005.Google Scholar
[38]Li, Y. and Huang, P., A coupled lattice Boltzmann model for advection and anisotropic dispersion problem in shallow water. Adv. Wat. Res., 31(12):1719–1730, 2008.Google Scholar
[39]Miller, J. J. H., On the location of zeros of certaines classes of polynomials with application to numerical analysis. J. Inst. Math. Appl., 8:397–406, 1971.Google Scholar
[40]Pan, C., Luo, L.-S., and Miller, C. T., An evaluation of lattice Boltzmann schemes for porous media simulation. J. Comp. Fluids, 35(4):898–909, 2006.CrossRefGoogle Scholar
[41]Qian, Y., d’Humières, D., and Lallemand, P., Lattice BGK models for Navier-Stokes equation. Europhys. Lett., 17:479–484, 1992.Google Scholar
[42]Rasin, I., Saucci, S. and Miller, W., A multi-relaxation lattice kinetic method for passive scalar diffusion, J. Comput. Phys., 206:453–462, 2005.Google Scholar
[43]Rheinländer, M., Stability and multiscale analysis of an advective Lattice Boltzmann scheme. Progress Comp. Fluid Dynamics, 8(1-4):56–68, 2008.CrossRefGoogle Scholar
[44]Servan-Camas, B. and Tsai, F. T. C., Lattice Boltzmann method for two relaxation times for advection-diffusion equation: third order analysis and stability analysis. Adv. Wat. Res., 31:1113–1126, 2008.Google Scholar
[45]Servan-Camas, B. and Tsai, F. T. C., Saltwater intrusion modeling in heterogeneous confined aquifers using two-relaxation-time Lattice Boltzmann method. J. Comput. Phys., 228:236–256, 2009.Google Scholar
[46]Servan-Camas, B. and Tsai, F. T. C., Two-relaxation-time lattice Boltzmann method for the anisotropic dispersive Henry problem. Water Resources Research, 46:W02515–25, 2009.Google Scholar
[47]van der Sman, R. G. M., MRT Lattice Boltzmann schemes for confined suspension flows. Comp. Phys. Commun., 181:1562–1569,2010.Google Scholar
[48]Suga, S., Stability and accuracy of Lattice Boltzmann Schemes for anisotropic advection-diffusion equations. Int. J. Mod. Phys. C, 20(4):633–650, 2009.Google Scholar
[49]Suga, S., An accurate multi-level finite difference scheme for 1D diffusion equations derived from the Lattice Boltzmann method. J. Stat. Phys., 140:494–503.Google Scholar
[50]Wolfram, S., software package “Mathematica 6”.Google Scholar
[51]Yoshida, H. and Nagaoka, M., Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation. J. Comput. Phys., 229:7774–7795, 2010.Google Scholar
[52]Zhang, X., Bengough, A. G., Deeks, L. K., Crawford, J. W., and Young, I. M.A lattice BGK model for advection and anisotropic dispersion equation. Adv. Wat. Res., 25:1–8, 2002.Google Scholar