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A general fourth-order mesoscopic multiple-relaxation-time lattice Boltzmann model and its macroscopic finite-difference scheme for two-dimensional diffusion equations

Published: 18 July 2024 Publication History

Abstract

In this work, we first develop a general mesoscopic multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the two-dimensional diffusion equation with the constant diffusion coefficient and source term, where the D2Q5 (five discrete velocities in two-dimensional space) lattice structure is considered. Then we exactly derive the corresponding macroscopic finite-difference scheme of the MRT-LB model. Additionally, we also propose a proper MRT-LB model for the diffusion equation with a linear source term, and obtain a macroscopic six-level finite-difference scheme. After that, we conduct the accuracy and stability analysis on the finite-difference scheme and the mesoscopic MRT-LB model, and find that at the diffusive scaling, both of them can achieve a fourth-order accuracy in space based on the Taylor expansion. The stability analysis also shows that they are both unconditionally stable. Finally, some numerical experiments are conducted, and the numerical results are also consistent with our theoretical analysis.

Highlights

A general fourth-order MRT-LB model is developed for the two-dimensional diffusion equations.
The macroscopic finite-difference scheme of the MRT-LB model is obtained theoretically.
The accuracy and stability of the MRT-LB model and finite-difference scheme are analyzed.

References

[1]
S. Chen, G. Doolen, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech. 30 (1998) 329.
[2]
S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford University Press, Oxford, 2001.
[3]
C.K. Aidun, J.R. Clausen, Lattice-Boltzmann method for complex flows, Annu. Rev. Fluid Mech. 42 (2010) 439.
[4]
Z. Guo, C. Shu, Lattice Boltzmann Method and Its Applications in Engineering, World Scientific Publishing, Singapore, 2013.
[5]
T. Krüger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, E.M. Viggen, The Lattice Boltzmann Method: Principles and Practice, Springer, Switzerland, 2017.
[6]
H. Wang, X. Yuan, H. Liang, Z. Chai, B. Shi, A brief review of the phase-field-based lattice Boltzmann method for multiphase flows, Capillary 2 (2019) 33.
[7]
C. Huber, B. Chopard, M. Manga, A lattice Boltzmann model for coupled diffusion, J. Comput. Phys. 229 (20) (2010) 7956.
[8]
M.G. Ancona, Fully-Lagrangian and lattice-Boltzmann methods for solving systems of conservation equations, J. Comput. Phys. 115 (1) (1994) 107.
[9]
S. Suga, An accurate multi-level finite difference scheme for 1D diffusion equations derived from the lattice Boltzmann method, J. Stat. Phys. 140 (3) (2010) 494.
[10]
Y. Lin, N. Hong, B. Shi, Z. Chai, Multiple-relaxation-time lattice Boltzmann model-based four-level finite-difference scheme for one-dimensional diffusion equations, Phys. Rev. E 104 (1) (2021).
[11]
G. Silva, Discrete effects on the source term for the lattice Boltzmann modelling of one-dimensional reaction–diffusion equations, Comput. Fluids 251 (2023).
[12]
R.G.M. Van der Sman, M.H. Ernst, Convection-diffusion lattice Boltzmann scheme for irregular lattices, J. Comput. Phys. 160 (2) (2000) 766.
[13]
I. Ginzburg, Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation, Adv. Water Resour. 28 (11) (2005) 1171.
[14]
I. Rasin, S. Succi, W. Miller, A multi-relaxation lattice kinetic method for passive scalar diffusion, J. Comput. Phys. 206 (2) (2005) 453.
[15]
B. Shi, Z. Guo, Lattice Boltzmann model for nonlinear convection-diffusion equations, Phys. Rev. E 79 (1) (2009).
[16]
B. Chopard, J.L. Falcone, J. Latt, The lattice Boltzmann advection-diffusion model revisited, Eur. Phys. J. Spec. Top. 171 (1) (2009) 245.
[17]
H. Yoshida, M. Nagaoka, Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation, J. Comput. Phys. 229 (20) (2010) 7774.
[18]
I. Ginzburg, Multiple anisotropic collisions for advection-diffusion lattice Boltzmann schemes, Adv. Water Resour. 51 (2013) 381.
[19]
Z. Chai, T.S. Zhao, Lattice Boltzmann model for the convection-diffusion equation, Phys. Rev. E 87 (6) (2013).
[20]
Z. Chai, B. Shi, Z. Guo, A multiple-relaxation-time lattice Boltzmann model for general nonlinear anisotropic convection-diffusion equations, J. Sci. Comput. 69 (1) (2016) 355.
[21]
O. Aursjø, E. Jettestuen, J.L. Vinningland, A. Hiorth, An improved lattice Boltzmann method for simulating advective-diffusive processes in fluids, J. Comput. Phys. 332 (2017) 363.
[22]
L. Li, Multiple-time-scaling lattice Boltzmann method for the convection-diffusion equation, Phys. Rev. E 99 (6) (2019).
[23]
J. Michelet, M.M. Tekitek, M. Berthier, Multiple relaxation time lattice Boltzmann schemes for advection-diffusion equations with application to radar image processing, J. Comput. Phys. 471 (2022).
[24]
S. Dellacherie, Construction and analysis of lattice Boltzmann methods applied to a 1D convection-diffusion equation, Acta Appl. Math. 131 (1) (2014) 69.
[25]
S. Cui, N. Hong, B. Shi, Z. Chai, Discrete effect on the halfway bounce-back boundary condition of multiple-relaxation-time lattice Boltzmann model for convection-diffusion equations, Phys. Rev. E 93 (4) (2016).
[26]
Y. Chen, Z. Chai, B. Shi, Fourth-order multiple-relaxation-time lattice Boltzmann model and equivalent finite-difference scheme for one-dimensional convection-diffusion equations, Phys. Rev. E 107 (2023).
[27]
M. Hirabayashi, Y. Chen, H. Ohashi, The lattice BGK model for the Poisson equation, JSME Int. J. Ser. B 44 (1) (2001) 45.
[28]
Z. Chai, B. Shi, A novel lattice Boltzmann model for the Poisson equation, Appl. Math. Model. 32 (10) (2008) 2050.
[29]
Z. Chai, H. Liang, R. Du, B. Shi, A lattice Boltzmann model for two-phase flow in porous media, SIAM J. Sci. Comput. 41 (4) (2019).
[30]
Q. Li, Z. Zheng, S. Wang, J. Liu, A multi-level finite difference scheme for one-dimensional Burgers equation derived from the lattice Boltzmann method, J. Appl. Math. 2012 (2012) 1.
[31]
Chen, Y.; Liu, X.; Chai, Z.; Shi, B. (2023): A Cole-Hopf transformation based fourth-order multiple-relaxation-time lattice Boltzmann model for the coupled Burgers' equations. arXiv preprint arXiv:2309.02825.
[32]
Y.H. Qian, D. d′Humière, P. Lallemand, Lattice BGK models for Navier-Stokes equation, Europhys. Lett. 17 (6) (1992) 479.
[33]
Z. Chai, B. Shi, Multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: modeling, analysis, and elements, Phys. Rev. E 102 (2) (2020).
[34]
Z. Chai, T.S. Zhao, Nonequilibrium scheme for computing the flux of the convection-diffusion equation in the framework of the lattice Boltzmann method, Phys. Rev. E 90 (1) (2014).
[35]
P. Lallemand, L.S. Luo, Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E 61 (2000) 6546.
[36]
C. Pan, L.-S. Luo, C.T. Miller, An evaluation of lattice Boltzmann schemes for porous medium flow simulation, Comput. Fluids 35 (8) (2006) 898.
[37]
L.-S. Luo, W. Liao, X. Chen, Y. Peng, W. Zhang, Numerics of the lattice Boltzmann method: effects of collision models on the lattice Boltzmann simulations, Phys. Rev. E 83 (2011).
[38]
S. Chapman, T.G. Cowling, The Mathematical Theory of Nonuniform Gases, Cambridge University Press, Cambridge, 1970.
[39]
E. Ikenberry, C. Truesdell, On the pressures and the flux of energy in a gas according to Maxwell's kinetic theory, J. Ration. Mech. Anal. 5 (1) (1956) 1.
[40]
W.-A. Yong, W. Zhao, L.-S. Luo, Theory of the lattice Boltzmann method: derivation of macroscopic equations via the Maxwell iteration, Phys. Rev. E 93 (3) (2016).
[41]
D.J. Holdych, D.R. Noble, J.G. Georgiadis, R.O. Buckius, Truncation error analysis of lattice Boltzmann methods, J. Comput. Phys. 193 (2) (2004) 595.
[42]
A. Wagner, Thermodynamic consistency of liquid-gas lattice Boltzmann simulations, Phys. Rev. E 74 (5) (2006).
[43]
D. d′Humière, I. Ginzburg, Viscosity independent numerical errors for lattice Boltzmann models: from recurrence equations to magic collision numbers, Comput. Math. Appl. 58 (5) (2009) 823.
[44]
I. Ginzburg, Truncation errors, exact and heuristic stability analysis of two-relaxation-times lattice Boltzmann schemes for anisotropic advection-diffusion equation, Commun. Comput. Phys. 11 (5) (2012) 1439.
[45]
F. Dubois, Equivalent partial differential equations of a lattice Boltzmann scheme, Comput. Math. Appl. 55 (7) (2008) 1441.
[46]
F. Dubois, Third order equivalent equation of lattice Boltzmann scheme, Discrete Contin. Dyn. Syst. 23 (1–2) (2009) 221.
[47]
F. Dubois, Nonlinear fourth order Taylor expansion of lattice Boltzmann schemes, Asymptot. Anal. 127 (4) (2022) 297.
[48]
M. Junk, A finite difference interpretation of the lattice Boltzmann method, Numer. Methods Partial Differ. Equ. 17 (4) (2001) 383.
[49]
T. Inamuro, A lattice kinetic scheme for incompressible viscous flows with heat transfer, Philos. Trans. R. Soc. Lond. A 360 (1792) (2002) 477.
[50]
E.C. Du Fort, S.P. Frankel, Stability conditions in the numerical treatment of parabolic differential equations, Math. Comput. 7 (43) (1953) 135.
[51]
Y.-K. Kwok, K.-K. Tam, Stability analysis of three-level difference schemes for initial-boundary problems for multidimensional convective-diffusion equations, Commun. Numer. Methods Eng. 9 (7) (1993) 595.
[52]
R. Fučík, R. Straka, Equivalent finite difference and partial differential equations for the lattice Boltzmann method, Comput. Math. Appl. 90 (2021) 96.
[53]
R. Fučík, P. Eichler, J. Klinkovsky, R. Straka, T. Oberhuber, Lattice Boltzmann method analysis tool (LBMAT), Numer. Algorithms 93 (2023) 1509–1525,.
[54]
T. Bellotti, B. Graille, M. Massot, Finite difference formulation of any lattice Boltzmann scheme, Numer. Math. 152 (1) (2022) 1.
[55]
T. Bellotti, Truncation errors and modified equations for the lattice Boltzmann method via the corresponding finite difference schemes, Modél. Math. Anal. Numér. 57 (3) (2023) 1225–1255.
[56]
Z. Chai, C. Huang, B. Shi, Z. Guo, A comparative study on the lattice Boltzmann models for predicting effective diffusivity of porous media, Int. J. Heat Mass Transf. 98 (2016) 687.
[57]
P.J. Dellar, Non-hydrodynamic modes and general equations of state in lattice Boltzmann equations, Physica A 362 (1) (2006) 132–138.
[58]
J.J.H. Miller, On the location of zeros of certain classes of polynomials with applications to numerical analysis, J. Inst. Math. Appl. 8 (3) (1971) 397.
[59]
E.J. Routh, A Treatise of the Stability of a Given State of Motion, Macmillan, London, 1877.
[60]
A. Hurwitz, Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Math. Ann. 46 (2) (1895) 273.
[61]
F.R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers, New York, 1959.
[62]
S.-H. Hou, Classroom note: a simple proof of the Leverrier-Faddeev characteristic polynomial algorithm, SIAM Rev. 40 (3) (1998) 706–709.
[63]
Bellotti, T. (2023): Initialisation from lattice Boltzmann to multi-step finite difference methods: modified equations and discrete observability. arXiv preprint arXiv:2302.07558.

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Published In

cover image Journal of Computational Physics
Journal of Computational Physics  Volume 509, Issue C
Jul 2024
579 pages

Publisher

Academic Press Professional, Inc.

United States

Publication History

Published: 18 July 2024

Author Tags

  1. Lattice Boltzmann method
  2. Finite-difference scheme
  3. Diffusion equations

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