A comparative analysis on the schemes for exact lattice Boltzmann (LB) evolution equation is presented in this paper. It includes two classical exact LB schemes, i.e., Bosch-Karlin (BK) scheme and He-Luo (HL) scheme, and the present Taylor-expansion (TE) scheme. TE scheme originates from the extension of BK scheme. The mathematical mechanism and the equilibrium distribution evolution behind these exact schemes have been detailedly addressed. After that, an analysis is carried out to discuss the cause of the LB equation difference among the schemes, which offers an insight of the exactness in these schemes and brings up their continuity precondition. At last, the schemes are systematically addressed for their pros and cons in the further development of LB equations.
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LIU H J, ZOU C, SHI B C, et al. Thermal lattice-BGK model based on large-eddy simulation of turbulent natural convection due to internal heat generation [J]. International Journal of Heat and Mass Transfer, 2006. 49: 4672–4680.
SHI B C, GUO Z L. Thermal lattice BGK simulation of turbulent natural convection due to internal heat generation [J]. International Journal of Modern Physics B, 2003. 17(1/2): 173–177.
LADD A J C, VERBERG R. Lattice-Boltzmann simulations of particle-fluid suspensions [J]. Journal of Statistical Physics, 2001. 104(5/6): 1191–1251.
SHEIKHOLESLAMI M, GORJI-BANDPY M, GANJI D D. Lattice Boltzmann method for MHD natural convection heat transfer using nanofluid [J]. Powder Technology, 2014. 254: 82–93.
LIU X L, CHENG P. Lattice Boltzmann simulation for dropwise condensation of vapor along vertical hydrophobic flat plates [J]. International Journal of Heat and Mass Transfer, 2013. 64: 1041–1052.
SEMMA E, EI GANAOUI M, BENNACER R, et al. Investigation of flows in solidification by using the lattice Boltzmann method [J]. International Journal of Thermal Sciences, 2008. 47: 201–208.
MOUNTRAKIS L, LORENZ E, MALASPINAS O, et al. Parallel performance of an IB-LBM suspension simulation framework [J]. Journal of Computational Science, 2015. 9: 45–50.
MENG X H, GUO Z L. Localized lattice Boltzmann equation model for simulating miscible viscous displacement in porous media [J]. International Journal of Heat and Mass Transfer, 2016. 100: 767–778.
LIU Q, HE Y L. Lattice Boltzmann simulations of convection heat transfer in porous media [J]. Physica A, 2017. 465: 742–753.
CHAI Z H, LIANG H, DU R, et al. A lattice Boltzmann model for two-phase flow in porous media [J]. SIAM Journal on Scientific Computing, 2019. 41(4): B746-B772.
GUO Z L, SHU C. Lattice Boltzmann method and its applications in engineering [M]. Singapore: World Scientific Publishing, 2013.
SUCCI S. The lattice Boltzmann equation for fluid dynamics and beyond [M]. Oxford: Oxford University Press, 2001.
KRGER T, KUSUMAATMAJA H, KUZMIN A, et al. The lattice Boltzmann method: Principles and practice [M]. Switzerland: Springer, 2017.
FRISCH U, HASSLACHER B, POMEAU Y. Lattice-gas automata for the Navier-Stokes equation [J]. Physical Review Letters, 1986. 56(14): 1505–1508.
BHATNAGAR P L, GROSS E P, KROOK M. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems [J]. Physical Review, 1954. 94(3): 511–525.
BOSCH F, KARLIN I V. Exact lattice Boltzmann equation [J]. Physical Review Letters, 2013. 111:090601.
HE X Y, CHEN S Y, DOOLEN G D. A novel thermal model for the lattice Boltzmann method in incompressible limit [J]. Journal of Computational Physics, 1998. 146: 282–300.
HE X Y, LUO L S. Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation [J]. Physical Review E, 1997. 56(6): 6811–6817.
YE H F, KUANG B, YANG Y H. Derivation of lattice Boltzmann equation via analytical characteristic integral [J]. Chinese Physics B, 2019. 28(1): 014701.
SHAN X W. The mathematical structure of the lattices of the lattice Boltzmann method [J]. Journal of Computational Science, 2016. 17: 475–481.
HWANG Y H. Macroscopic model and truncation error of discrete Boltzmann method [J]. Journal of Computational Physics, 2016. 322: 52–73.
YONG W A, ZHAO W F, LUO L S. Theory of the lattice Boltzmann method: Derivation of macroscopic equations via the Maxwell iteration [J]. Physical Review E, 2016. 93:033310.
DELLAR P J. An interpretation and derivation of the lattice Boltzmann method using Strang splitting [J]. Computers and Mathematics with Applications, 2013. 65: 129–141.
ARFKEN G, WEBER H, HARRIS F E. Mathematical methods for physicists: A comprehensive guide [M]. 7th ed. Cambridge, Massachusetts, USA: Academic Press, 2011.
YE Huanfeng would like to express his gratitude to Dr. GAN Zecheng for helpful discussion.
Foundation item: the National Science and Technology Major Project of China (No. 2017ZX06002002)
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Ye, H., Kuang, B. & Yang, Y. Derivations of Exact Lattice Boltzmann Evolution Equation. J. Shanghai Jiaotong Univ. (Sci.) (2020) doi:10.1007/s12204-020-2158-3
- lattice Boltzmann (LB) method
- exact LB evolution equation
- Taylor-expansion (TE) scheme
- O 242.1