A block triple-relaxation-time lattice Boltzmann model for nonlinear anisotropic convection-diffusion equations

Abstract A block triple-relaxation-time (B-TriRT) lattice Boltzmann model for general nonlinear anisotropic convection–diffusion equations (NACDEs) is proposed, and the Chapman–Enskog analysis shows that the present B-TriRT model can recover the NACDEs correctly. There are some striking features of the present B-TriRT model: firstly, the relaxation matrix of B-TriRT model is partitioned into three relaxation parameter blocks, rather than a diagonal matrix in general multiple-relaxation-time (MRT) model; secondly, based on the analysis of half-way bounce-back (HBB) scheme for Dirichlet boundary conditions, we obtain an expression to determine the relaxation parameters; thirdly, the anisotropic diffusion tensor can be recovered by the relaxation parameter block that corresponds to the first-order moment of non-equilibrium distribution function. A number of simulations of isotropic and anisotropic convection–diffusion equations are conducted to validate the present B-TriRT model. The results indicate that the present model has a second-order accuracy in space, and is also more accurate and more stable than some available lattice Boltzmann models.

[1]  X. Shan.  Central-moment-based Galilean-invariant multiple-relaxation-time collision model. , 2019, Physical review. E.

[2]  Cyrus K. Aidun,et al.  Lattice-Boltzmann Method for Complex Flows , 2010 .

[3]  Rongzong Huang,et al.  A modified multiple-relaxation-time lattice Boltzmann model for convection-diffusion equation , 2014, J. Comput. Phys..

[4]  D. d'Humières,et al.  Study of simple hydrodynamic solutions with the two-relaxation-times lattice Boltzmann scheme , 2008 .

[5]  Zhenhua Chai,et al.  General propagation lattice Boltzmann model for nonlinear advection-diffusion equations. , 2018, Physical review. E.

[6]  S. Succi,et al.  Discrete Boltzmann trans-scale modeling of high-speed compressible flows. , 2018, Physical review. E.

[7]  Baochang Shi,et al.  Modified lattice Boltzmann scheme for nonlinear convection diffusion equations , 2012 .

[8]  Z. Chai,et al.  Comparative study of natural convection melting inside a cubic cavity using an improved two-relaxation-time lattice Boltzmann model , 2019, International Journal of Heat and Mass Transfer.

[9]  Ning Pan,et al.  Lattice Boltzmann modeling of the effective thermal conductivity for fibrous materials , 2007 .

[10]  P. Lallemand,et al.  Theory of the lattice boltzmann method: dispersion, dissipation, isotropy, galilean invariance, and stability , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Z. Chai,et al.  Generalized modification in the lattice Bhatnagar-Gross-Krook model for incompressible Navier-Stokes equations and convection-diffusion equations. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Bastien Chopard,et al.  A lattice Boltzmann model for coupled diffusion , 2010, J. Comput. Phys..

[13]  Bastien Chopard,et al.  Lattice Boltzmann method with regularized pre-collision distribution functions , 2006, Math. Comput. Simul..

[14]  I. Karlin,et al.  Lattice Boltzmann method for thermal flow simulation on standard lattices. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Bruce D. Jones,et al.  Multiphase lattice Boltzmann simulations for porous media applications , 2014, Computational Geosciences.

[16]  G. Doolen,et al.  Discrete Boltzmann equation model for nonideal gases , 1998 .

[17]  Z. Chai,et al.  A comparative study on the lattice Boltzmann models for predicting effective diffusivity of porous media , 2016 .

[18]  L. Luo,et al.  Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model , 1997 .

[19]  C. Shu,et al.  Lattice kinetic scheme for the incompressible viscous thermal flows on arbitrary meshes. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Xiaowen Shan,et al.  Multicomponent lattice-Boltzmann model with interparticle interaction , 1995, comp-gas/9503001.

[21]  Cass T. Miller,et al.  An evaluation of lattice Boltzmann schemes for porous medium flow simulation , 2006 .

[22]  Z. Chai,et al.  A lattice Boltzmann analysis of the conjugate natural convection in a square enclosure with a circular cylinder , 2019, Applied Mathematical Modelling.

[23]  I. Ginzburg.  Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation , 2005 .

[24]  S Succi,et al.  Regularized lattice Bhatnagar-Gross-Krook model for two- and three-dimensional cavity flow simulations. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Irina Ginzburg,et al.  Lattice Boltzmann modeling with discontinuous collision components: Hydrodynamic and Advection-Diffusion Equations , 2007 .

[26]  Irina Ginzburg,et al.  Truncation Errors, Exact and Heuristic Stability Analysis of Two-Relaxation-Times Lattice Boltzmann Schemes for Anisotropic Advection-Diffusion Equation , 2012 .

[27]  Zhenhua Chai,et al.  Regularized lattice Boltzmann model for a class of convection-diffusion equations. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  W. Tao,et al.  A critical review of the pseudopotential multiphase lattice Boltzmann model: Methods and applications , 2014 .

[29]  Aleksander S Popel,et al.  An immersed boundary lattice Boltzmann approach to simulate deformable liquid capsules and its application to microscopic blood flows , 2007, Physical biology.

[30]  Chenghai Sun,et al.  Lattice-Boltzmann models for high speed flows , 1998 .

[31]  Masato Yoshino,et al.  A numerical method for incompressible non-Newtonian fluid flows based on the lattice Boltzmann method , 2007 .

[32]  Erlend Magnus Viggen,et al.  The Lattice Boltzmann Method , 2017 .

[33]  Mohammad Hossein Saadat,et al.  Lattice Boltzmann model for compressible flows on standard lattices: Variable Prandtl number and adiabatic exponent. , 2019, Physical review. E.

[34]  Zhenhua Chai,et al.  Nonequilibrium scheme for computing the flux of the convection-diffusion equation in the framework of the lattice Boltzmann method. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  D. d'Humières,et al.  Two-relaxation-time Lattice Boltzmann scheme: About parametrization, velocity, pressure and mixed boundary conditions , 2008 .

[36]  John W. Crawford,et al.  A lattice BGK model for advection and anisotropic dispersion equation , 2002 .

[37]  Zhaoli Guo,et al.  A Localized Mass-Conserving Lattice Boltzmann Approach for Non-Newtonian Fluid Flows , 2015 .

[38]  Zhifang Lin,et al.  Lattice Boltzmann method for simulating the viscous flow in large distensible blood vessels. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Z. Chai,et al.  Regularized lattice Boltzmann simulation of double-diffusive convection of power-law nanofluids in rectangular enclosures , 2016 .

[40]  Changfeng Ma,et al.  Lattice Boltzmann method for the generalized Kuramoto–Sivashinsky equation , 2009 .

[41]  Zhenhua Chai,et al.  A Lattice Boltzmann Model for Two-Phase Flow in Porous Media , 2018, SIAM J. Sci. Comput..

[42]  Z. Chai,et al.  Lattice Boltzmann model for the convection-diffusion equation. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  早川 款達郎,et al.  打ち切り誤差--Truncation Errors (きる ) , 1993 .

[44]  Abdul-Majid Wazwaz,et al.  The tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations , 2005, Appl. Math. Comput..

[45]  Jianhua Lu,et al.  General bounce-back scheme for concentration boundary condition in the lattice-Boltzmann method. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[46]  Irina Ginzburg,et al.  Generic boundary conditions for lattice Boltzmann models and their application to advection and anisotropic dispersion equations , 2005 .

[47]  W. Yong,et al.  On a two-relaxation-time D2Q9 lattice Boltzmann model for the Navier-Stokes equations , 2018 .

[48]  Zhenhua Chai,et al.  A comparative study of local and nonlocal Allen-Cahn equations with mass conservation , 2018, International Journal of Heat and Mass Transfer.

[49]  Shiyi Chen,et al.  A Novel Thermal Model for the Lattice Boltzmann Method in Incompressible Limit , 1998 .

[50]  Hongwei Zheng,et al.  A lattice Boltzmann model for multiphase flows with large density ratio , 2006, J. Comput. Phys..

[51]  Erlend Magnus Viggen,et al.  The Lattice Boltzmann Method: Principles and Practice , 2016 .

[52]  Bastien Chopard,et al.  The lattice Boltzmann advection-diffusion model revisited , 2009 .

[53]  Takaji Inamuro,et al.  Lattice Boltzmann methods for viscous fluid flows and for two-phase fluid flows , 2006 .

[54]  T. Inamuro,et al.  Numerical simulation of the dispersion of aggregated Brownian particles under shear flows , 2013 .

[55]  Bruce M. Boghosian,et al.  Efficient lattice Boltzmann models for the Kuramoto–Sivashinsky equation , 2017, Computers & Fluids.

[56]  Chuguang Zheng,et al.  A Lattice BGK Scheme with General Propagation , 2002, J. Sci. Comput..

[57]  Zhenhua Chai,et al.  Non-Newtonian Effect on Hemodynamic Characteristics of Blood Flow in Stented Cerebral Aneurysm , 2013 .

[58]  B. Chopard,et al.  Lattice Boltzmann Simulations of Blood Flow: Non-Newtonian Rheology and Clotting Processes , 2005 .

[59]  J. Boon.  The Lattice Boltzmann Equation for Fluid Dynamics and Beyond , 2003 .

[60]  Danna Zhou,et al.  d. , 1934, Microbial pathogenesis.

[61]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[62]  Hiroaki Yoshida,et al.  Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation , 2010, J. Comput. Phys..

[63]  Zhenhua Chai,et al.  A unified lattice Boltzmann model for some nonlinear partial differential equations , 2008 .

[64]  P. Philippi,et al.  High-order regularization in lattice-Boltzmann equations , 2017 .

[65]  Zhenhua Chai,et al.  Lattice Boltzmann model for high-order nonlinear partial differential equations. , 2009, Physical review. E.

[66]  C. Shu,et al.  Lattice Boltzmann Method and Its Applications in Engineering , 2013 .

[67]  Yang Li,et al.  Axisymmetric lattice Boltzmann model for multiphase flows with large density ratio , 2018, International Journal of Heat and Mass Transfer.

[68]  Shiyi Chen,et al.  Lattice Boltzmann computations for reaction‐diffusion equations , 1993 .

[69]  Raoyang Zhang,et al.  Efficient kinetic method for fluid simulation beyond the Navier-Stokes equation. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[70]  Irina Ginzburg,et al.  Multiple anisotropic collisions for advection-diffusion Lattice Boltzmann schemes , 2013 .

[71]  Xiao-dong Wang,et al.  Lattice kinetic scheme for the Navier-Stokes equations coupled with convection-diffusion equations , 2018, Physical Review E.

[72]  Chen,et al.  Lattice Boltzmann model for compressible fluids , 1992 .

[73]  Zhenhua Chai,et al.  Discrete effect on the halfway bounce-back boundary condition of multiple-relaxation-time lattice Boltzmann model for convection-diffusion equations. , 2016, Physical review. E.

[74]  Takaji Inamuro,et al.  A lattice kinetic scheme for incompressible viscous flows with heat transfer , 2002, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[75]  Baochang Shi,et al.  Lattice Boltzmann model for nonlinear convection-diffusion equations. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[76]  C. Shu,et al.  Simplified thermal lattice Boltzmann model for incompressible thermal flows. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[77]  Chuguang Zheng,et al.  Thermal lattice Boltzmann equation for low Mach number flows: decoupling model. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[78]  I. Karlin,et al.  Gibbs' principle for the lattice-kinetic theory of fluid dynamics. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[79]  Byron Goldstein,et al.  Lattice Boltzmann Simulation of Diffusion-Convection Systems with Surface Chemical Reaction , 2000 .

[80]  Zhenhua Chai,et al.  A Multiple-Relaxation-Time Lattice Boltzmann Model for General Nonlinear Anisotropic Convection–Diffusion Equations , 2016, J. Sci. Comput..

[81]  Dieter Wolf-Gladrow,et al.  A lattice Boltzmann equation for diffusion , 1995 .

[82]  Shiyi Chen,et al.  LATTICE BOLTZMANN METHOD FOR FLUID FLOWS , 2001 .

[83]  Xiaomei Yu,et al.  A lattice Boltzmann model for reaction dynamical systems with time delay , 2006, Appl. Math. Comput..

[84]  Z. Chai,et al.  Lattice Boltzmann simulation of melting in a cubical cavity with a local heat-flux source , 2018, International Journal of Heat and Mass Transfer.