Lattice Boltzmann method with two relaxation times for advection–diffusion equation: Third order analysis and stability analysis

Serván-Camas, Borja; Tsai, Frank T.-C.

doi:10.1016/j.advwatres.2008.05.001

Cited by 85 publications

(55 citation statements)

References 22 publications

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“…In general, the models of the LBM can be classified into three kinds based on the collision operator: the singlerelaxation-time model [or the Bhatnagar-Gross-Krook (BGK) model] [8][9][10][11][12][13][14][15][16]28], the two-relaxation-time (TRT) model [17][18][19]29], and the MRT model (or generalized model) [20][21][22][23]30,31]. In this paper, the MRT model is considered for its superiority over the BGK model in terms of stability and accuracy [21,32].…”

Section: Multiple-relaxation-time Lattice Boltzmann Model For Thementioning

confidence: 99%

“…Over the past decades, the lattice Boltzmann equation (LBM), as a mesoscopic numerical method based on the simplified kinetic models, has gained great success in the study of various fluid flow systems with complex physical phenomena and geometry structures [2][3][4][5][6][7] and has also been applied to solve the CDE [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. It is noticed that most of the previous investigations on the CDE with the LBM only focused on the accuracy and convergence rate of the LBM in depicting the scalar variable C [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Recently, the computation of the flux has also received increasing attention in the study of heat transfer in turbulent Rayleigh-Bénard convection [24,25] and in porous media with an emphasis on obtaining effective physical properties of fluids in a porous medium [26,27].…”

Section: Introductionmentioning

confidence: 99%

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Nonequilibrium scheme for computing the flux of the convection-diffusion equation in the framework of the lattice Boltzmann method

Chai

Zhao

2014

Phys. Rev. E

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In this paper, we propose a local nonequilibrium scheme for computing the flux of the convection-diffusion equation with a source term in the framework of the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). Both the Chapman-Enskog analysis and the numerical results show that, at the diffusive scaling, the present nonequilibrium scheme has a second-order convergence rate in space. A comparison between the nonequilibrium scheme and the conventional second-order central-difference scheme indicates that, although both schemes have a second-order convergence rate in space, the present nonequilibrium scheme is more accurate than the central-difference scheme. In addition, the flux computation rendered by the present scheme also preserves the parallel computation feature of the LBM, making the scheme more efficient than conventional finite-difference schemes in the study of large-scale problems. Finally, a comparison between the single-relaxation-time model and the MRT model is also conducted, and the results show that the MRT model is more accurate than the single-relaxation-time model, both in solving the convection-diffusion equation and in computing the flux.

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Section: Multiple-relaxation-time Lattice Boltzmann Model For Thementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonequilibrium scheme for computing the flux of the convection-diffusion equation in the framework of the lattice Boltzmann method

Chai

Zhao

2014

Phys. Rev. E

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“…We also underline that our results in L ∞ are stronger than any results in L 2 , and that they are obtained by using convexity properties of the proposed LBM schemes and, thus, without using any discrete Fourier transform (up to our knowledge, L 2 -stability results are obtained with von Neumann analysis in the LBM realm: see for example [5,11,12,13,15,16,17,18]). In particular, our approach allows us to study the stability with non-periodic boundary conditions and to obtain discrete maximum principles, which is not possible with a von Neumann analysis.…”

Section: Introductionmentioning

confidence: 96%

“…This approach is possible because of the simplicity of the D1Q2 lattice. Nevertheless, it is not obvious that our approach would be possible for other lattices such as the D1Q3 lattice studied (among other lattices) in [5,13,15,16,17].…”

Section: Introductionmentioning

confidence: 99%

“…LBM schemes are now often used to solve many type of Partial Differential Equations (PDEs) system. Among them, we find the heat equation with or without phase change [4,5,6,7], the heat equation with radiative source term [8], the hyperbolic heat equation (also named telegraph equation) with or without radiative source term [9], the Richard equation for porous media [10], the advection-diffusion equation [11,12,13,14,15,16,17], the advection equation [18], the incompressible Navier-Stokes system [19] eventually applied in porous media with heat and mass transfer [20] or in a diphasic situation [21,22] or with a free-surface [23], the Bingham model for viscoplastic flows [24]. Among the reasons which justify the use of LBM schemes, we can cite its algorithmic simplicity, its time explicit nature, its scalability when the algorithm is parallelized.…”

Section: Introductionmentioning

confidence: 99%

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Construction and Analysis of Lattice Boltzmann Methods Applied to a 1D Convection-Diffusion Equation

Dellacherie

2013

Acta Appl Math

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We construct and we analyze two LBM schemes build on the D1Q2 lattice to solve the 1D (linear) convectiondiffusion equation. We obtain these LBM schemes by showing that the 1D convection-diffusion equation is the fluid limit of a discrete velocity kinetic system. Then, we show in the periodic case that these LBM schemes are equivalent to a finite difference type scheme named LFCCDF scheme. This allows us, firstly, to prove the convergence in L ∞ of these schemes, and to obtain discrete maximum principles for any time step in the case of the 1D diffusion equation with different boundary conditions. Secondly, this allows us to obtain most of these results for the Du Fort-Frankel scheme for a particular choice of the first iterate. We also underline that these LBM schemes can be applied to the (linear) advection equation and we obtain a stability result in L ∞ under a classical CFL condition. Moreover, by proposing a probabilistic interpretation of these LBM schemes, we also obtain Monte-Carlo algorithms which approach the 1D (linear) diffusion equation. At last, we present numerical applications justifying these results.

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Lattice Boltzmann simulation of asymptotic longitudinal mass dispersion in reconstructed random porous media

et al. 2018

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To research macroscopic mass transport characteristics of porous media, a lattice Boltzmann method (LBM) approach was utilized to calculate asymptotic longitudinal mass dispersion. In this study, a D2Q9 model with multi‐relaxation‐time (MRT) collision operator, which is appropriate for incompressible flow with a high Péclet number without refining the lattice, was chosen. With respect to the microstructure of porous media, random placement (RP) method was applied to obtain randomly positioned particles. Based on the exhausted numerical results presented in the study, a new correlation of longitudinal mass dispersion was established. By comparing with available experimental data in the literature, reasonable agreements are observed in a wide porosity range from 0.3 to 0.7, indicating the validity of the proposed correlation. © 2018 American Institute of Chemical Engineers AIChE J, 64: 2770–2780, 2018

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Lattice Boltzmann method with two relaxation times for advection–diffusion equation: Third order analysis and stability analysis

Cited by 85 publications

References 22 publications

Nonequilibrium scheme for computing the flux of the convection-diffusion equation in the framework of the lattice Boltzmann method

Nonequilibrium scheme for computing the flux of the convection-diffusion equation in the framework of the lattice Boltzmann method

Construction and Analysis of Lattice Boltzmann Methods Applied to a 1D Convection-Diffusion Equation

Lattice Boltzmann simulation of asymptotic longitudinal mass dispersion in reconstructed random porous media

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