Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation

Yoshida, Hiroaki; Nagaoka, Makoto

doi:10.1016/j.jcp.2010.06.037

Cited by 294 publications

(286 citation statements)

References 61 publications

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“…I, in developing LB models for the CDE, many previous investigators focused on the accuracy and convergence rate of the model in describing the scalar variable φ [12][13][14][15][16][17][18][19][20][21]. More recently, the heat or mass flux, as another important physical variable, has also received increasing attention in predicting effective physical properties of porous media [33].…”

Section: A Local Scheme For the Heat And Mass Fluxesmentioning

confidence: 99%

“…The lattice Boltzmann method (LBM), as a kinetic-based numerical method, has made a great progress in the study of fluid flows for its advantage in dealing with complex boundaries [5][6][7][8][9], and has also been extended to solve the CDE [10][11][12][13][14][15][16][17][18][19][20][21][22]. Dawson et al [10] first applied the LBM to the study of solvent flow where an LB model was used to solve the N-S equations for fluid flow, while another was adopted to solve the CDE for the concentration field; the method has also been extended to investigate thermal flows [11].…”

Section: Introductionmentioning

confidence: 99%

“…However, as pointed out by Ginzburg [16], this BGK-typed LB model cannot have a mass conserving equilibrium distribution function once the relaxation times differ with each other. To overcome this problem inherent in the BGK-typed LB model, the two-relaxation-time and multiple-relaxation-time LB models have also been proposed in a more general way to solve the anisotropic CDE [16][17][18][19]. Although many LB models have been proposed for the CDE, the Chapman-Enskog analysis shows that the CDE can only be recovered exactly from these models under some unrealistic assumptions (e.g., the velocity must be a constant).…”

Section: Introductionmentioning

confidence: 99%

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Lattice Boltzmann model for the convection-diffusion equation

2013

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We propose a lattice Boltzmann (LB) model for the convection-diffusion equation (CDE) and show that the CDE can be recovered correctly from the model by the Chapman-Enskog analysis. The most striking feature of the present LB model is that it enables the collision process to be implemented locally, making it possible to retain the advantage of the lattice Boltzmann method in the study of the heat and mass transfer in complex geometries. A local scheme for computing the heat and mass fluxes is then proposed to replace conventional nonlocal finite-difference schemes. We further validate the present model and the local scheme for computing the flux against analytical solutions to several classical problems, and we show that both the model for the CDE and the computational scheme for the flux have a second-order convergence rate in space. It is also demonstrated the present model is more accurate than existing LB models for the CDE.

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Section: A Local Scheme For the Heat And Mass Fluxesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lattice Boltzmann model for the convection-diffusion equation

2013

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“…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%

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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“…(1) with an isotropic diffusion coefficient of c = (2τ − 1)/8 (see [4] for the proof). To extend the model for anisotropic diffusion, like in the heart, the matrix A is replaced by A = M −1 SM [9], where…”

Section: Lattice-boltzmann Model Of Cardiac Electrophysiologymentioning

confidence: 99%

LBM-EP: Lattice-Boltzmann Method for Fast Cardiac Electrophysiology Simulation from 3D Images

Rapaka

Mansi

Georgescu

et al. 2012

Medical Image Computing and Computer-Assisted Intervention – MICCAI 2012

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Abstract. Current treatments of heart rhythm troubles require careful planning and guidance for optimal outcomes. Computational models of cardiac electrophysiology are being proposed for therapy planning but current approaches are either too simplified or too computationally intensive for patient-specific simulations in clinical practice. This paper presents a novel approach, LBM-EP, to solve any type of mono-domain cardiac electrophysiology models at near real-time that is especially tailored for patient-specific simulations. The domain is discretized on a Cartesian grid with a level-set representation of patient's heart geometry, previously estimated from images automatically. The cell model is calculated node-wise, while the transmembrane potential is diffused using Lattice-Boltzmann method within the domain defined by the levelset. Experiments on synthetic cases, on a data set from CESC'10 and on one patient with myocardium scar showed that LBM-EP provides results comparable to an FEM implementation, while being 10 − 45 times faster. Fast, accurate, scalable and requiring no specific meshing, LBM-EP paves the way to efficient and detailed models of cardiac electrophysiology for therapy planning.

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Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation

Cited by 294 publications

References 61 publications

Lattice Boltzmann model for the convection-diffusion equation

Lattice Boltzmann model for the convection-diffusion equation

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

LBM-EP: Lattice-Boltzmann Method for Fast Cardiac Electrophysiology Simulation from 3D Images

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