Lattice Boltzmann model for the complex Ginzburg-Landau equation

Zhang, Jianying; Yan, Guangwu

doi:10.1103/physreve.81.066705

Cited by 23 publications

(22 citation statements)

References 80 publications

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“…2, the red line with symbols represents the result of the new model proposed in this paper, and the green one is the result of the model in Ref. [59]. This figure provides a qualitative trend of the numerical order of convergence.…”

Section: A Single Spiral Wavementioning

confidence: 89%

“…1, a spiral wave in the center is formed. The patterns are results of (a) Alternative Direction Implicit (ADI) scheme, (b) the one-order LBM in Reference [59], and (c) the model in this paper at time t = 50. The numerical results are in good agreement.…”

Section: A Single Spiral Wavementioning

confidence: 98%

“…Performing the Taylor expansion and Chapman-Enskog analysis on the lattice Boltzmann equation (4), a series of partial differential equations on the equilibrium distribution can be obtained [59], and based on these equations and distribution functions (10)-(11), (14)- (19), the macroscopic equation can be recovered as…”

Section: The Complex Ginzburg-landau Equation With the Fourth-order Tmentioning

confidence: 99%

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Numerical Studies Based on Higher-Order Accuracy Lattice Boltzmann Model for the Complex Ginzburg-Landau Equation

Zhang

Yan

2011

J Sci Comput

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In this paper, a higher-order accuracy lattice Boltzmann model for the complex Ginzburg-Landau equation is proposed. In order to obtain higher-order accuracy of truncation error and to overcome the drawbacks of "error rebound" in the previous models, a new assumption of additional distribution is presented to improve the accuracy of the model for the complex partial differential equation with nonlinear source term. As results, the complex Ginzburg-Landau equation is recovered with the fourth-order accuracy of truncation error. Based on this model, the problems of a single spiral wave in two-dimensional (2D) space and a single scroll in three-dimensional (3D) space are implemented to test the lattice Boltzmann scheme. The comparisons between the LBM results and the Alternative Direction Implicit results are given in detail. The numerical examples show that assumptions of source term can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the complex Ginzburg-Landau equation.

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Section: A Single Spiral Wavementioning

confidence: 89%

Section: A Single Spiral Wavementioning

confidence: 98%

Section: The Complex Ginzburg-landau Equation With the Fourth-order Tmentioning

confidence: 99%

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Numerical Studies Based on Higher-Order Accuracy Lattice Boltzmann Model for the Complex Ginzburg-Landau Equation

Zhang

Yan

2011

J Sci Comput

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“…By chosing appropriate collision operator or equilibrium distribution, the lattice Boltzmann model is able to recover the PDE of interest. Recently, it has been developed to simulate linear and nonlinear PDE such as Laplace equation [5], Poisson equation [6,7], the shallow equation [8], Burgers equation [9], Korteweg-de Vires equation [10], Wave equation [11,12], reaction-diffusion equation [13,14], convection-diffusion equation [15,16] and even some complex equations [17,18]. The LB models for advection and anisotropic dispersion equation have been proposed [19,20], among them the model by Ginzburg [21] is generic.…”

Section: Introductionmentioning

confidence: 99%

Lattice Boltzmann Model for Nonlinear Heat Equations

Zheng

Wang

et al. 2012

Lecture Notes in Computer Science

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Abstract. In this paper, a lattice Boltzmann scheme with an amending function for the nonlinear heat equations with the form ∂ t φ = α∇ 2 φ + ψ(φ) which directly to solve some important nonlinear equations, including Fisher equation, NewellWhitehead equation and FitzHugh-Nagumo equation is proposed . Detailed simulations of these equations are performed, and it is found that the numerical results agree well with the analytical solutions or the numerical solutions reported in previous studies.

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“…. , b with b the number of links that connect the center site to its neighbors (here a D2Q5 lattice is adopted, i.e., b = 4), then the corresponding lattice Boltzmann equation can be expressed as, [41][42][43] …”

Section: Lattice Boltzmann Modelsmentioning

confidence: 99%

Lattice Boltzmann models for the grain growth in polycrystalline systems

Zheng

Chen

et al. 2016

AIP Advances

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In the present work, lattice Boltzmann models are proposed for the computer simulation of normal grain growth in two-dimensional systems with/without immobile dispersed second-phase particles and involving the temperature gradient effect. These models are demonstrated theoretically to be equivalent to the phase field models based on the multiscale expansion. Simulation results of several representative examples show that the proposed models can effectively and accurately simulate the grain growth in various single- and two-phase systems. It is found that the grain growth in single-phase polycrystalline materials follows the power-law kinetics and the immobile second-phase particles can inhibit the grain growth in two-phase systems. It is further demonstrated that the grain growth can be tuned by the second-phase particles and the introduction of temperature gradient is also an effective way for the fabrication of polycrystalline materials with grained gradient microstructures. The proposed models are useful for the numerical design of the microstructure of materials and provide effective tools to guide the experiments. Moreover, these models can be easily extended to simulate two- and three-dimensional grain growth with considering the mobile second-phase particles, transient heat transfer, melt convection, etc.

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Lattice Boltzmann model for the complex Ginzburg-Landau equation

Cited by 23 publications

References 80 publications

Numerical Studies Based on Higher-Order Accuracy Lattice Boltzmann Model for the Complex Ginzburg-Landau Equation

Numerical Studies Based on Higher-Order Accuracy Lattice Boltzmann Model for the Complex Ginzburg-Landau Equation

Lattice Boltzmann Model for Nonlinear Heat Equations

Lattice Boltzmann models for the grain growth in polycrystalline systems

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