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Finite-difference lattice Boltzmann model for nonlinear convection-diffusion equations

Author

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  • Wang, Huili
  • Shi, Baochang
  • Liang, Hong
  • Chai, Zhenhua

Abstract

In this paper, a finite-difference lattice Boltzmann (LB) model for nonlinear isotropic and anisotropic convection-diffusion equations is proposed. In this model, the equilibrium distribution function is delicately designed in order to recover the convection-diffusion equation exactly. Different from the standard LB model, the temporal and spatial steps in this model are decoupled such that it is convenient to study convection-diffusion problem with the non-uniform grid. In addition, it also preserves the advantage of standard LB model that the complex-valued convection-diffusion equation can be solved directly. The von Neumann stability analysis is conducted to discuss the stability region which can be used to determine the free parameters appeared in the model. To test the performance of the model, a series of numerical simulations of some classical problems, including the diffusion equation, the nonlinear heat conduction equation, the Sine-Gordon equation, the Gaussian hill problem, the Burgers–Fisher equation, and the nonlinear Schrödinger equation, have also been carried out. The results show that the present model has a second-order convergence rate in space, and generally it is also more accurate than the standard LB model.

Suggested Citation

  • Wang, Huili & Shi, Baochang & Liang, Hong & Chai, Zhenhua, 2017. "Finite-difference lattice Boltzmann model for nonlinear convection-diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 334-349.
    • Handle: RePEc:eee:apmaco:v:309:y:2017:i:c:p:334-349
    DOI: 10.1016/j.amc.2017.04.015
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    References listed on IDEAS

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    1. Sheng, Q. & Khaliq, A.Q. M. & Voss, D.A., 2005. "Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 68(4), pages 355-373.
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    Cited by:

    1. Cheichan, Mohammed S. & Kashkool, Hashim A. & Gao, Fuzheng, 2019. "A weak Galerkin finite element method for solving nonlinear convection-diffusion problems in two dimensions," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 149-163.
    2. Krivovichev, Gerasim V., 2018. "Linear Bhatnagar–Gross–Krook equations for simulation of linear diffusion equation by lattice Boltzmann method," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 102-119.

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