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Finite-difference lattice Boltzmann model for nonlinear convection-diffusion equations. (English) Zbl 1411.76146
Summary: 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.

MSC:
76M28 Particle methods and lattice-gas methods
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
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