Lattice Boltzmann magnetohydrodynamics

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Abstract
Lattice gas and lattice Boltzmann methods are recently developed numerical schemes for simulating a variety of physical systems. In this paper a new lattice Boltzmann model for modeling two‐dimensional (2‐D) incompressible magnetohydrodynamics (MHD) is presented. The current model fully utilizes the flexibility of the lattice Boltzmann method in comparison with previous lattice gas and lattice Boltzmann MHD models, reducing the number of moving directions from 36 in other models to 12 only. To increase computational efficiency, a simple single time relaxation rule is used for collisions, which directly controls the transport coefficients. The bidirectional streaming process of the particle distribution function in this paper is similar to the original model [H. Chen and W. H. Matthaeus, Phys. Rev. Lett. 58, 1845 (1987), S. Chen et al., Phys. Rev. Lett. 67, 3776 (1991)], but has been greatly simplified, affording simpler implementation of boundary conditions and increasing the feasibility of extension into a workable three‐dimensional (3‐D) model. Analytical expressions for the transport coefficients are presented. Also, as example cases, numerical calculation for the Hartmann flow is performed, showing a good agreement between the theoretical prediction and numerical simulation, and a sheet‐pinch simulation is performed and compared with the results obtained with a spectral method.