Lattice Boltzmann Magnetohydrodynamics

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 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 bi-directional 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., {\bf 58}, 1845(1987), S.Chen, H.Chen, D.Mart\'{\i}nez and W.H.Matthaeus, Phys. Rev. Lett. {\bf 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 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