Double-D2Q9 lattice Boltzmann models with extended equilibrium for two-dimensional magnetohydrodynamic flows

Abstract

The vast majority of the existing lattice Boltzmann methods (LBMs) suggest to relax relevant quantities to a second-order truncated equilibrium state. Despite its simplicity and popularity, this choice does not fully exploit the potential of any lattice discretization. In this paper, an extended equilibrium state is adopted to evaluate the suitability of different LBMs (i.e., the Bhatnagar-Gross-Krook, the multiple-relaxation-time in terms of raw and central moments, and the simplified one) to simulate two-dimensional magnetohydrodynamic flows by means of the D2Q9 velocity space. Two sets of particle distribution functions are employed: one for the flow field and the other for the magnetic one. Even if the minimal five-velocities discretization is sufficient to represent the evolution of the latter, a nine-velocities model enhances the capability to enforce the divergence-free condition of the magnetic field, as shown. Therefore, a double-D2Q9 approach is herein devised. Eventually, the computational cost involved by all the schemes is discussed both in terms of virtual memory and run time. Interestingly, the simplified LBM for magnetohydrodynamic flows is herein presented for the first time.