One-stage simplified lattice Boltzmann method for two- and three-dimensional magnetohydrodynamic flows

In this paper, we propose a new simplified lattice Boltzmann method (SLBM) for magnetohydrodynamic flows that outperforms the classical one in terms of accuracy, while preserving its advantages. A very recent paper [De Rosis et al., “Double-D2Q9 lattice Boltzmann models with extended equilibrium for two-dimensional magnetohydrodynamic flows,” Phys. Fluids 33, 035143 (2021)] demonstrated that the SLBM enforces the divergence-free condition of the magnetic field in an excellent manner and involves the lowest amount of virtual memory. However, the SLBM is characterized by the poorest accuracy. Here, the two-stage algorithm that is typical of the SLBM is replaced by a one-stage procedure following the approach devised for non-conductive fluids in a very recent effort [Delgado-Gutierrez et al., “A single-step and simplified graphics processing unit lattice Boltzmann method for high turbulent flows,” Int. J. Numer. Methods Fluids 93, 2339 (2021)]. The Chapman–Enskog expansion formally demonstrates the consistency of the present scheme. The resultant algorithm is very compact and easily implemented. Given all these features, we believe that the proposed approach is an excellent candidate to perform numerical simulations of two- and three-dimensional magnetohydrodynamic flows.