In the design of fusion reactors, liquid metal represents an optimal solution for the material of the blanket and the first wall, given its superior thermal conductivity and chemical stability. The significant magnetohydrodynamics (MHD) effects exhibited by liquid metals in the strong magnetic field environment of fusion reactors require the development of highly accurate, efficient, and robust simulation algorithms capable of accurately representing the complex interactions between the fluid and the magnetic field. Taking advantage of the lattice Boltzmann method (LBM), this paper develops an LBM algorithm for MHD effect. The algorithm is based on a double distribution function model with coupled computation of the flow field and magnetic field. Furthermore, the collision terms in the LBM are processed by the latest recursive regularized approach to filter out unphysical modes associated with the non-equilibrium state distribution function, thus enhancing the robustness of the algorithm. The numerical experimental results demonstrate that the algorithm has good accuracy and robustness, which preliminarily verifies its effectiveness and potential in dealing with MHD problems.