A Linear Energy Stable Scheme for Two-Phase Magnetohydrodynamic Diffuse Interface Model

In this paper, we study a numerical approximation of the Cahn–Hilliard–magnetohydrodynamic diffuse interface model. First, based on the idea of Lagrange multiplier, we introduce an auxiliary variable, and reformulate the Cahn–Hilliard–magnetohydrodynamic system. Second, we construct a linear, decoupled, and first-order semi-discrete scheme for this system by combining the pressure-projection method and some subtle implicit–explicit treatments for nonlinear coupling terms. We also prove that the proposed scheme is unconditionally energy stable. Furthermore, error estimates are provided through rigorous theoretical analysis. Finally, we validate the efficiency of the scheme through numerical simulations.