Error estimates for the highly efficient and energy stable schemes for the 2D/3D two-phase MHD

In this paper, we mainly focus on the rigorous convergence analysis of two fully decoupled, unconditionally energy-stable methods for the diffuse interface two-phase magnetohydrodynamics (MHD) model. The two methods consist of the semi-implicit stabilization method and the invariant energy quadratization (IEQ) method, which are both applied to the phase field system. In addition, the pressure correction method is used for the saddle point system, and appropriate implicit-explicit treatments are employed for the nonlinear coupled terms. We prove the unconditional energy stability of the two schemes. In addition, we mainly establish the error estimates based on the bounds of $\left\|\phi^{k}\right\|_{L^{\infty}}$ and $\left\|\textbf{b}^{k}\right\|_{L^{\infty}}$. Several numerical examples are presented to test the accuracy and stability of the proposed methods.