Error estimates of semi-implicit numerical scheme for a diffuse interface model of two-phase magnetohydrodynamic flows

In this paper, we carry out a rigorous error analysis of the fully discrete semi-implicit numerical scheme proposed in Yang et al. (2019) for the diffuse interface model of two-phase magnetohydrodynamics (MHD) flows with different viscosities and electric conductivities in two and three-dimensional cases. The nonlinear and strong coupled properties and the variable coefficients of the model itself bring the major analytical difficulties in the error estimates. Based on three projection operators, including Stokes projection, Maxwell projection and Ritz projection, we select appropriate test functions, apply the Lipschitz continuous properties of the variable coefficients, and develop the strategies of utilizing intermediate terms to address the major difficulties caused by the model itself. Finally, we establish both the spatial and temporal convergence rates.