Zhaowei Wang

In this paper, we construct error analysis of second‐order decoupled scheme for the two‐phase magnetohydrodynamic diffuse interface model. First, we construct a second‐order time‐discrete scheme in which the time derivative term is treated by the second‐order backward difference formulation, the scalar auxiliary variable (SAV) method is used to linearize the nonlinear energy potential, and the other nonlinear terms are treated by semi‐implicit method. Importantly, we use Gauge–Uzawa method to decouple velocity and pressure. The proposed scheme have two distinct features: It does not need to give an initial value of the pressure; it does not require artificial boundary conditions on pressure, thus saving additional computational cost. Second, the unconditional stability of the second‐order scheme is proved. Then, through rigorous error analysis, we provide suboptimal convergence orders. Finally, some numerical experiments demonstrate the accuracy and effectiveness of the proposed scheme. It is also proved that the proposed scheme is superior to other classical second‐order scheme.

In this paper, we construct an optimal convergence analysis of second‐order stabilized scheme for the thermally coupled incompressible magnetohydrodynamic (MHD) system. First, we construct first‐ and second‐order time‐discrete schemes in which the time derivative term is treated by the first‐order backward Euler method and the second‐order backward difference formulation, respectively. And the nonlinear terms are treated by semi‐implicit method. Importantly, we use the Gauge–Uzawa method to decouple velocity and pressure. The proposed schemes have the following two distinct features: they do not need to give an initial value of the pressure, and they do not require artificial boundary conditions on the pressure. Second, the unconditional energy stability of the two schemes is proved. Then, through rigorous error analysis, we provide optimal convergence orders for all unknowns. Finally, some numerical experiments demonstrate the accuracy and effectiveness of the proposed schemes.

In this paper, we first establish a novel first-order, fully decoupled, unconditionally stable time discretization scheme for the MHD system with variable density. This scheme successfully decouples all the coupling terms by combining the gauge-Uzawa method and the scalar auxiliary variable (SAV) method. And we prove its unconditional energy stability. Then we give the first-order finite element scheme and its implementation. Furthermore, we perform a rigorous error analysis of the proposed numerical scheme. Finally, we perform some numerical experiments to demonstrate the effectiveness of the decoupling scheme.