A Second‐Order Fully Decoupled Scheme and Its Optimal Finite Element Error Estimate for a Two‐Phase Magnetohydrodynamics Model

ABSTRACT

This paper proposes a fully decoupled, linear, second‐order accurate, fully discrete numerical scheme with unconditional energy stability for the reformulated two‐phase magnetohydrodynamics model. The reformulation itself is achieved by unifying the scalar auxiliary variable (SAV) and zero‐energy‐contribution (ZEC) methods via a time‐dependent auxiliary variable. The scheme achieves second‐order temporal accuracy by second‐order backward differential formula (BDF2), realizes decoupling and linearity among the phase, flow, and magnetic fields via the unified SAV–ZEC method and a fully explicit treatment of the nonlinear terms and their coefficients, and attains intra‐field decoupling within the flow field by the second‐order pressure‐correction projection method. The unconditional energy stability and optimal error estimate of the scheme are established. One simplification in the phase‐field analysis involves embedding the inverse of discrete Laplacian into the test functions. This embedding induces an – norm transformation, which allows the associated terms to be estimated directly. One technique to achieve second‐order temporal accuracy involves an auxiliary function that incorporates a first‐order temporal pressure approximation for canceling the lower‐order temporal terms. The introduction of a Stokes quasi‐projection operator similarly prevents the loss of the spatial convergence rate. Four numerical examples are presented to verify the efficacy of the proposed scheme.