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

TL;DR

This work tackles the challenge of numerically solving diffuse-interface two-phase magnetohydrodynamics by introducing two linear, fully decoupled time-stepping schemes that are unconditionally energy-stable. The first method uses semi-implicit stabilization, while the second employs the invariant energy quadratization (IEQ) approach; both enable decoupled subproblems and rely on pressure correction for incompressibility. The authors establish unconditional energy stability and rigorous error estimates, deriving key bounds on $\|\phi^k\|_{L^{\infty}}$ and $\|\mathbf{b}^k\|_{L^{\infty}}$ to support convergence without time-step restrictions, and validate the theory with numerical experiments on smooth solutions, spinodal decomposition, and Boussinesq-type setups. Overall, the results provide robust, efficient, and scalable schemes for simulating 2D/3D two-phase MHD with reliable stability and accuracy guarantees.

Abstract

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 $\left\|φ^{k}\right\|_{L^{\infty}}$ and $\left\|\textbf{b}^{k}\right\|_{L^{\infty}}$. Several numerical examples are presented to test the accuracy and stability of the proposed methods.

Figures (5)

• Figure 1: The algorithm energy of Scheme I (a), Scheme III (b); The mass of phase field for Scheme I (c) and Scheme III (d).
• Figure 2: The dynamics of spinodal decomposition examples for Scheme I with $\varepsilon$=0.01, $\lambda$=0.01$, \bar{\phi}$=-0.05, $\mu$=1.
• Figure 3: The dynamics of spinodal decomposition examples for Scheme III with $\varepsilon$=0.01, $\lambda$=0.01$, \bar{\phi}$=-0.05, $\mu$=1.
• Figure 4: Snapshots of phase field without the Lorentz force term.
• Figure 5: Snapshots of phase field with $\mu$=0.001.

Theorems & Definitions (30)

• Remark 2.1
• Theorem 2.1
• proof
• Remark 3.1
• Remark 3.2
• Theorem 3.1
• proof
• Remark 3.3
• Lemma 3.1
• Lemma 3.2