The conservative Allen–Cahn equation with a nonlocal Lagrange multiplier satisfies mass conservation and energy dissipation property. A challenge to numerically solving the equation is how to treat the nonlinear and nonlocal terms to preserve mass conservation and energy stability without compromising accuracy. To resolve this problem, we first apply the convex splitting idea to not only the term corresponding to the Allen–Cahn equation but also the nonlocal term. A wise implementation of the convex splitting for the nonlocal term ensures numerically exact mass conservation. And we combine the convex splitting with the specially designed implicit–explicit Runge–Kutta method. We show analytically that the scheme is uniquely solvable and unconditionally energy stable by using the fact that the scheme guarantees exact mass conservation. Numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed scheme.
Bibliographical noteFunding Information:
The authors thank the reviewers for the constructive and helpful comments on the revision of this article. This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science and ICT of Korea (MSIT) (Nos. 2019R1A6A1A11051177, 2019R1C1C1011112, 2020R1C1C1A01013468).
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
- Conservative Allen–Cahn equation
- Convex splitting
- High-order time accuracy
- Mass conservation
- Unconditional energy stability
- Unconditional unique solvability