A High-Order and Unconditionally Energy Stable Scheme for the Conservative Allen-Cahn Equation with a Nonlocal Lagrange Multiplier

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.