A Linear Unconditionally Maximum Bound Principle‐Preserving BDF2 Scheme for the Mass‐Conservative Allen‐Cahn Equation

ABSTRACT The mass‐conservative Allen‐Cahn (MCAC) equation is a critical component in phase field modeling, which provides a means to describe the evolution of phase boundaries and interfaces satisfying the maximum bound principle (MBP) while ensuring mass conservation. In this paper, we propose and analyze a linear second‐order unconditionally MBP preserving and mass‐conservative numerical scheme for the MCAC equation based on the stabilizing method with two stabilization terms. More precisely, we use the central finite difference method for spatial discretization and the second‐order backward differentiation formulation (BDF2) with variable time steps. Under mild constraints on the adjacent time‐step ratios and the two stabilizing parameters, we obtain that the proposed numerical scheme is unconditionally mass‐conserving and MBP‐preserving. Further, the error estimate of the proposed numerical scheme with the discrete ‐norm is successfully derived, along with a uniform bound of a mild modified discrete energy. Finally, ample numerical experiments are carried out to validate the theoretical results and demonstrate the performance of the method.