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It is well known that the classic Allen-Cahn equation satisfies the maximumbound principle (MBP), that is, the absolute value of its solution is uniformlybounded for all time by certain constant under suitable initial and boundaryconditions. In this paper, we consider numerical solutions of the modifiedAllen-Cahn equation with a Lagrange multiplier of nonlocal and local effects,which not only shares the same MBP as the original Allen-Cahn equation but alsoconserves the mass exactly. We reformulate the model equation with a linearstabilizing technique, then construct first- and second-order exponential timedifferencing schemes for its time integration. We prove the unconditional MBPpreservation and mass conservation of the proposed schemes in the time discretesense and derive their error estimates under some regularity assumptions.Various numerical experiments in two and three dimensions are also conducted toverify the theoretical results.

Unconditionally stable exponential time differencing schemes for the mass-conserving Allen-Cahn equation with nonlocal and local effects