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

Abstract

It is well known that the classic Allen–Cahn equation satisfies the maximum bound principle (MBP), that is, the absolute value of its solution is uniformly bounded for all time by certain constant under suitable initial and boundary conditions. In this paper, we consider numerical solutions of the modified Allen–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 also conserves the mass exactly. We reformulate the model equation with a linear stabilizing technique, then construct first- and second-order exponential time differencing schemes for its time integration. We prove the unconditional MBP preservation and mass conservation of the proposed schemes in the time discrete sense and derive their error estimates under some regularity assumptions. Various numerical experiments in two and three dimensions are also conducted to verify the theoretical results.