In comparison with the Cahn–Hilliard equation, the classic Allen-Cahn equation satisfies the maximum bound principle (MBP) but fails to conserve the mass along the time. In this paper, we consider the MBP and corresponding numerical schemes for the modified Allen–Cahn equation, which is formed by introducing a nonlocal Lagrange multiplier term to enforce the mass conservation. We first study sufficient conditions on the nonlinear potentials under which the MBP holds and provide some concrete examples of nonlinear functions. Then we propose first and second order stabilized exponential time differencing schemes for time integration, which are linear schemes and unconditionally preserve the MBP in the time discrete level. Convergence of these schemes is analyzed as well as their energy stability. Various two and three dimensional numerical experiments are also carried out to validate the theoretical results and demonstrate the performance of the proposed schemes.

与Cahn–Hilliard方程相比，经典的Allen-Cahn方程满足最大边界原理（MBP），但不能随时间节省质量。在本文中，我们考虑了MBP和相应的数值方案，用于改进的Allen-Cahn方程，该方程是通过引入非局部拉格朗日乘数项来实施质量守恒而形成的。我们首先研究MBP所具有的非线性电势的充分条件，并提供一些非线性函数的具体示例。然后，我们提出了用于时间积分的一阶和二阶稳定指数时间差分方案，它们是线性方案，无条件地将MBP保留在时间离散级别上。分析了这些方案的收敛性以及它们的能量稳定性。