Unconditional MBP preservation and energy stability of the stabilized exponential time differencing schemes for the vector-valued Allen–Cahn equations

Abstract

The vector-valued Allen–Cahn equations have been extensively applied to simulate the multiphase flow models. In this paper, we consider the maximum bound principle (MBP) and corresponding numerical schemes for the vector-valued Allen–Cahn equations. We firstly formulate the stabilized equations via utilizing the linear stabilization technique, and then focus on the bounding constant of the nonlinear function based on the fact that the extremes of a constrained problem will occur in the bounded and convex domain. Later the first- and second-order stabilized exponential time differencing schemes are adopted for temporal integration, which are linear and unconditionally preserve the discrete MBP in the time discrete sense. Moreover, the proposed schemes can be proven to dissipate the original energy instead of the modified energy. Their convergence analysis is also presented. Various numerical examples in two and three dimensions are performed to verify these theoretical results and demonstrate the efficiency of the proposed schemes.