A second-order maximum bound principle preserving operator splitting method for the Allen–Cahn equation with applications in multi-phase systems

In this paper, a highly efficient space–time operator splitting finite element method is presented to solve the two- and three-dimensional Allen-Cahn equations. The main advantage of the proposed method is that it reduces the high storage requirements and complexity of the high-dimensional computation by splitting the high-dimensional problem into a series of one-dimensional subproblems. The proposed method is space–time second-order and can be performed in parallel. Moreover, a bound preserving least-distance modification technique is developed to force the discrete maximum bound principle in solving each one-dimensional subproblem. Finally, numerical simulations including the two- and multi-phase separations, mean curvature flows and dendritic crystal growth in two and three dimensions are provided to demonstrate the validity and accuracy of the proposed method. • A highly efficient second-order space–time operator splitting method is presented to solve the two- and three-dimensional Allen-Cahn equations in parallel. • A new least-distance modification technique is developed to force the discrete maximum bound principle of the numerical solution. • The proposed numerical method is applied to simulate multi-phase separation and dendritic crystal growth in two and three dimensions.