Efficient numerical scheme for solving the Allen-Cahn equation

This article presents an efficient and robust algorithm for the numerical solution of the Allen-Cahn equation, which represents the motion of antiphase boundaries. The proposed algorithm is based on the diagonally implicit fractional-step θ scheme for time discretization and the conforming finite element method for space discretization. For the steady-state solution, both uniform and adaptive grids are used to illustrate the effectiveness of adaptive grids over uniform grids. For the unsteady solution, the diagonally implicit fractional-step θ scheme is compared with other time discretization schemes in terms of computational cost and temporal error estimation accuracy. Numerical examples are presented to illustrate the capabilities of the proposed algorithm in solving nonlinear partial differential equations.