A reduced order method for Allen–Cahn equations

Abstract

In this article, we present a reduced order method for modeling and computing Allen–Cahn equations. A global basis method is used in the discretized system of the Allen–Cahn equations and Proper Orthogonal Decomposition (POD) method is utilized to reduce the global basis. To treat the difficulty of nonlinearity for Allen–Cahn equations, we apply Discrete Empirical Interpolation method (DEIM) to the nonlinear term from the discretization system. A reduced order method is developed by integrating POD and DEIM. It is well-known that the Allen–Cahn equations have a nonlinear stability property, i.e., the free-energy functional decreases with respect to time. The discretized Allen–Cahn system modeled by the POD–DEIM reduced order method can inherit the nonlinear stability of the continuous model. The computation efficiency is significantly enhanced by using the reduced order method. A few numerical results are presented to illustrate the performance of the reduced order method for deterministic Allen–Cahn equations and stochastic Allen–Cahn equations.