In this article, we study the reduced-order modelling for Allen-Cahn equation. First, a collection of phase field data, i.e., an ensemble of snapshots of at some
time instances is obtained from numerical simulation using a time-space discretization.
The full discretization makes use of a temporal scheme based on the scalar auxiliary
variable approach and a spatial spectral Galerkin method. It is shown that the time
stepping scheme is unconditionally stable. Then a reduced order method is developed
using by proper orthogonal decomposition (POD) and discrete empirical interpolation method (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. Our
numerical experiments show that the discretized Allen-Cahn system resulting from
the POD-DEIM method inherits this favorable property by using the scalar auxiliary
variable approach. A few numerical results are presented to illustrate the performance
of the proposed reduced order method. In particular, the numerical results show that
the computational efficiency is significantly enhanced as compared to directly solving
the full order system.