Reduced-Order Modelling for the Allen-Cahn Equation Based on Scalar Auxiliary Variable Approaches (Journal of Mathematical Study, 2019)
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 sometime instances is obtained from numerical simulation using a time-space discretization.The full discretization makes use of a temporal scheme based on the scalar auxiliaryvariable approach and a spatial spectral Galerkin method. It is shown that the timestepping scheme is unconditionally stable. Then a reduced order method is developedusing by proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM). It is well-known that the Allen-Cahn equations have a nonlinearstability property, i.e., the free-energy functional decreases with respect to time. Ournumerical experiments show that the discretized Allen-Cahn system resulting fromthe POD-DEIM method inherits this favorable property by using the scalar auxiliaryvariable approach. A few numerical results are presented to illustrate the performanceof the proposed reduced order method. In particular, the numerical results show thatthe computational efficiency is significantly enhanced as compared to directly solvingthe full order system.
Keywords: Allen-Cahn equation; scalar auxiliary variable; proper orthogonal decomposition; discrete empirical interpolation method.