A reduced-order energy-stability-preserving finite difference iterative scheme based on POD for the Allen-Cahn equation

We focus on proposing and analyzing a reduced-order finite difference (ROFD) iterative scheme based on a proper orthogonal decomposition (POD) technique for Allen-Cahn equations with a small perturbation parameter and strong nonlinearity. We prove the discrete maximum-principle (DMP) preserving and discrete energy-stability (DES) preserving of the ROFD iterative solutions under a reasonable restriction on the time step size. We also analyze the convergence of the ROFD iterative solutions for Allen-Cahn equations. Numerical tests are presented to verify our proposed ROFD method, such as error estimates, convergence rates, DMP-preserving and DES-preserving. Moreover, it is shown that the computing time of ROFD iterative schemes with very few unknowns is much less than that of usual finite difference (FD) iterative schemes.