Energy stable model order reduction for the 1 Allen - Cahn equation

The Allen-Cahn equation is a gradient system, where the free-energy func4 tional decreases monotonically in time. We develop an energy stable reduced order 5 model (ROM) for a gradient system, which inherits the energy decreasing property 6 of the full order model (FOM). For the space discretization we apply a discontinuous 7 Galerkin (dG) method and for time discretization the energy stable average vector 8 field (AVF) method. We construct ROMs with proper orthogonal decomposition 9 (POD)-greedy adaptive sampling of the snapshots in time and evaluating the non10 linear function with greedy discrete empirical interpolation method (DEIM). The 11 computational efficiency and accuracy of the reduced solutions are demonstrated 12 numerically for the parametrized Allen-Cahn equation with Neumann and periodic 13 boundary conditions. 14