Energy stable model order reduction for the Allen-Cahn equation

TL;DR

This paper presents an energy stable reduced order model for the Allen-Cahn equation, combining structure-preserving discretizations and adaptive sampling to ensure efficiency and accuracy in parametrized settings.

Contribution

It introduces an energy stable ROM framework for the Allen-Cahn equation using dG, AVF, POD, and DEIM, preserving the energy decreasing property of the full model.

Findings

The ROM maintains the energy decreasing property of the full model.

Numerical tests show high accuracy and efficiency for parametrized problems.

The method is effective with Neumann and periodic boundary conditions.

Abstract

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