Energy Stable Discontinuous Galerkin Finite Element Method for the Allen-Cahn Equation

Abstract

Allen--Cahn equation with constant and degenerate mobility, and with polynomial and logarithmic energy functionals is discretized using symmetric interior penalty discontinuous Galerkin (SIPG) finite elements in space. We show that the energy stable average vector field (AVF) method as the time integrator for gradient systems like the Allen-Cahn equation satisfies the energy decreasing property for the fully discrete scheme. The numerical results for one and two dimensional Allen-Cahn equation with periodic boundary condition, using adaptive time stepping, reveal that the discrete energy decreases monotonically, the phase separation and metastability phenomena can be observed and the ripening time is detected correctly.