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

TL;DR

This paper develops an energy-stable discontinuous Galerkin finite element method combined with an AVF time integrator for the Allen-Cahn equation, ensuring energy decrease and accurately capturing phase separation phenomena.

Contribution

It introduces a novel combination of SIPG spatial discretization with AVF time integration for the Allen-Cahn equation, guaranteeing energy stability in the fully discrete scheme.

Findings

Discrete energy decreases monotonically in simulations.

The method accurately captures phase separation and metastability.

Ripening time is correctly detected in numerical experiments.

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.