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

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KARASÖZEN B., Uzunca M., Sariaydin-Filibelioglu A., YÜCEL H.

INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS, vol.15, no.3, 2018 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 15 Issue: 3
  • Publication Date: 2018
  • Doi Number: 10.1142/s0219876218500135
  • Journal Name: INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Keywords: Allen-Cahn equation, gradient systems, discontinuous Galerkin method, unconditional energy stability, time adaptivity, CONVEX SPLITTING SCHEMES, NUMERICAL SIMULATIONS, HILLIARD EQUATION, MEAN-CURVATURE, GRADIENT FLOWS, APPROXIMATION, 2ND-ORDER, DISCRETIZATION, MODEL
  • Middle East Technical University Affiliated: Yes

Abstract

In this paper, we investigate numerical solution of Allen-Cahn equation with constant and degenerate mobility, and with polynomial and logarithmic energy functionals. We discretize the model equation by symmetric interior penalty Galerkin (SIPG) method in space, and by average vector field (AVF) method in time. We show that the energy stable AVF method as the time integrator for gradient systems like the Allen-Cahn equation satisfies the energy decreasing property for fully discrete scheme. Numerical results reveal that the discrete energy decreases monotonically, the phase separation and metastability phenomena can be observed, and the ripening time is detected correctly.