Linearly implicit methods for Allen-Cahn equation


Uzunca M., KARASÖZEN B.

Applied Mathematics and Computation, vol.450, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 450
  • Publication Date: 2023
  • Doi Number: 10.1016/j.amc.2023.127984
  • Journal Name: Applied Mathematics and Computation
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, INSPEC, Public Affairs Index, zbMATH, Civil Engineering Abstracts
  • Keywords: Allen -Cahn equation, Gradient systems, Energy dissipation, Linearly implicit methods, SCHEMES, INTEGRATION
  • Middle East Technical University Affiliated: Yes

Abstract

It is well known that the Allen-Cahn equation satisfies a nonlinear stability property, i.e., the free-energy functional decreases in time. Linearly implicit integrators have been developed for energy-preserving methods for conservative systems with polynomial Hamiltonians, which are based on the concept of polarization. In this paper, we construct linearly implicit methods for gradient flows preserving the energy dissipation by polarizing the free-energy functional. Two-step linearly implicit methods are derived for the Allen-Cahn equation inheriting energy dissipation law. Numerical experiments for one-, two-, and three-dimensional Allen-Cahn equations demonstrate the energy dissipation and the accuracy of the linearly implicit methods.