The analysis of an efficient numerical scheme for the Allen–Cahn equations using the Galerkin method

Pius W.M. Chin

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we propose an efficient numerical scheme for the Allen–Cahn equations. We show theoretically using the Galerkin method and the compactness theorem that the solution of the afore-mentioned equation exists and is unique in appropriate spaces with the interaction length parameter α well controlled. We further, show numerically that the proposed scheme is stable and converge optimally in the L2 as well as the H1-norms with its numerical solution preserving all the qualitative properties of the exact solution. With the help of an example and a carefully chosen α, we use numerical experiments to justify the validity of the proposed scheme.

Original languageEnglish
Article number106061
JournalCommunications in Nonlinear Science and Numerical Simulation
Volume105
DOIs
Publication statusPublished - Feb 2022
Externally publishedYes

Keywords

  • Allen–Cahn equations
  • Galerkin method
  • Non-standard finite difference method
  • Nonlinear equation
  • Optimal rate of convergence
  • Stability

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