AN ENERGY STABLE AND MAXIMUM BOUND PRESERVING SCHEME WITH VARIABLE TIME STEPS FOR TIME FRACTIONAL ALLEN-CAHN EQUATION

Hong Lin Liao, Tao Tang, Tao Zhou*

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

41 Citations (Scopus)

Abstract

In this work, we propose a Crank-Nicolson-type scheme with variable steps for the time fractional Allen-Cahn equation. The proposed scheme is shown to be unconditionally stable (in a variational energy sense) and is maximum bound preserving. Interestingly, the discrete energy stability result obtained in this paper can recover the classical energy dissipation law when the fractional order α \rightarrow 1. That is, our scheme can asymptotically preserve the energy dissipation law in the α \rightarrow 1 limit. This seems to be the first work on a variable time-stepping scheme that can preserve both the energy stability and the maximum bound principle. Our Crank-Nicolson scheme is built upon a reformulated problem associated with the Riemann-Liouville derivative. As a byproduct, we build up a reversible transformation between the L1-type formula of the Riemann-Liouville derivative and a new L1-type formula of the Caputo derivative with the help of a class of discrete orthogonal convolution kernels. This is the first time such a discrete transformation is established between two discrete fractional derivatives. We finally present several numerical examples with an adaptive time-stepping strategy to show the effectiveness of the proposed scheme.

Original languageEnglish
Pages (from-to)A3503-A3526
Number of pages24
JournalSIAM Journal on Scientific Computing
Volume43
Issue number5
DOIs
Publication statusPublished - 2021

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • adaptive time stepping
  • asymptotic preserving
  • energy stability
  • maximum principle
  • time-fractional Allen-Cahn equation

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