An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen-Cahn equation

TL;DR

This paper introduces an unconditionally stable, maximum bound preserving scheme with variable time steps for the time fractional Allen-Cahn equation, which asymptotically recovers classical energy dissipation as the fractional order approaches 1.

Contribution

It is the first scheme to simultaneously preserve energy stability and maximum bound principle with variable time steps for this equation.

Findings

Scheme is unconditionally stable in energy sense.

Scheme preserves maximum bound principle.

Numerical examples demonstrate effectiveness with adaptive time-stepping.

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 $\alpha \rightarrow 1.$ That is, our scheme can asymptotically preserve the energy dissipation law in the $\alpha \rightarrow 1$ limit. This seems to be the first work on variable time-stepping scheme that can preserve both the energy stability and the maximum bound principle. Our Crank-Nicolson scheme is build upon a reformulated problem associated with the Riemann-Liouville derivative. As a by product, 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 \textit{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.