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

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 $\alpha \to 1.$ That is, our scheme can asymptotically preserve the energy dissipation law in the $\alpha \to 1$ 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.

arXiv comment26 pages, 28 figures, 4 tables

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• The variable-step L1 scheme preserving a compatible energy law for time-fractional Allen-Cahn equation
• A decreasing upper bound of energy for time-fractional phase-field equations
• Compatible $L^2$ norm convergence of variable-step L1 scheme for the time-fractional MBE mobel with slope selection
• Energy stability of variable-step L1-type schemes for time-fractional Cahn-Hilliard model
• Asymptotically compatible energy of variable-step fractional BDF2 formula for time-fractional Cahn-Hilliard model