Energy stability of variable-step L1-type schemes for time-fractional Cahn-Hilliard model

Abstract

The positive definiteness of discrete time-fractional derivatives is fundamental to the numerical stability (in the energy sense) for time-fractional phase-field models. A novel technique is proposed to estimate the minimum eigenvalue of discrete convolution kernels generated by the nonuniform L1, half-grid based L1 and time-averaged L1 formulas of the fractional Caputo's derivative. The main discrete tools are the discrete orthogonal convolution kernels and discrete complementary convolution kernels. Certain variational energy dissipation laws at discrete levels of the variable-step L1-type methods are then established for time-fractional Cahn-Hilliard model.They are shown to be asymptotically compatible, in the fractional order limit $\alpha \to 1$, with the associated energy dissipation law for the classical Cahn-Hilliard equation. Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the effectiveness of the proposed methods.

26 pages, 25 figures, 10 tables

References in corpus (2)

• An energy stable and maximum bound preserving scheme with variable time steps for time fractional Allen-Cahn equation
• Numerical Energy Dissipation for Time-Fractional Phase-Field Equations

Cited by in corpus (3)

• Positive definiteness of real quadratic forms resulting from the variable-step approximation of convolution operators
• Asymptotically compatible energy and dissipation law of the nonuniform L2-$1_{\sigma }$ scheme for time fractional Allen-Cahn model
• Asymptotically compatible energy of variable-step fractional BDF2 formula for time-fractional Cahn-Hilliard model