Convergence analysis of variable steps BDF2 method for the space fractional Cahn-Hilliard model

TL;DR

This paper develops and analyzes a variable-step BDF2 numerical scheme for the space fractional Cahn-Hilliard equation, ensuring energy dissipation, convergence, and mass conservation, with numerical validation and adaptive time-stepping.

Contribution

It introduces a rigorously analyzed variable-step BDF2 scheme for the fractional Cahn-Hilliard model, including convergence proof and energy dissipation properties.

Findings

The scheme is energy dissipative under certain step ratios.

Convergence of the fully discrete scheme is rigorously proved.

Numerical experiments confirm accuracy and energy dissipation.

Abstract

An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation, involving the fractional Laplacian, derived from a gradient flow in the negative order Sobolev space $H^{-\alpha}$, $\alpha\in(0,1)$. The Fourier pseudo-spectral method is applied for the spatial approximation. The proposed scheme inherits the energy dissipation law in the form of the modified discrete energy under the sufficient restriction of the time-step ratios. The convergence of the fully discrete scheme is rigorously provided utilizing the newly proved discrete embedding type convolution inequality dealing with the fractional Laplacian. Besides, the mass conservation and the unique solvability are also theoretically guaranteed. Numerical experiments are carried out to show the accuracy and the energy dissipation both for various interface widths. In particular, the multiple-time-scale evolution of the solution is captured by an adaptive time-stepping strategy in the short-to-long time simulation.