A Non-uniform Time-stepping Convex Splitting Scheme for the Time-fractional Cahn-Hilliard Equation

TL;DR

This paper introduces a novel non-uniform time-stepping convex splitting scheme for the time-fractional Cahn-Hilliard equation, achieving high accuracy, stability, and efficiency through advanced discretization techniques.

Contribution

It presents a new numerical algorithm combining non-uniform time-stepping, convex splitting, and spectral methods for improved simulation of the time-fractional Cahn-Hilliard equation.

Findings

Second-order temporal accuracy

Spectral spatial accuracy

Unconditionally energy stable scheme

Abstract

In this paper, a non-uniform time-stepping convex-splitting numerical algorithm for solving the widely used time-fractional Cahn-Hilliard equation is introduced. The proposed numerical scheme employs the $L1^+$ formula for discretizing the time-fractional derivative and a second-order convex-splitting technique to deal with the non-linear term semi-implicitly. Then the pseudospectral method is utilized for spatial discretization. As a result, the fully discrete scheme has several advantages: second-order accurate in time, spectrally accurate in space, uniquely solvable, mass preserving, and unconditionally energy stable. Rigorous proofs are given, along with several numerical results to verify the theoretical results, and to show the accuracy and effectiveness of the proposed scheme. Also, some interesting phase separation dynamics of the time-fractional Cahn-Hilliard equation has been investigated.