ENERGY STABILITY OF NUMERICAL METHODS AND SHARP INTERFACE LIMITS FOR THE TIME-FRACTIONAL CAHN–HILLIARD EQUATION

Abstract

The Cahn—Hilliard (CH) equation is an important equation for modeling phase separation with energy dissipation. Recently, the time-fractional Cahn—Hilliard (TFCH) equation has been introduced to model phase separation with memory effect. It is natural to investigate the dissipation property and other related topics for the TFCH equation. In this thesis, dissipative functional and energy stability are proposed for the TFCH and its numerical schemes. Error estimates are established for the L1 scheme. By using the method of asymptotic expansion, sharp interface limits for the TFCH equation are also derived to study the coarsening rate. Extensions to the time-fractional Allen—Cahn equation are also studied.