An implicit difference scheme for the time-fractional Cahn–Hilliard equations

Abstract

In this paper, an efficient finite difference scheme is developed for solving the time-fractional Cahn–Hilliard equations which is the well-known representative of phase-field models. The time Caputo derivative is approximated by the popular L1 formula. The stability and convergence of the finite difference scheme in the discrete L2-norm are proved by the discrete energy method. To compare and observe the phenomenon of solution, a generalized difference scheme based on the graded mesh in time is also given. The dynamics of the solution and accuracy of the schemes are verified numerically. Numerical experiments show that the solution of the time-fractional Cahn-Hilliard equation always tends to be in an equilibrium state with the increase of time for different values of order α∈(0,1), which is consistent with the phase separation phenomenon.