A modified Crank-Nicolson finite difference method preserving maximum-principle for the phase-field model

Abstract

In this paper, we mainly study a new Crank-Nicolson finite difference (FD) method with a large time step for solving the nonlinear phase-field model with a small parameter disturbance. To this end, we first introduce an artificial stability term to build a modified Crank-Nicolson FD (MCNFD) scheme, and then prove that the MCNFD scheme satisfies the intrinsic maximum principle of the phase-field model. Secondly, we discuss that the MCNFD scheme satisfies the intrinsic energy stability of the phase-field model, and obtain the optimal error estimation of the MCNFD solution based on the conclusion of the discrete maximum principle. Finally, two numerical examples of one-dimensional and two-dimensional phase-field models are given to verify that the solution of the MCNFD scheme satisfies the above theoretical results.