Numerical Simulation Of The Cahn-Hilliard Equation By Large Time-stepping Methods

The Cahn-Hilliard equation was originally introduced to describe phase separation ofthe molten binary alloy which is rapidly quenched to lower temperature, and then extensivelyapplied to other fields such as Spinodal decomposition, Fickian diffusion, and two-phase?ow. Both the fourth and the nonlinear terms make the Cahn-Hilliard equation stiff anddifficult to solve numerically. In this paper, we study a so-called"large time-steppingmethod", recently proposed and investigated by several authors. Specially, we focus on theimpact of the stabilization term (called"A-term") on the numerical solutions. The mainresults of this work are as follow: Firstly, by using a energy estimation method, we establishthe well-posedness of the Cahn-Hilliard equation with periodic boundary conditions. Thisis a generalization of an existing result for the Neumann boundary conditions. Secondly,by combining the large time-stepping finite-difference in time and Fourier spectral methodin space, a series of numerical experiments is carried out to investigate the in?uence ofthe A-term on the simulation results. Our numerical simulation show that the A-term,although stabilize the calculations, has profound impact on the long term behavior of thenumerical solutions. Moreover, it is found that this in?uence depends on the diffusioncoefficient. Finally, in order to make evident of the in?uence of the A-term, an error esti-mate for the first-order scheme is derived. A sufficient condition for convergence is provided.