A stable and conservative finite difference scheme for the Cahn-Hilliard equation

Summary.

We propose a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes a phase separation phenomenon. Numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill-posed problem. We design a new difference scheme based on a general strategy proposed recently by Furihata and Mori. The new scheme inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation. The decrease of the total energy implies boundedness of discretized Sobolev norm of the solution. This in turn implies, by discretized Sobolev's lemma, boundedness of max norm of the solution, and hence the stability of the solution. An error estimate for the solution is obtained and the order is $O( (\Delta x)^2 + (\Delta t)^2)$. Numerical examples demonstrate the effectiveness of the proposed scheme.