The following two papers, a semi-implicit time discretization of gradient flows is given and analyzed. It is shown that the scheme is unconditionally gradient stable and that the discrete equations are uniquely solvable for all time steps. The key feature of the method is a separation of the contractive and expansive terms of the equation across the time step. The Cahn-Hilliard equation is used as an example.

These schemes allow the Cahn-Hilliard equation to be solved on a workstation instead of a supercompter.

An unconditionally stable one step scheme for gradient systems, Submitted SIAM J. of Scientific Computing

Unconditionally gradient stable time marching the Cahn-Hilliard equation, in Computational and Mathematical Models of Microstructural Evolution, Eds. J.W. Bullard, R. Kalia, M. Stoneham, L-Q. Chen, The Materials Research Society, 1998

Finally, the following link contains Matlab code to solve the Cahn-Hilliard equation and demonstrates the power of the method. There are two easy ways to execute the code, first, save the text in the link to a file called ch.m, and then once matlab has initialized type ch. Or, second cut and paste the text into matlab.