Numerical computations of coarsening in the one-dimensional Cahn-Hilliard model of phase separation

Time dependent solutions of the Cahn-Hilliard equation are studied numerically. In particular heteroclinic orbits, which connect different equilibrium solutions at t = -∞ and t = +∞, are sought. Thus boundary value problems in space-time are computed. This computation requires an investigation of the stability of equilibria, since projections onto the stable and unstable manifolds determine the boundary conditions at t = -∞ and t = +∞. This stability analysis is then followed by solution of the appropriate boundary value problem in space-time. The results obtained cannot be found by standard initial value simulations. By specifying the two steady states at t = ±∞ appropriately it is possible to find orbits reflecting a given degree of coarsening over the time evolution. This gives a clear picture of the dynamic coarsening admissible in the equation. It also provides an understanding of orbits on the global attractor for the equation