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.
© 1994 Elsevier B.V. Received 29 September 1993, Revised 1 May 1994. The work of Bai and Spence is supported by the UK Science Engineering Research Council. The work of Stuart is funded by the Office of Naval Research under contract N00014-92-J-1876 and the National Science Foundation under contract number DMS-9201727.