J.W. Cahn, S.N. Chow and E.S. Van Vleck,

``Spatially Discrete Nonlinear Diffusion Equations,''

(1995) Rocky Mount. J. Math. 25 pp. 87-118.

ABSTRACT

We consider spatially discrete nonlinear diffusion equations that are similar in
form to the Cahn-Hilliard and Cahn-Allen equations. Since these equations
are spatially discrete, solutions exist even for negative gradient energy
coefficients. In order to study these equations analytically on finite subsets of
one, two andthree dimensional lattices we propose a discrete variational
calculus. It is shown that under very general boundary conditions these
equations possess a gradient structure. We prove the existence of a global
attractor and show that when all equilibria are hyperbolic the global attractor
consists of the equilibria and the connecting orbits between the equilibria.
The equilibria of specific one, two and three dimensional equations are
studied.We exhibit constant, two-periodic and three-periodic equilibrium
solutions and study their stability properties. Numerical methods for solving
the time dependent equations are proposed. We employ a fully implicit time
integration scheme and solve the equations on a massively parallel SIMD
machine. To take advantage of the structure of our problem and the data
parallel computing environment we solve the linear systems using the iterative
methods CGS and CGNR. Finally, we exhibit the results of our numerical
simulations. The numerical results show the robust pattern formation that
exists for different values of the parameters.