C.P. Grant and E.S. Van Vleck,

``Slowly-Migrating Transition Layers for the Discrete Allen-Cahn and Cahn-Hilliard Equations,''

(1995) Nonlinearity 8 pp. 861-876.

ABSTRACT

It has recently been proposed that spatially discretized versions of the
Allen-Cahn and Cahn-Hilliard equations for modeling phase transitions have
certain theoretical and phenomenological advantages over their continuous
counterparts. This paper deals with one-dimensional discretizations and
examines the extent to which dynamical metastability, which manifests itself
in the original partial differential equations in the form of solutions with
slowly-moving transition layers, is also present for the discrete equations. It is
shown that, in fact, there are transition-layer solutions that evolve at a speed
bounded by $C_1 \eps (1 + C_2/(n \eps))^{-C_3 n + C_4}$ for all $n \geq n_0$
and $\eps \leq \eps_0$, where $1/n$ is the spatial mesh size, $\eps$ is the
interaction length, and $n_0$ and $\eps_0$ are constants.