Slowly-Migrating Transition Layers for the Discrete Allen-Cahn and Cahn-Hilliard Equations
Grant, Christopher P.
Van Vleck, Erik S.
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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 [equation omitted], where 1/n is the spatial mesh size and ε is the interaction length.