Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling

We present a computationally efficient scheme of modeling the phase-ordering dynamics of thermodynamically unstable phases. The scheme utilizes space-time discrete dynamical systems, viz., cell dynamical systems (CDS). Our proposal is tantamount to proposing new Ansa$iumltze for the kinetic-level description of the dynamics. Our present exposition consists of two parts: part I (this paper) deals mainly with methodology and part II [S. Puri and Y. Oono, Phys. Rev. A (to be published)] gives detailed demonstrations. In this paper we provide a detailed exposition of model construction, structural stability of constructed models (i.e., insensitivity to details), stability of the scheme, etc. We also consider the relationship between the CDS modeling and the conventional description in terms of partial differential equations. This leads to a new discretization scheme for semilinear parabolic equations and suggests the necessity of a branch of applied mathematics which could be called qualitative numerical analysis.