Stable finite difference scheme for a model equation of phase separation

摘要

A stable, conservative, and practical finite difference scheme is presented in order to solve numerically a model equation of phase separation in some binary alloys. The model system is introduced by Eguchi, Oki, and Matsumura in the attempt of theoretically describing such phenomena, and extends the well-known Cahn–Hilliard equation; even numerical solutions are hard to obtain for this dynamics. Our scheme utilizes a Lyapunov functional, that is, the free energy of the system, whose effectiveness is demonstrated by numerical implementations.