Abstract
We present an unconditionally stable finite difference method forsolving the viscous Cahn--Hilliard equation. We prove theunconditional stability of the proposed scheme by using the decreaseof a discrete functional. We present numerical results that validatethe convergence and unconditional stability properties of themethod. Further, we present numerical experiments that highlight thedifferent temporal evolutions of the Cahn--Hilliard and viscousCahn--Hilliard equations.
Highlights
IntroductionWe consider a finite difference scheme for the viscous Cahn– Hilliard (vCH) equation φt(x, t) = ∆μ(x, t), (1)
We consider a finite difference scheme for the viscous Cahn– Hilliard equation φt(x, t) = ∆μ(x, t), (1)μ(x, t) = F (φ(x, t)) − 2∆φ(x, t) + νφt(x, t), (2)where Ω ⊂ Rd (d = 1, 2, 3) is a domain
The CH equation is a diffuse interface model for describing the spinodal decomposition in binary alloys [3], and the viscous Cahn– Hilliard (vCH) equation is considered as a phenomenological continuum model for phase separation coupling with a slowly relaxing variable [2, 29]
Summary
We consider a finite difference scheme for the viscous Cahn– Hilliard (vCH) equation φt(x, t) = ∆μ(x, t), (1). Note that if ν = 0, the vCH equation becomes the Cahn–Hilliard (CH) equation. The CH equation is a diffuse interface model for describing the spinodal decomposition in binary alloys [3], and the vCH equation is considered as a phenomenological continuum model for phase separation coupling with a slowly relaxing variable [2, 29]. The sharp interface limit of the CH equation is the Mullins–Sekerka model [11, 30]. The viscosity term νφt can be interpreted as describing the influences of internal microforces [19].
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