Cahn-Hilliard算子分裂方法是稳定的_Theoperator-splittingmethodforCahn-H

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arXiv:2107.01418v1 [math.NA] 3 Jul 2021

The operator-splitting method for Cahn-Hilliard

is stable

Dong Li

∗

Chaoyu Quan

†

Abstract

We prove energy stability of a standard operator-splitting method for

the Cahn-Hilliard equation. We establish uniform bound of Sobolev norms

of the numer ical solution and convergence of the splitting approximation.

This is the ﬁr st unconditional energy stability result for the operator-

splitting method for the Cahn -Hilliard equation. Our analysis can be

extended to many other models.

1 Introduction

We consider numerical solutions of the Cahn-Hilliard ([2]) equation:

(

∂

u = ∆(−ν∆u + f(u)), (t, x) ∈ (0, ∞) × Ω,



t=0

= u

(1.1)

where u = u(t, x) is a real-valued function corresponding to the concentration

diﬀerence in a binary system. The parameter ν > 0 is usually called t he mobility

coeﬃcient which is taken to a constant here for simplicity. The nonlinear term

f(u) is derived from a standard do uble well potential, namely:

f(u) = u

− u = F

′

(u), F (u) =

− 1)

Due to this speciﬁc choice of equal-well double potential, the minima of the

potential are located at u = ±1 which corresponds diﬀerent phases or states. The

∗

Department of Ma thematics, the Hong Kong University of Science & Technology, Clear

Water Bay, Hong Kong. Email: mpdongli@gmail.com.

†

SUSTech International Center for Mathematics, Southern University of Science and Tech-

nology, Shenzhen, China. Email: quancy@sustech.edu.cn.

We deﬁne S

(τ) : w → u as the solution operator to the f ollowing problem:

u − w

= ∆(w

− w). (1.9)

In yet other words,

u = S

(τ)w = w + τ∆(w

− w). (1.10)

This is one of the simplest discretization on the timer interval [0, τ] for the exact

problem

(

∂

u = ∆(u

− u), t > 0;



t=0

= w.

(1.11)

By using the operator-splitting, the solution of the original equation from time t

to time t + τ is approximated as

u(t + τ, x) ≈



(τ)S

(τ)u



(t, x). (1.12)

The main purpose of this note is establish stability of the above operator-

splitting algorithm. Prior to our work, there were very few rigor ous results on the

analysis of the operator-splitting type algorithms for the Cahn-Hilliard equation

and similar models. In [24], Weng, Zhai and Feng considered a viscous Cahn-

Hilliard model of the for m

(1 − α)∂

u = ∆(−ǫ

∆u + f(u) + α∂

u), (1.13)

where 0 < α < 1. They considered a fast explicit Strang splitting and estab-

lished stability and converg ence under t he assumption that A = k∇u

num

∞

B = ku

num

∞

are bounded, and satisfy a technical condition 6A + 8 − 24B > 0

(see Theorem 1 on pp. 7 of [24]), where u

num

denotes the numerical solution. In

[23], Gidey and Reddy considered a convective Cahn-Hillia r d model of the form

∂

u − γ∇ · h(u) + ǫ

∆

u = ∆(f(u)), (1.14)

where h(u) =

, u

). They considered operato r -splitting of (1.14) into hyper-

bolic part, nonlinear diﬀusion part and diﬀusion part respectively, and obtained

various conditional results concerning certain weak solutions. In [22], Cheng,

Kurganov, Qu and Tang considered the Strang splitting for the Cahn-Hilliard

equation and molecular beam epitaxy type models. Some conditional results

were given in [22] but rigorous analysis of energy stability has remained open.

The purpose of this note is to establish a new theoretical framework for the rig-

orous analysis of energy stability and higher-order Sobolev-norm stability for the

operator- splitting method applied to these diﬃcult equations. Our ﬁrst result

establishes unifor m Sobolev control of the numerical solution for a ll time.

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