Journal of Computational Physics 230 (2011) 7441–7455

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Journal of Computational Physics

journal homepage: www.elsevier .com/locate / jcp

A conservative numerical method for the Cahn–Hilliard equation incomplex domains

Jaemin Shin, Darae Jeong, Junseok Kim ⇑Department of Mathematics, Korea University, Seoul 136-701, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 January 2011Received in revised form 28 April 2011Accepted 7 June 2011Available online 17 June 2011

Keywords:Cahn–Hilliard equationDegenerate mobilityMultigrid methodPhase separationComplex domain

0021-9991/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.jcp.2011.06.009

⇑ Corresponding author. Tel.: +82 2 3290 3077; faE-mail addresses: cfdkim@korea.ac.kr, cfdkim@dURL: http://math.korea.ac.kr/~cfdkim/ (J. Kim).

We propose an efficient finite difference scheme for solving the Cahn–Hilliard equationwith a variable mobility in complex domains. Our method employs a type of uncondition-ally gradient stable splitting discretization. We also extend the scheme to compute theCahn–Hilliard equation in arbitrarily shaped domains. We prove the mass conservationproperty of the proposed discrete scheme for complex domains. The resulting discretizedequations are solved using a multigrid method. Numerical simulations are presented todemonstrate that the proposed scheme can deal with complex geometries robustly. Fur-thermore, the multigrid efficiency is retained even if the embedded domain is present.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

In this paper, we consider an efficient finite difference scheme for the Cahn–Hilliard (CH) equation with a variable mobil-ity in complex domains.

@/ðx; tÞ@t

¼ r � ½Mð/ðx; tÞÞrlð/ðx; tÞÞ�; x 2 X; 0 < t 6 T; ð1Þ

lð/ðx; tÞÞ ¼ F 0ð/ðx; tÞÞ � �2D/ðx; tÞ; ð2Þ

where X � Rdðd ¼ 1;2;3Þ is a domain, / = mA �mB is a difference of molar fractions mA and mB of species A and B, and M(/) isa nonnegative diffusional mobility. F(/) is the free energy density of a homogeneous fluid and � is a positive constant. The CHequation has the total free energy functional:

Eð/Þ ¼Z

XFð/Þ þ �

2

2jr/j2

� �dx: ð3Þ

The CH equation was originally introduced to describe the complicated phase separation and coarsening phenomena calledthe spinodal decomposition in binary alloys [5,6]. We also define the quantity c as the molar fraction mA, i.e., c = mA. Note that/ = 2c � 1. Typical forms of the free energy are given as [11]

Fð/Þ ¼ ð1� /2Þ2

4and FðcÞ ¼ c2ð1� cÞ2

4: ð4Þ

. All rights reserved.

x: +82 2 929 8562.ongguk.edu (J. Kim).

−1.5 −1 −0.5 0 0.5 1 1.50

0.05

0.1

0.15

0.2

0.25

φ : concentration

F(φ)

(a)

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

φ : concentration

M(φ)

(b)

Fig. 1. (a) Free energy F(/). (b) Variable diffusional mobility M(/).

7442 J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455

The free energy has a double-well potential that has local minima at the minimum and maximum of the concentration asshown in Fig. 1(a). And we take variable mobilities of the form

Mð/Þ ¼ 1� /2 and MðcÞ ¼ cð1� cÞ ð5Þ

which are degenerated at the minimum and maximum of the concentration as shown in Fig. 1(b) [7]. This mobility signif-icantly lowers the long-range diffusion across bulk regions with the enhanced diffusion in the interfacial region. The systemis completed by taking an initial condition and the natural and no-flux boundary conditions n�r/ = n�rl = 0 on @X, where nis a unit normal vector to @X.

In the original derivation of the CH equation, a concentration dependent mobility appeared [4–6]. Various numericalmethods have been applied to solve the CH equation with a constant mobility [1,9,12,14,15,17,22,25,27,32]. However, rel-atively few authors have considered the CH equation with a concentration dependent mobility [2,3,16,19–21,26,30,31,33].

In this paper, we propose an efficient finite difference scheme for the CH equation with a variable mobility. Furthermore,we also extend the scheme to compute the CH equation in arbitrarily shaped domains. The application of Cartesian mesh forsolving problems with complex geometry has been used in the past decade. Many authors have studied the Cartesian gridmethod and the finite volume method for solving the Poisson equation with an irregular domain [8,10,18,23,29]. Cartesiangrid embedded boundary methods have advantages over unstructured grid methods because of simpler grid generation.

This paper is organized as follows. In Section 2, we describe the numerical solution of the CH equation with a variablemobility and the details of the treatment of arbitrarily shaped domains. The numerical results showing the robustnessand superiority of the proposed scheme with a variable mobility are described in Section 3. Conclusions are presented inSection 4.

2. Numerical methods

2.1. Numerical discretization

2.1.1. The Cahn–Hilliard equation in a rectangular domainIn this section, we present fully discrete schemes for the CH equation in a rectangular domain. For simplicity of exposi-

tion, we shall discretize the CH Eqs. (1) and (2) in two-dimensional space X = (a,b) � (c,d). Three-dimensional discretizationis analogously defined. Let Nx and Ny be positive even integers, Dx = (b � a)/Nx and D y = (d � c)/Ny be the uniform mesh sizes.We assume h = Dx = D y. Also let Xh = {(xi,yj): xi = a + (i � 0.5)h, yj = c + (j � 0.5)h, 1 6 i 6 Nx,1 6 j 6 Ny} be the set of cell-cen-ters. Let /n

ij and lnij be approximations of /(xi,yj, tn) and l(xi,yj, tn), where tn = nDt and Dt is the time step. We use zero Neu-

mann boundary condition for / and l at the domain boundary, for example

Dx/12;j¼ Dx/Nxþ1

2;j¼ Dy/i;12

¼ Dy/i;Nyþ12¼ 0;

where the discrete differentiation operators are

Dx/iþ12;j¼

/iþ1;j � /ij

hand Dy/i;jþ1

2¼

/i;jþ1 � /ij

h:

And we use the notationrd/ij ¼ Dx/iþ12;j;Dy/i;jþ1

2

� �to represent the discrete gradient of / at cell-edges. Correspondingly, the

divergence at cell-centers, using values from cell-edges, is rd � Dx/iþ12;j;Dy/i;jþ1

2

� �ij¼ Dx/iþ1

2;j� Dx/i�1

2;jþ Dy/i;jþ1

2�

�Dy/i;j�1

2Þ=h.

Then we define the discrete Laplacian by Dd/ij:¼rd�rd/ij and the discrete l2 inner products by

J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455 7443

ðc;dÞh :¼ h2XNx

i¼1

XNy

j¼1

cijdij;

rdc;rddð Þe :¼ h2XNx

i¼0

XNy

j¼1

Dxciþ12;j

Dxdiþ12;jþXNx

i¼1

XNy

j¼0

Dyci;jþ12Dydi;jþ1

2

!:

We also define discrete norms as jcj22 = (c,c)h and jcj21 ¼ ðrdc;rdcÞe.In Ref. [20], a Crank–Nicolson (CN) method for the CH equation was proposed as follows:

cnþ1ij � cn

ij

Dt¼ rd � MðcÞnþ

12

ij rdlnþ1

2ij

� �; ð6Þ

lnþ12

ij ¼ 12

F 0ðcnþ1ij Þ þ F 0ðcn

ij� �

� �2

2Ddðcnþ1

ij þ cnijÞ; ð7Þ

where MðcÞnþ12

iþ12;j

:¼ M ðcnij þ cn

iþ1;j þ cnþ1ij þ cnþ1

iþ1;jÞ=4� �

. This CN method has second-order accuracy in time and space. However, it

has a severe time step restriction.In this paper, we present a semi-implicit time and centered difference space discretization of Eqs. (1) and (2). For the

phase-field variable c, we have the following discretization:

cnþ1ij � cn

ij

Dt¼ rd � MðcÞnijrdl

nþ12

ij

� �; ð8Þ

lnþ12

ij ¼ 12

cnþ1ij � �2Ddcnþ1

ij þ F 0ðcnijÞ �

12

cnij ð9Þ

and for the phase-field variable /, we have the following one:

/nþ1ij � /n

ij

Dt¼ rd � Mð/Þnijrdl

nþ12

ij

� �; ð10Þ

lnþ12

ij ¼ 2/nþ1ij � �2Dd/

nþ1ij þ F 0ð/n

ijÞ � 2/nij; ð11Þ

where

rd � Mð/Þnijrdlnþ1

2ij

� �:¼

Mniþ1

2;jlnþ1

2iþ1;j � lnþ1

2ij

� ��Mn

i�12;j

lnþ12

ij � lnþ12

i�1;j

� �h2 þ

Mni;jþ1

2lnþ1

2i;jþ1 � lnþ1

2ij

� ��Mn

i;j�12

lnþ12

ij � lnþ12

i;j�1

� �h2 : ð12Þ

Here Mniþ1

2;j¼ M ð/n

iþ1;j þ /nijÞ=2

� �and the other terms are similarly defined. We note that if the mobility M(/) is constant, then

the discrete schemes (8)–(11) become unconditionally gradient stable linear splitting schemes [13].During computation, the values of the concentration c can be negative or greater than one and also / has a similar case.

For physical relevance of the solution, we want the non-negative mobility (M(�) P 0), therefore we use M(c) = jc(1 � c)j andM(/) = j1 � /2j for numerical evaluations with the concentration dependent mobility.

Fig. 2. Shaded area Xin is embedded in X and Xout is defined as X nXin ¼ X \Xcin .

7444 J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455

2.1.2. The Cahn–Hilliard equation in a non-rectangular domainNow, we give a numerical scheme for the CH equation in a non-rectangular domain with homogeneous Neumann bound-

ary conditions. Let Xin be an arbitrarily shaped open domain which is embedded in X. Also, let C = oXin be the arbitrarilyshaped domain boundary (see Fig. 2).

To solve Eqs. (10) and (11) in an arbitrarily shaped domain Xin, we propose the following numerical scheme:

/nþ1ij � /n

ij

Dt¼ rd � GijMð/n

ijÞrdlnþ1

2ij

� �; ð13Þ

lnþ12

ij ¼ 2/nþ1ij � �2rd � Gijrd/

nþ1ij

� �þ F 0ð/n

ijÞ � 2/nij: ð14Þ

Here the boundary control function G is defined in Xh by

Gðx; yÞ ¼1 if ðx; yÞ 2 Xh

in;

0 if ðx; yÞ 2 Xhout ¼ X nXh

in;

(ð15Þ

where Xhin is an open region that consists of interior points of the union of the closed cells whose centers are inside the inter-

face C. For example, in Fig. 3 the value of function G at points (�,�) in Xhout is zero. The value of function G at point (�, �) in Xh

in

is one.As we refine the mesh, i.e., h ? 0, the arbitrarily shaped domain Xin is approximated by Xh

in (see Fig. 4).

2.2. Conservation property of the total mass

In this section, we show that our scheme inherits the conservation of total mass in a complex domain. We want to showthat

X

ij2Xin/nþ1

ij ¼X

ij2Xin/n

ij for all n ¼ 0;1; . . . : ð16Þ

We start by summing Eq. (13) in the whole numerical domain X, i.e.,

Xij2X

/nþ1ij � /n

ij

Dt¼ 1

h2

Xij2X

Giþ12;j

Mniþ1

2;jlnþ1

2iþ1;j � lnþ1

2ij

� �� Gi�1

2;jMn

i�12;j

lnþ12

ij � lnþ12

i�1;j

� �h

þGi;jþ12Mn

i;jþ12

lnþ12

i;jþ1 � lnþ12

ij

� �� Gi;j�1

2Mn

i;j�12

lnþ12

ij � lnþ12

i;j�1

� �i

¼ 1

h2

XNy

j¼1GNxþ1

2;jMn

Nxþ12;j

lnþ12

Nxþ1;j � lnþ12

Nx ;j

� �� G1

2;jMn

12;j

lnþ12

1j � lnþ12

0;j

� �h i

þ 1

h2

XNx

i¼1Gi;Nyþ1

2Mn

i;Nyþ12

lnþ12

i;Nyþ1 � lnþ12

i;Ny

� �� Gi;12

Mni;12

lnþ12

i1 � lnþ12

i0

� �h i: ð17Þ

Then, by applying zero Neumann boundary condition of lnþ12

ij on oX, we obtain

Xij2X

/nþ1ij � /n

ij

Dt¼ 0; i:e:;

Xij2X

/nþ1ij ¼

Xij2X

/nij: ð18Þ

Fig. 3. Shaded area Xhin is embedded in X and Xh

out is defined as X \Xhin

c. Note that the value of function G at points (�,�) in Xh

out is zero.

Fig. 4. Convergence of Xhin to Xin as we refine the mesh. Mesh sizes are shown below each figure.

J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455 7445

Next, we consider summation of Eq. (13) in the outside numerical domain Xout, i.e.,

Fig. 5.

Xij2Xout

/nþ1ij � /n

ij

Dt¼ 1

h2

Xij2Xout

Giþ12;j

Mniþ1

2;jlnþ1

2iþ1;j � lnþ1

2ij

� �� Gi�1

2;jMn

i�12;j

lnþ12

ij � lnþ12

i�1;j

� �þ Gi;jþ1

2Mn

i;jþ12

lnþ12

i;jþ1 � lnþ12

ij

� �h�Gi;j�1

2Mn

i;j�12

lnþ12

ij � lnþ12

i;j�1

� �i¼ 0; ð19Þ

where we have used the assumption (15) that G is zero in Xout. By using the above property (18), we obtain

Xij2Xin

/nþ1ij ¼

Xij2X

/nij �

Xij2Xout

/nþ1ij ¼

Xij2Xin

/nij þ

Xij2Xout

/nij �

Xij2Xout

/nþ1ij ¼

Xij2Xin

/nij � Dt

Xij2Xout

/nþ1ij � /n

ij

Dt¼X

ij2Xin/n

ij;

where we have used Eq. (19) in the last equality. Therefore, we proved the conservation property of mass, Eq. (16), in a com-plex domain Xin.

2.3. Numerical algorithm

In this section, we show the way to control a matrix in finer and coarser grids with the boundary control function G. Andwe describe the multigrid method and implementation in detail to solve the resulting coupled system of equations.

First we set the boundary control matrix function G as defined in Eq. (15) to represent a complex domain. In Figs. 3 and 5,the cell-edge (�,�) values are used in the smoothing step and the cell-center (�) values are used in the restriction and inter-polation operators. Denote that Gh is a discrete boundary control function defined in Xh. In the coarser grid XH, the controlfunction GH is different from Gh because XH is defined as shown in Fig. 5. The details about the coarser domain XH are ex-plained by the restriction and interpolation operators. Note that as a set we have the following inclusion: Xh

in � XHin.

Next, we describe a full approximation storage (FAS) multigrid method to solve the discrete system (13) and (14) at theimplicit time level. A pointwise Gauss–Seidel relaxation is used as the smoother in the multigrid method. See the referencetext [28] for additional details and backgrounds. The algorithm of the multigrid method for solving the discrete CH system is:First, let us rewrite Eqs. (13) and (14) as follows.

Illustration of the coarsening process for the multigrid method. Shaded area XHin is also embedded in X but is not generally same to Xh

in , i.e., Xhin � XH

in .

7446 J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455

Lð/nþ1;lnþ12Þ¼ ðnn;wnÞ; where Lð/nþ1;lnþ1

2Þ

¼ /nþ1ij =Dt�rd � GijMð/n

ijÞrdlnþ1

2ij

� �; �2rd � Gijrd/

nþ1ij

� ��2/nþ1

ij þlnþ12

ij

� �and the source term is ðnn;wnÞ¼ /n

ij=Dt;F 0ð/nijÞ�2/n

ij

� �:

In the following description of one FAS cycle, XH is coarser than Xh by factor 2. Given the number m1 and m2 of pre-and post-smoothing relaxation sweeps, an iteration step for the multigrid method using the V-cycle is formally written as in Fig. 6[28]. In Fig. 6, f/nþ1;m;lnþ1

2;mg and f/̂nþ1;mþ1; l̂nþ12;mþ1g are the approximations of /n+1(xi,yj) and lnþ1

2ðxi; yjÞ before and afteran FAS cycle.

The FAS multigrid algorithm in Fig. 6 consists of the following components.

� Smoothing:

f�/nþ1;m; �lnþ12;mg ¼ SMOOTHm1 ð/nþ1;m;lnþ1

2;m; Lh; nn;wnÞ;

which means performing m1-smoothing steps with the initial approximations /nþ1;m;lnþ12;m, source terms nn, wn, and SMOOTH

relaxation operator to get the approximations �/nþ1;m and �lnþ12;m. One SMOOTH relaxation operator step consists of solving the

system (22) and (23) given below by 2 � 2 matrix inversion for each i and j. Here, we derive the smoothing operator in twodimensions. Rewriting Eq. (13), we get

/nþ1;mij

Dtþ

Giþ12;j

Mniþ1

2;jþ Gi�1

2;jMn

i�12;jþ Gi;jþ1

2Mn

i;jþ12þ Gi;j�1

2Mn

i;j�12

h2 lnþ12;m

ij

¼ nnij þ

Giþ12;j

Mniþ1

2;jlnþ1

2;miþ1;j þ Gi�1

2;jMn

i�12;jlnþ1

2;mi�1;j

h2 þGi;jþ1

2Mn

i;jþ12lnþ1

2;mi;jþ1 þ Gi;j�1

2Mn

i;j�12lnþ1

2;mi;j�1

h2 : ð20Þ

Also, by rewriting Eq. (14), we obtain

� �2

h2 Giþ12;jþGi�1

2;jþGi;jþ1

2þGi;j�1

2

� �þ2

� �/nþ1;m

ij þlnþ12;m

ij ¼wnij��2

h2 Giþ12;j

/nþ1;miþ1;j þGi�1

2;j/nþ1;m

i�1;j þGi;jþ12/nþ1;m

i;jþ1 þGi;j�12/nþ1;m

i;j�1

� �:

ð21Þ

Fig. 6. The FAS multigrid cycle (Xh,XH) two-grid method.

J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455 7447

Next, we replace /nþ1;mkl and lnþ1

2;mkl in Eqs. (20) and (21) with �/nþ1;m

kl and �lnþ12;m

kl if k 6 i and l 6 j, otherwise with /nþ1;mkl and

lnþ12;m

kl , i.e.,

�/nþ1;mij

Dtþ

Giþ12;j

Mniþ1

2;jþ Gi�1

2;jMn

i�12;jþ Gi;jþ1

2Mn

i;jþ12þ Gi;j�1

2Mn

i;j�12

h2�lnþ1

2;mij

¼ nnij þ

Giþ12;j

Mniþ1

2;jlnþ1

2;miþ1;j þ Gi�1

2;jMn

i�12;j

�lnþ12;m

i�1;j

h2 þGi;jþ1

2Mn

i;jþ12lnþ1

2;mi;jþ1 þ Gi;j�1

2Mn

i;j�12�lnþ1

2;mi;j�1

h2 : ð22Þ

� �2

h2 Giþ12;jþGi�1

2;jþGi;jþ1

2þGi;j�1

2

� �þ2

� ��/nþ1;m

ij þ �lnþ12;m

ij ¼wnij��2

h2 Giþ12;j

/nþ1;miþ1;j þGi�1

2;j�/nþ1;m

i�1;j þGi;jþ12/nþ1;m

i;jþ1 þGi;j�12

�/nþ1;mi;j�1

� �:

ð23Þ

H

� Restriction: A restriction operator Ih is defined as

dHij ¼ IH

h dhij ¼

1Rij

Gh2i�1;2j�1dh

2i�1;2j�1 þ Gh2i�1;2jd

h2i�1;2j þ Gh

2i;2j�1dh2i;2j�1 þ Gh

2i;2jdh2i;2j

� �;

where Rij ¼ Gh2i�1;2j�1 þ Gh

2i�1;2j þ Gh2i;2j�1 þ Gh

2i;2j and if Rij = 0, then we define dHij ¼ 0 (see Fig. 7(a)). For example, Fig. 7(c) shows

notations and symbols on fine and coarse grids. Restriction from level-h to level-H is done as follows:

dH11 ¼

1R11

Gh11dh

11 þ Gh12dh

12 þ Gh21dh

21 þ Gh22dh

22

� �¼ 1

4dh

11 þ dh12 þ dh

21 þ dh22

� �;

dH12 ¼ 0; dH

21 ¼13

dh31 þ dh

32 þ dh34

� �; and dH

22 ¼12

dh33 þ dh

43

� �:

� Interpolation: The interpolation operator IhH (see Fig. 7(b)) represents the prolongation of corrections from level-H to

level-h and is defined as

dh2i�1;2j�1

dh2i�1;2j

dh2i;2j�1

dh2i;2j

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼ Ih

HdHij ¼

Gh2i�1;2j�1dH

ij for }

Gh2i�1;2jd

Hij for

Gh2i;2j�1dH

ij for M

Gh2i;2jd

Hij for � :

8>>>>><>>>>>:

ð24Þ

For example, interpolation from level-H and level-h is done as follows (see Fig. 7(c)):

dh11

dh12

dh21

dh22

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

Gh11dH

11

Gh12dH

11

Gh21dH

11

Gh22dH

11

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

dH11

dH11

dH11

dH11

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;;

dh13

dh14

dh23

dh24

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

Gh13dH

12

Gh14dH

12

Gh23dH

12

Gh24dH

12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

0

0

0

0

8>>>><>>>>:

9>>>>=>>>>;;

dh31

dh32

dh41

dh42

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

Gh31dH

21

Gh32dH

21

Gh41dH

21

Gh42dH

21

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

dH21

dH21

0

dH21

8>>>>><>>>>>:

9>>>>>=>>>>>;;

dh33

dh34

dh43

dh44

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

Gh33dH

22

Gh34dH

22

Gh43dH

22

Gh44dH

22

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;¼

dH22

0

0

dH22

8>>>>><>>>>>:

9>>>>>=>>>>>;:

By the above restriction and interpolation, the proposed discretization has an effect that none of grid points in Xout partic-ipates in the evolution of values in Xin.

3. Numerical experiments

In this section, we perform the following tests: showing the relation between the � value and the width of the transitionlayer, calculating the accuracy of the proposed method, demonstrating the multigrid efficiency, comparing stabilities of theCrank–Nicolson and the proposed scheme, investigating the effect of constant and variable mobilities in 2D and 3D, validat-ing the total mass conservation and the total energy dissipation, and spinodal decomposition on complex domains such as a

Fig. 7. (a) Restriction. (b) Interpolation. (c) Gh = 1 on gray region and Gh = 0 on white region.

7448 J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455

wavy curved channel, sequential T-junctions, a sphere, and a wavy curved tube. Note that we will use either / or c in ournumerical experiments because two equations with different phase-field variables are equivalent.

3.1. The relation between the � value and the width of the transition layer

In the first numerical experiment, we consider the relation between the � value and the width of the transition layer forthe CH equation. Across the interfacial region, the concentration field varies from 0.05 to 0.95 over a distance of approxi-mately 4

ffiffiffi2p

tanh�1ð0:9Þ for c and varies from �0.9 to 0.9 over a distance of approximately 2ffiffiffi2p

tanh�1ð0:9Þ for /. Therefore,if we want this value to be approximately m grid points, the � value needs to be taken as follows, respectively:

�m ¼mh

4ffiffiffi2p

tanh�1ð0:9Þfor c and �m ¼

mh

2ffiffiffi2p

tanh�1ð0:9Þfor /:

To confirm those formula, we run simulations with the initial data /(x,y,0) = �0.4 + 0.001rand (x,y) in a domain X = (0,64)� (0,64) with a 64 � 64 mesh and h = Dt = 1. Here, the random number, rand (x,y), is uniformly distributed between �1 and1. Shaded region indicates the embedded domain. Fig. 8(a) and (b) show contours from / = �0.9 to / = 0.9 increased by 0.2with �3 and �10 at t = 5000, respectively. From the two figures, we can see that the interfacial transition layer is about 3 and10 grid points, respectively.

3.2. Convergence test

We should demonstrate the convergence of the proposed scheme. The initial state for the test is /(x,y,0) = 0.8cos(2px)-cos(2p y) in a domain Xin = (�1,1) � (�1,1) � (�0.5,0.5) � (�0.5,0.5) (see Fig. 9). To calculate the accuracy of the proposedmethod, we perform a number of simulations on a set of increasingly finer grids. The numerical solutions are computed onthe grids h = 1/2n+1 for n = 5, 6, 7, 8, and 9. For each case, the calculation is run up to time t = 0.01 with the time stepD t = 0.001/4n�5 and a fixed � = 0.04. We define the error of a grid to be the discrete l2-norm of the difference between thatgrid and the average of the next finer grid cells covering it as follows. eh=h

2 ij:¼ /hij � ð/h

22i;2jþ /h

22i�1;2jþ /h

22i;2j�1þ /h

22i�1;2j�1Þ=4.

Fig. 8. The concentration contours at the nine levels from / = �0.9 to / = 0.9 increased by 0.2 with (a) �3 and (b) �10.

Fig. 9. The initial condition /(x,y,0) with a small obstacle on X = (�1,1) � (�1,1).

Table 1The errors and rates of convergence for concentration /.

Case 32–64 Rate 64–128 Rate 128–256 Rate 256–512

l2-error 1.44E�2 1.89 3.89 E�3 2.12 8.94E�4 2.05 2.16E�4

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

10−6

10−4

10−2

100

102

fine grid iterations

resi

dual

1 level2 levels3 levels4 levels

Fig. 10. The number of fine grid iterations needed to converge to a residual norm of 1.0E-6.

J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455 7449

Table 2The number of the fine grid iterations(FGI) for each level.

Level 1 2 3 4 5 6 7 8

FGI 9407 3289 809 221 65 29 29 29

Table 3Stability constraint of Dt for the CN scheme and the proposed scheme.

Mesh size Scheme

Crank–Nicolson scheme Proposed scheme

h = 1/64 Dt 6 0.001120 Dt 6 0.777h = 1/128 Dt 6 0.000270 Dt 6 0.914h = 1/256 Dt 6 0.000672 Dt 6 1.140h = 1/512 Dt 6 0.000166 Dt 6 0.879

7450 J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455

And the rate of convergence is defined as the ratio of successive errors as log2ðkeh=h2k2=keh

2=h4k2Þ. Then Table 1 shows the errors

and rates of convergence. The results suggest that the scheme is first-order accurate in time and second-order in space as weexpect from the discretization.

3.3. Multigrid efficiency

To demonstrate the efficiency of the multigrid method, we perform the similar simulation as in [34]. The initial state andcomputational domain are same to the convergence test case. The parameters are �3, h = 1/256, Dt = 1.0h, and 512 � 512mesh grid. All available vcycle levels are considered. Fig. 10 shows the maximum value of residual versus the number of finegrid iterations needed to converge to a residual norm of 1.0E-6 (we only plot four different levels for clarity). And in Table 2the number of the fine grid iterations is listed for all levels. Significant improvement in the convergence is observed as weincrease the vcycle level.

3.4. Stability comparison of two schemes

We compare the stability constraints for the CN scheme (6) and (7) and the proposed scheme (8) and (9). For eachscheme, we calculate the maximum Dt so that stable solutions can be computed up to T = 100Dt. The initial concentrationis taken to be c(x,y,0) = 0.5 + 0.5rand (x,y) in the computational domain X = (0,1) � (0,1). Table 3 shows the results of thestability constraints of Dt with two schemes. We can see that the proposed scheme can use larger time steps. If the timestep is reasonably small, then the stability of the proposed scheme seems independent of space step size.

3.5. The effect of constant and variable mobilities

3.5.1. Two-dimensional caseWe examine the evolution of a random perturbation with the small magnitude about a mean composition and also con-

sider numerical experiments highlighting the difference between the variable mobility M(c) = c(1 � c) and the constantmobility M(c) 0.25 which is the maximum value of the variable mobility M(c). The initial condition is taken to bec(x,y,0) = 0.25 + 0.001rand (x,y) in the computational domain X = (0,1) � (0,1). And we take the simulation parameters:�4, h = 1/128, and Dt = 0.5h. We stop the numerical computations when the discrete l2 -norm of the difference between(n + 1)th andnth time step solutions becomes less than 10�6, that is, kcn+1 � cnk2 6 10�6. In the constant mobility case,we start the simulation with c(x,y,7.813) from the variable mobility case.

In Fig. 11, the first row and the second row represent time evolutions of phase-fields with the constant and the variablemobilities, respectively. The final numerical solutions are stationary numerical solutions according to the stopping criteria.In the bottom row of figure, dark regions are nucleated. As the remaining regions grow, a numerical equilibrium state isreached. The variable mobility generally reduces the bulk diffusion, which corresponds to the interface-diffusion-controlleddynamics, i.e., diffusion process takes place along the interface between the two phases. This fact is made clear by comparingthe results with the constant mobility. In the case of the constant mobility, the evolution leads to a single disk.

3.5.2. Three-dimensional caseNext, we investigate the effect of constant and variable mobilities on the three-dimensional domain. We simulate the

evolution on a 643 mesh and use the following parameters: � = 0.01, h = 1/64, and Dt = h. The initial condition is taken tobe c(x,y,z,0) = 0.25 + 0.2rand (x,y,z) in the computational domain X = (0,1) � (0,1) � (0,1). In the constant mobility case,we start the simulation with c(x,y,z,62.5) from the variable mobility case. In Fig. 12, evolutions of the concentration c(x,y,z, t)

Fig. 11. Evolution of the concentration c(x,y, t) with a constant mobility M(c) 0.25 (the top row) and a variable mobility M(c) = c(1 � c) (the bottom row).The times are shown below each figure.

Fig. 12. Evolution of the concentration c(x,y,z, t) with a constant mobility M(c) 0.25 (the top row) and a variable mobility M(c) = c(1 � c) (the bottom row).The times are shown below each figure.

J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455 7451

with a constant mobility M(c) 0.25 (the top row) and a variable mobility M(c) = c(1 � c) (the bottom row) are shown. In thecase of the variable mobility, two nucleated components are the numerical steady state solution. However, in the case of theconstant mobility (the top row), only one component is shown.

3.6. The decrease of the total energy and the conservation of the average concentration

3.6.1. Rectangular domain caseIn order to demonstrate that the numerical scheme inherits the energy decreasing property, we consider the evolution of

the discrete total energy. We define a discrete energy functional by

EhðcnÞ ¼ ðFðcnÞ;1Þh þ�2

2jcnj21: ð25Þ

The initial state is taken to be c(x,y,0) = 0.25 + 0.001rand (x,y) in the computational domain X = (0,1) � (0,1). Parametersh = 1/128, �4, and Dt = 0.1 are taken. In Fig. 13, the time evolution of the non-dimensional discrete total energyEhðcnÞ=Ehðc0Þ (solid line) is shown. We can see that the total discrete energy is non-increasing and the average concentrationis preserved. And this numerical result agrees well with the total energy dissipation property. Also, the inscribed small fig-ures are the concentration fields at the indicated times.

0 1000 2000 3000 40000

0.2

0.4

0.6

0.8

1

Time

total energymass

Fig. 13. The temporal evolution of non-dimensional discrete total energy EhðcnÞ=Ehðc0Þ (solid line) of the numerical solutions with the initial condition,c(x,y,0) = 0.25 + 0.001 rand (x,y).

7452 J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455

3.6.2. Non-rectangular domain caseNow, we apply the numerical scheme for arbitrarily domains. The initial state is taken to be /(x,y,0) = 0.25 + 0.001rand

(x,y) in the computational domain X = (0,1) � (0,1) with the mesh size 256 � 256 and the temporal step size Dt = 0.1/256.The interface C is a circle whose radius is 0.45 and center is located at (0.5,0.5). Xin is the enclosed domain by the interfaceC. To demonstrate the discrete energy dissipation property in an arbitrary domain, we define a discrete energy functional by

Fig. 14(x,y,0)

Ehð/nÞ ¼ ðGFð/nÞ;1Þh þ�2

2jG/nj21: ð26Þ

Fig. 14 shows that the discrete total energy is monotonically decreasing and the average concentration is conserved. The in-scribed small figures are the evolution of the concentration field.

3.7. Evolution in complex domains

3.7.1. Spinodal decomposition in a wavy curved channelWe also apply our proposed scheme to a more complicated domain. First, we define the interface C which has the upper

bound u(x) and the lower bound l(x) in the computational domain X = (0,4) � (0,1). That is,

uðxÞ ¼ 0:05ffiffiffixp

sinð3pxÞ þ 0:75; ð27ÞlðxÞ ¼ 0:01x2 cosð2pxÞ � 0:04 sinð6pxÞ þ 0:3: ð28Þ

We simulate the evolution on a 1024 � 256 mesh size. The spatial size h = 1/256, the temporal size Dt = 0.5h, and �4 are used.The initial condition is taken to be /(x,y,0) = �0.35 + 0.1rand (x,y). Fig. 15 shows the evolution of the concentration /(x,y, t)

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Time

total energymass

. The time dependent non-dimensional discrete total energy Ehð/nÞ=Ehð/0Þ (solid line) of the numerical solutions with the initial condition, /= 0.25 + 0.001 rand (x,y).

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

(a)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

(b)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

(c)

Fig. 15. Evolution of the concentration /(x,y, t) with a variable mobility M(/) = 1 � /2 in two-dimensional domain with the boundary control function G.The times of each figure are (a) t = 0.98, (b) t = 3.91, and (c) t = 9.77.

Fig. 16. Spinodal decomposition in sequential T-junctions.

J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455 7453

with a variable mobility M(/) = 1 � /2 and the boundary control function G. The times of each figure are (a) t = 0.98, (b)t = 3.91, and (c) t = 9.77.

3.7.2. Spinodal decomposition in sequential T-junctionsMicrofluidic technology offers capabilities for the precise handling of small fluid volumes dispersed as droplets. Sequen-

tial application of the geometrically mediated T-junction is used to facilitate the precise conversion of large initial slugs ofthe dispersed phase into small droplets comparable in size to the channel [24]. We test our scheme on sequential T-junc-

Fig. 17. Evolution of the concentration c(x,y,z, t) with an initial condition c(x,y,z,0) = 0.25 + 0.2rand (x,y,z). The times are shown below each figure.

Fig. 18. Evolution of the concentration c(x,y,z, t) with a variable mobility M(c) = c(1 � c) in a three-dimensional domain. The times are shown below eachfigure.

7454 J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455

tions. The numerical parameters are h = 1/512, Dt = 0.1h, and �4. The initial state is /(x,y,0) = �0.4 + 0.001rand (x,y) in thecomputational domain X = (0,3) � (0,2). In Fig. 16, we see spinodal decomposition in sequential T-junctions at timet = 0.0098.

3.7.3. Spinodal decomposition in a sphereWe extend the numerical experiment to the three-dimensional space. The surface C is a sphere whose radius is 0.45 and

center is (0.5, 0.5, 0.5). In this computation, we choose �4, h = 1/64, and Dt = h. The initial condition is taken to bec(x,y,z,0) = 0.25 + 0.2rand (x,y,z) on 643 mesh grid. In Fig. 17, Xin is inside mesh. For visibility, we plot only the longitudeand latitude of the boundary of the domain Xin. This simulation also represents that the separation and small nucleationoccur at an early stage (t = 2.34). After this initial phase separation, the coarsening process is developed slowly(t = 96.88,312.5).

3.7.4. Spinodal decomposition in a wavy curved tubeNext, for a more complex domain simulation in 3D, we use the following surface:

rðh; zÞ ¼ ð0:01z2 cosð2pzÞ � 0:04 sinð6pzÞ þ 0:3Þð0:9� 0:3 cosðhþ pzÞÞ for 0 6 h 6 2p; 2 6 z 6 4;

where r, h, and z are distance from, azimuthal angle about, and distance along axis, respectively (see Fig. 18 (a)). The numer-ical parameters are 128 � 128 � 256 mesh size, h = 1/128, Dt = h, and �3. The initial condition is c(x,y,z,0) = 0.25 + 0.002rand(x,y,z) in the computational domain X = (�0.5,0.5) � (�0.5,0.5) � (2,4). Fig. 18(b)–(d) show the evolution of the concentra-tion c(x,y,z, t) with a variable mobility M(c) = 1 � c2. The times are shown below each figure.

J. Shin et al. / Journal of Computational Physics 230 (2011) 7441–7455 7455

4. Conclusions

We considered an efficient finite difference scheme for the Cahn–Hilliard equation with a variable mobility of a model forphase separation in a binary mixture. The numerical method is based on a type of the unconditionally gradient stable split-ting discretization. We also extended the scheme to compute the Cahn–Hilliard equation in complex domains. And weproved the mass conservation property of the proposed discrete scheme for complex domains. The resulting discretizedequations are solved by a multigrid method. Numerical simulations were presented to show the robustness and the supe-riority of the proposed scheme compared to other Crank–Nicolson scheme. The most salient feature is that although thealgorithm is simple, it is applicable to many complex domains. Also, the algorithm is well suited to the multigrid method.The method described here is a specific application of a general formulism for constructing conservative finite differencemethods for problems with complex embedded domains. Therefore, it is applicable to other partial differential equationsin irregular domains.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education, Science and Technology (No. 2010-0003989). The authors also wish to thank thereviewers for the constructive and helpful comments on the revision of this article.

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