First and second order operator splitting methods for the phase field crystal equation

Contents lists available atScienceDirect

Journal of Computational Physics

www.elsevier.com/locate/jcp

First and second order operator splitting methods for the phase field crystal equation

Hyun Geun Lee

^a

, Jaemin Shin

^a

, June-Yub Lee

^b

^,∗

aInstituteofMathematicalSciences,EwhaWomansUniversity,Seoul120-750,RepublicofKorea bDepartmentofMathematics,EwhaWomansUniversity,Seoul120-750,RepublicofKorea

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

Articlehistory:

Received16October2014

Receivedinrevisedform8May2015 Accepted30June2015

Availableonline3July2015

Keywords:

Phasefieldcrystal Operatorsplittingmethod Firstandsecondorderconvergences Fourierspectralmethod

Inthispaper, wepresent operator splitting methods for solving the phasefield crystal equationwhich is a model for the microstructural evolution of two-phasesystems on atomiclengthanddiffusivetimescales.Acoreideaofthemethodsistodecomposethe originalequationintolinearandnonlinearsubequations,inwhichthelinearsubequation has a closed-form solution in the Fourier space. We apply a nonlinear Newton-type iterativemethodtosolve the nonlinearsubequation attheimplicit timelevel and thus aconsiderablylargetimestepcanbeused.Bycombiningthesesubequations,weachieve thefirst- andsecond-orderaccuracyintime.Wepresentnumericalexperimentstoshow theaccuracyandefficiencyoftheproposedmethods.

©^{2015ElsevierInc.}All rights reserved.

1. Introduction

Materialpropertiesatthemeso- andmacro-scalesaretoalargeextentcontrolledbycomplexmicrostructuresexhibiting topological defects, such as vacancies, grain boundaries, and dislocations. An understanding of formation andevolution of these defects is of great interest, and defects pose significant challenges to modeling andsimulation because of the complexitytheyintroduce.Recently,anewmodelcalledthephasefieldcrystal(PFC)modelforsimulatingdefectshasbeen proposedbyElderetal.[1,2].Thismodeldescribesthemicrostructureoftwo-phasesystemsonatomiclengthscalesbuton diffusive timescales, leadingto significantcomputationalsavings comparedtomolecular dynamicssimulationswhichare limitedbyatomiclengthscalesandfemtosecondtimescales.InthePFCmodel,aphase-fieldformulationisintroducedthat accountsfortheperiodicstructureofacrystallatticethroughafreeenergyfunctionalofSwift–Hohenbergtype[3]

E (φ) :=







1

4

φ

⁴

+

¹

− 

2

φ

²

− |∇φ|

²

+

¹ 2

(φ)

²



dx

,

(1)

where

φ :  ⊂ R

^d

→ R (

^d

=

¹

,

2

,

3

)

isthedensityfield,



isapositiveconstantwithphysicalsignificance,and

∇

^and



are thegradientandLaplacianoperators,respectively.ThePFCequation isderived fromtheenergyfunctional

E(φ)

underthe constraintofmassconservation:

∂φ

∂

t

= ∇ · (

^M

(φ)∇ μ ) ,

(2)

*

Correspondingauthor.

E-mailaddress:[email protected](J.-Y. Lee).

http://dx.doi.org/10.1016/j.jcp.2015.06.038 0021-9991/©^{2015ElsevierInc.}All rights reserved.

whereM

(φ) >

0 isamobilityfunctionand

μ

isthechemicalpotential definedas

μ _:=

^δ_δφ^E

_{= φ}

³

_{+ (}

1

−  )φ +

²

φ + 

²

φ

. δE

δφ denotes thevariational derivative of

E

^{with respectto}

φ

. Weassume that

φ

,

φ

, and

μ

are periodic on



.Because (2) isof gradient type, it is easy to seethat theenergy functional

E(φ)

^is non-increasing intime. Taking M

(φ) =

^{1 for} convenience,weobtainthePFCequation

∂φ

∂

t

= (φ

³

+ (

¹

−  )φ +

²

φ + 

²

φ).

(3)

ThePFCequationisasixth-ordernonlinearpartialdifferentialequation.Itisnoteasytogetananalyticsolutioningen- eral,therefore,accurateandefficientnumericalalgorithmsareessentialinthecomputersimulations.Variouscomputational algorithms [1,2,4–8] have beenapplied to solve the PFC equation numerically. In [1,2], Elderet al.use an explicit Euler methodwhichisknowntobeunstablefortimestep



t aboveathresholdproportionalto

(

x

)

⁶where



x isgridspacing.

Thus,theexplicitEulermethodiscomputationallyexpensivetoevolvelargesystems.

In[4],ChengandWarrenproposeamethodwhichimprovesthetimesteprestrictionconsiderablylargerbysplittingthe lineartermsintobackward andforwardpieces whiletreating thenonlineartermexplicitly.In[5],Backofenetal. present asemi-implicitfiniteelementmethodwhichisa backwardEulermethod,whereasthenonlinearterm

φ

³ inthechemical potential

μ

islinearizedvia

(φ

ⁿ⁺¹

)

³

≈

³

(φ

ⁿ

)

²

φ

ⁿ⁺¹

−

²

(φ

ⁿ

)

³.In[6],Huetal. presentfirst-orderone-stepandsecond-order two-stepmethodswhereaconsiderablylargetime stepcan beusedbycomputingthenonlineartermatanimplicittime level.However,aneffectivetimestepbecomessmallerthanthespecifiedtimestep,astheauthorsobservedforlargertime steps.

WeherepresentaccurateandefficientoperatorsplittingmethodsforsolvingthePFCequationthatarefirst- andsecond- ordertimeaccurate.Operatorsplittingschemeshavebeenandcontinuetobeusedformanytypesofevolutionequations [9–12].Itiseasytoconstructafirst-ordersolution A

(

tⁿ⁺¹

)

oftimeevolutionequation

∂

A

∂

t

=

^f1

(

A

) +

^f2

(

A

)

bycomputing A

(

tⁿ

+ 

^t

) ∼ = 

S

₁^t

◦ S

₂^t



A

(

tⁿ

)

where

S

₁^{t and}

S

₂^{t aretheevolutionoperatorsfor ∂}_∂^A_t

=

^f1

(

A

)

and ^∂_∂^A_t

=

^f2

(

A

)

,respectively.Thenasecond-orderscheme canbederivedsimplybysymmetrizingthefirst-orderscheme[10]:

A

(

tⁿ

+ 

^t

) ∼ = 

S

₁^t/2

◦ S

₂^t

◦ S

₁^t/2



A

(

tⁿ

).

AcoreideaoftheproposedmethodsistodecomposethePFCequationintolinearandnonlinearsubequations,inwhichthe linearsubequationhasaclosed-formsolutionintheFourierspace. WeapplyanonlinearNewton-typeiterativemethodto solvethenonlinearsubequationattheimplicittimelevelandthusaconsiderablylargetimestepcanbeused.Inparticular, wecombineahalf-time linearsolveranda second-ordernonlinearsolverfollowedby afinal half-timelinearsolverfora second-ordermethod.

This paper is organized as follows. In Section 2, we propose new operator splitting methods for the PFC equation.

Numerical experimentsshowingthe accuracyandefficiencyofthe proposed methodsare presentedinSection 3. Finally, conclusionsaredrawninSection4.

2. Operatorsplittingmethodsforthephasefieldcrystalequation

We considerthe PFCequation (3) intwo-dimensional space

 = [

⁰

,

L1

] × [

⁰

,

L2

]

^{. LetN}1 and N2 be positiveintegers, h1

=

^L1

/

N1 andh2

=

^L2

/

N2 be uniformgridsizes, and



t bethetime stepsize. Wedenotex_l1

=

^l1h1 and y_l2

=

^l2h2 for l1

=

⁰

,

1

, . . . ,

N1

−

^{1 andl}2

=

⁰

,

1

, . . . ,

N2

−

^1.Let

φ

_lⁿ

1l2 betheapproximationof

φ (

xl1

,

yl2

,

tⁿ

)

,wheretⁿ

=

ⁿ



t.

Forsimplicityofnotation,wedefinethe“linearoperator”

L

^{t asfollows}

L

^t

(φ (

tⁿ

)) := φ(

^tn

+ 

^t

),

(4)

where

φ (

tⁿ

+ 

^t

)

isasolutionofthelineardifferentialequation

∂φ

∂

t

= (

¹

−  )φ +

²



²

φ + 

³

φ

(5)

withan initial condition

φ (

tⁿ

)

.Tosolve the PFCequation withthe periodicboundary condition, we employthe discrete Fouriertransform

φ = F[φ]

^:

φ

k1k2

=

N

₁−¹ l₁=⁰

N

₂−¹ l₂=⁰

φ

l1l2e⁻ⁱ



x_l1ξ_k1+^y_l2ξ_k2

,

(6)

where

ξ

k1

=

²

π

k1

/

L1 and

ξ

k2

=

²

π

k2

/

L2 fork1

=

⁰

,

1

, . . . ,

N1

−

^{1 andk}2

=

⁰

,

1

, . . . ,

N2

−

^{1.Then, we have an analytical} formulafortheevolutionoperator

L

^{t inthediscreteFourierspace}

L

^t

(φ) = F

⁻¹



e^At

F

[

φ

]



,

(7)

where A

(

k1

,

k2

) = −(

¹

−  )

 ξ

_k²

1

+ ξ

_k²₂

 +

²



ξ

_k²

1

+ ξ

_k²₂



2

−  ξ

_k²

1

+ ξ

_k²₂



3

.Wealsodefinethe“nonlinearoperator”

N

^t asfollows

N

^t

(φ (

tⁿ

)) := φ(

^tn

+ 

^t

),

(8)

where

φ (

tⁿ

+ 

^t

)

isasolutionofthenonlineardifferentialequation

∂φ

∂

t

= φ

³ (9)

withaninitialcondition

φ (

tⁿ

)

.

Afirst-orderoperatorsplittingmethodforthePFCequationcanberepresentedas

φ

ⁿ⁺¹

= 

L

^t

◦ N

1t



φ

ⁿ

,

(10)

where

φ

ⁿ

∼ = φ(

^tn

)

and

φ

ⁿ⁺¹

∼ = φ(

^tn

+ 

^t

)

.Tosolvetheevolutionoperator

N

1t withthefirst-orderaccuracy,weapplythe backwardEulermethodtoEq.(9)andrepresenttheresultingequationtothedecouplingformbyintroducingtheauxiliary variable

ν

as

φ

ⁿ⁺¹

− φ

ⁿ

= 

t

 ν

ⁿ⁺¹

,

(11)

ν

ⁿ⁺¹

₌ ^ φ

ⁿ⁺¹



3

.

(12)

Becauseofthenonlinearityfor

φ

ⁿ⁺¹,welinearizethenonlineartermat

φ

^n,mas

 φ

ⁿ⁺¹



3

≈  φ

^n,m



3

+

³

 φ

^n,m



2



φ

ⁿ⁺¹

− φ

^n,m



(13) form

=

⁰

,

1

, · · ·

^{.Wethengetthenonlinear}Newton-typeiterativemethodas

I

−

^t



−

³

 φ

^n,m



2

I

 φ

^n,m+1

ν

^n,m+1



=

φ

ⁿ

−

²

 φ

^n,m



3



,

(14)

where

φ

^n,0

= φ

^{n.Wecaneliminate}

φ

^n,m+1in(14)andobtainthefollowingequationfor

ν

^n,m+1:

−

³



t

 φ

^n,m



2

 ν

^n,m+1

+ ν

^n,m+1

=

³

 φ

^n,m



2

φ

ⁿ

−

²

 φ

^n,m



3

.

(15)

Fromthenumericalsolution

ν

^n,m+1 of(15),wecanobtain

φ

^n,m+1

= φ

ⁿ

+ 

^t

 ν

^n,m+1

.

(16)

Andweset

φ

ⁿ⁺¹

= φ

^{n,m+1 (17)}

ifarelativel2-normoftheconsecutiveerror ^φ

n,m+1−φ^n,m

φ^n,m islessthantol.

Inaddition,asecond-orderoperatorsplittingmethodforthePFCequationcanberepresentedas

φ

ⁿ⁺¹

= 

L

^t/2

◦ N

2t

◦ L

^t/2



φ

ⁿ

.

(18)

Tosolvetheevolutionoperator

N

2t withthesecond-orderaccuracy,weapply theCrank–NicolsonmethodtoEq.(9)and representtheresultingequationtothedecouplingformbyintroducingtheauxiliaryvariable

ν

as

φ

ⁿ⁺¹

− φ

ⁿ

= 

^t

 ν

ⁿ⁺¹²

,

(19)

ν

ⁿ⁺¹²

₌

 φ

ⁿ⁺¹



3

+  φ

ⁿ



3

2

.

(20)

Replacing

(φ

ⁿ⁺¹

)

³by

 φ

^n,m



3

+

³

 φ

^n,m



2



φ

ⁿ⁺¹

− φ

^n,m



asin(13)yields

I

−

^t



−

³₂

 φ

^n,m



2

I

 φ

^n,m+1

ν

^n,m+1



=

φ

ⁿ

−  φ

^n,m



₃

+

¹₂

 φ

ⁿ



₃



(21) andwealsoapplynonlineariterationssimilarwith(15)–(17).

3. Numericalexperiments

Inthissection, wepresentexamplesto numericallydemonstratetheaccuracyandefficiencyoftheproposed operator splitting(OS)methodscomparedtothewell-knownenergystable(ES)methodsbyHuetal.[6].Firsttwonumericalexam- plesshowtheconvergenceandeffectivetimeoftwomethodsfora one-dimensionaltestproblem.Thethirdexampleshows timeevolutionofrandomperturbationin2D.Wethenperformasimulationofthegrowthofapolycrystalinasupercooled liquid,whichdemonstratesthefeasibilityoftheOSmethodincomputingtheevolutionoflargesystems.

Here webriefly review theES methodswhich willbe usedforthe numericalcomparisons.The first-orderESmethod canbesummarizedasfollows:

φ

ⁿ⁺¹

− φ

ⁿ

= 

t

 μ

ⁿ⁺¹

,

(22)

μ

ⁿ⁺¹

= (φ

ⁿ⁺¹

)

³

+ (

1

−  )φ

ⁿ⁺¹

+

2

φ

ⁿ

+  κ

ⁿ⁺¹

,

(23)

κ

ⁿ⁺¹

= φ

ⁿ⁺¹

.

(24)

Byapplyingthelinearizednonlinearterm(13)to(22)–(24),thefollowingnonlineariterationisconsidered,

⎡

⎣

^I

−

t



0

−

³

(φ

^n,m

)

²

− (

¹

−  )

I

−

−

^{0 I}

⎤

⎦

⎡

⎣ φ

^n,m+1

μ

^n,m+1

κ

^n,m+1

⎤

⎦ =

⎡

⎣ φ

ⁿ

2

φ

ⁿ

−

²

(φ

^n,m

)

³ 0

⎤

⎦ .

(25)

Andthesecond-orderESmethodis

φ

ⁿ⁺¹

− φ

ⁿ

= 

^t

 μ

ⁿ⁺¹²

,

(26)

μ

ⁿ⁺¹²

=

¹

4

(φ

ⁿ⁺¹

+ φ

ⁿ

)((φ

ⁿ⁺¹

)

²

+ (φ

ⁿ

)

²

) +

¹

− 

2

(φ

ⁿ⁺¹

+ φ

ⁿ

) +

³

φ

ⁿ

− φ

ⁿ⁻¹

+  κ

ⁿ⁺¹²

,

(27)

κ

ⁿ⁺¹²

₌

¹

2

(φ

ⁿ⁺¹

+ φ

ⁿ

),

(28)

where

φ

⁻¹

= φ

^{0.For(26)–(28),thefollowingnonlineariterationis}considered,

⎡

⎣

^I

−

^t



0

A₂₁ I

−

−

¹₂



0 I

⎤

⎦

⎡

⎣ φ

^n,m+1

μ

^n,m+1

κ

^n,m+1

⎤

⎦ =

⎡

⎣ φ

ⁿ

b₂

−

¹₂

φ

ⁿ

⎤

⎦ ,

(29)

whereA21

= −

¹₄



(φ

^n,m

)

²

+ (φ

ⁿ

)

²



−

¹₂

(

1

−  )

andb2

= −

¹₄



(φ

^n,m

)

²

+ (φ

ⁿ

)

²



φ

ⁿ

−

¹₂

(

1

−  )φ

ⁿ

+

³

φ

ⁿ

− φ

^{n−1.Onecanalso} applythenonlineariterations(25)and(29)untilarelativel2-normoftheconsecutiveerrorof

φ

^n,m+1 islessthantol.Then let

φ

ⁿ⁺¹ tobe

φ

^n,m+1.

3.1. Numericalconvergencewithasmoothtestfunctionin1D

WedemonstratetheconvergenceoftheOSmethodswithanumericalsolutionofthePFCequationwithinitialcondition

φ (

x

,

0

) =

⁰

.

07

−

⁰

.

02 cos



2

π (

x

−

¹²

)

32



+

⁰

.

02 cos²

 π (

x

+

¹⁰

)

32



−

0

.

01 sin²



4

π

x 32



(30) onadomain

 = [

⁰

,

32

]

^{.Thenumericalsolutionisevolvedtotime T}f

=

^{10 with}

 =

⁰

.

025.Thegridsizeisfixedtoh

=

¹

/

3 whichprovidesenoughspatialaccuracy(around10digits).

Toestimate theconvergenceratewithrespect toa timestep



t, simulationsare performedbyvarying thetime step



t

=

^Tf

/

2¹²

,

Tf

/

2¹¹

, . . . ,

Tf.We usetheiterativemethodto solvethediscretenonlinearsystem(11)–(12)and(19)–(20).

Thestoppingcriterionforthenonlinearm-iterationisthatarelativel₂-normoftheconsecutiveerrorislessthanatolerance (10⁻¹⁰inthisexample).Theinitialguessateachtimelevelistakenfromthesolutionattheprevioustimelevel.

Thenumberofm-iterationsaveragedoverthesimulationtime0

<

t

=

ⁿ



t

≤

^Tf isshownasafunctionoftimestep



t inFig. 1.In(14)and(21)fortheOSmethods,2–3iterationswereinvolvedinproceedingtothenexttimelevel.Webelieve that suchafastiterativeconvergencecanbe achievedsincethesuccessiveiteration (14)isaNewton-typeapproximation of(11)and(12)forthefirst-orderapproximation.Likewise, (21)isaNewton-typeapproximationof(19)and(20)forthe second-order.

Fig. 2shows therelativel₂-errorsat T_f

=

^{10 forvarioustime steps}



t.Here,the errorsarecomputedby comparison witha quadruplyover-resolvedreferencenumericalsolution.Both OSandESmethodsprovidethefirst- andsecond-order accuracyintime;however,theOSmethodsgive twoordersofmagnitudehigheraccuracywherebothOSandESmethods intheconvergenceregion.

Fig. 1. Number of m-iterations averaged over the simulation time.

Fig. 2. Relative l2-errors at Tf=10 for various time steps with=0.025 and h=1/3.

3.2. Analysisofeffectivetimestepwithasmoothtestfunctionin1D

In [4],Cheng andWarrendetermined aneffectivetime step



teff intheFourierspaceandobservedthata largetime step



t doesnot alwaystranslate intoasignificant amountofsystemevolution inthePFCequation.Similar timescaling effects havebeen reportedalsoforthe Allen–Cahn equation [11,13]. Tonumericallycalculatethe effectivetime step,we take the sameinitial condition (30)as in theprevious section on a domain

 = [

⁰

,

32

]

^{.The numerical solution is now} computeduptotime T_f

=

^{100 with}

 =

⁰

.

025 andh

=

¹

/

3.Thetolerancefortheiterativestepissettobe10⁻¹⁰.Andwe take thereferencesolutionwithasufficientlysmalltime step,say



t

=

^Tf

/

2¹²intheexperiments.Fig. 3showsthetime evolutions ofthe totalenergy with



t

=

^Tf

/

2⁶, T_f

/

2⁴, andT_f

/

2². First-orderOS andES methods are usedto generate theplots.UsingtheOSmethod,thetotalenergyatthesametimet arealmostthesamefordifferenttimesteps,whereas significantdifferencesemergewithlargertimestepsfortheESmethod.

Tomakethecomparisonquantitatively,wedefinetheeffectivetimeteff asshowninFig. 4,

E

^t

(

n



t

) = E

^ref

(

t_eff

),

where

E

^{t isthetotalenergyofthe computedsolution witha timestep}



t andwe takethe referencetotal energy

E

^ref fromthereferencesolution.

Fig. 5(a) showsthetime dilationratios R

(

t

) =

^teff

/

t usingdifferent time steps



t

=

^Tf

/

2¹⁰

,

Tf

/

2⁸

, . . . ,

Tf

/

2² forthe first-orderESandOSmethods,respectively.Forthefirst-orderESmethod,thedifferencebetweeneffectiveteff andnumer- icalt timesbecomesbiggerandbiggerasthetimestepincreases.Inthiscase,theratioatt

=

^Tf is R

(

T_f

) ≈

¹

−

²

·

¹⁰⁻²



t (seeFig. 5(b)).Ontheotherhand,forthefirst-orderOSmethod,theeffectivetimet_eff isalmostsameas thenumericaltime

Fig. 3. Time evolutions of the total energy with different time steps. First-order OS and ES methods are used to generate the plots.

Fig. 4. Definition of the effective time teff:E^t(nt)= E^ref(teff).

Fig. 5. Timedilationratio R(t)=^teff/t for0≤^t≤^Tf=^{100 usingdifferenttime steps}t=^Tf/2¹⁰,Tf/2⁸,. . . ,Tf/2². (a) R(t)forthe first-order,(b)

|^R(Tf)−¹|^forthefirst-order,(c)|^R(Tf)−¹|^forthesecond-ordermethod.

t regardlessoftimestepsizeandtheratiois R

(

T_f

) ≈

¹

+

⁴

·

¹⁰⁻⁵



t.Forthesecond-orderESmethod,weobservethatthe ratiodecreasesto R

(

Tf

) ≈

¹

−

¹

.

5

·

¹⁰⁻⁴

(

t

)

² (see Fig. 5(c)),butit isstilllargerthan thatobtainedbythefirst-orderOS method.Forthesecond-orderOSmethod,theratioisR

(

T_f

) ≈

¹

+

⁴

·

¹⁰⁻⁷

(

t

)

².

Fig. 6. TotalenergyE(t)ofthereferencesolutionandtotalenergyE(nt)computedbythefirst-orderOSmethodmarkedwith(·).Fourcomputational resultsclosesttothereferenceenergyE(^t)att=⁶⁰,120,180,240 arelabeledwithmarkers(,∗,◦^)andthecorrespondingdensityfieldsφ(nt)have beenshownabove.(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

3.3. Timeevolutionofrandomperturbationin2D

We now perform a 2D simulation of the PFC equation with

 =

⁰

.

025 and space grid size h

=

^{1 on a domain}

 = [

⁰

,

128

] × [

⁰

,

128

]

^{.Theinitialconditionissetto}

φ (

x

,

y

,

0

) =

0

.

07

+

0

.

07

·

rand

(

x

,

y

),

where rand

(

x

,

y

)

is a randomly chosen number between

−

^{1 and 1 butfixed throughoutall of the} experiments in this subsection.Thetoleranceforthem-iterativestepsissettobetol

=

^10−8.ThesimulationsareperformedbytheOSmethods

Fig. 7. TotalenergyE(t)ofthereferencesolutionandtotalenergyE(nt)computedbythesecond-orderOSmethodmarkedwith(·).Fourcomputational resultsclosesttothereferenceenergyE(^t)att=⁸⁰,160,240,320 arelabeledwithmarkers(,∗,◦^)andthecorrespondingdensityfieldsφ(nt)have beenshownabove.(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtothewebversionofthisarticle.)

withdifferenttimesteps



t

=

⁰

.

01

,

2

,

10,and20 andweconsiderthenumericalsolutionwith



t

=

⁰

.

01 asthereference solution.Figs. 6 and7showthetime evolutionsofthedensityfield

φ

computedbythefirst- andsecond-ordermethods, respectively. In the figure, each rows show the computational results withfixed



t and snapshots in each column are chosen to matchthe total energy ofthe first row(reference solution) ascloselyas possible.In each snapshots,the red, green,andblueregionsindicate

φ =

⁰

.

14,0.07,and0,respectively,andthenumericaltimeandtotalenergyareshownon thetopofthesubplots.

In [6],the authors observed that the numericaltimes required to reach the same energy levels using different time stepsaredramaticallydifferentforthefirst-orderESmethod.Thetimesforthesecond-orderESmethodarematchedmuch

Fig. 8. Heterogeneousnucleationoffivecrystallitesinasupercooledliquid.Thesnapshotsshowthedensityfieldφatdifferenttimes.Theparametersare

¯φ =0.285,=⁰.25,t=^1,andh=^1.Thesecond-orderOSmethodisusedinthecalculation.(Forinterpretationofthereferencestocolorinthisfigure, thereaderisreferredtothewebversionofthisarticle.)

better than thefirst-order computation, but a significant difference still exists. From Figs. 6 and7, we can see that the density fieldsobtainedat theclosest energy levels usingdifferenttime steps arequalitatively similar forboth first- and second-orderOSmethods,and,inparticular,thenumericaltimesrequiredtoreachtheclosestenergylevelsusingdifferent timestepsarenearlymatchedforthesecond-orderOSmethod.

3.4. Crystalgrowthinasupercooledliquidin2D

Wenowusethesecond-orderOSmethodtosimulatethegrowthofapolycrystalinasupercooledliquidwith

 =

⁰

.

25 onthecomputationaldomain

 = [

⁰

,

800

] × [

⁰

,

800

]

^.Theinitialconfigurationof

φ

showninFig. 8(a)consistsoffivesmall square patchesinahomogeneousenvironment

¯φ =

⁰

.

285.Weusethefollowingexpressiontodefinethedensityfield

φ

in thepatchwithamplitude A andfrequencymodeq,

φ (

xl

,

yl

) = ¯φ +

A



cos

(

qyl

)

cos

( √

3qxl

) −

0

.

5 cos

(

2qyl

) 

,

where x_l and y_l define a local system of Cartesian coordinates. We choose the parameters for five patches as

(

A

,

q

) = (

0

.

45

,

q0

)

,

(

0

.

45

,

0

.

25q0

)

,

(

0

.

45

,

4q0

)

,

(

0

.

9

,

q0

)

,

(

0

.

1

,

q0

)

where q0

=

⁰

.

1213

π

. The lower left and the upper right patch representlowerandhigherfrequencyanomalies,respectively,comparedtothecenterpatch. Theupperleftandthelower rightpatchhavethesamefrequencymodebuttheamplitudesarebiggerandsmaller,respectively.

Forthenumericalexperiment,wealsoset



t

=

^{1 andh}

=

^{1 andthetoleranceforthe}m-iterationissettobetol

=

^10−8. Figs. 8(b)–(f)show thesnapshotsofthedensityfield

φ

atdifferenttimes.Ineachfigure,thered, green,andblueregions indicate

φ =

⁰

.

6715,0,and

−

⁰

.

4746,respectively.Theinitialconfigurationevolvesintofivecrystallites,eachwithadifferent orientation anda well-definedliquid/crystal interface. As we can see inthe figures, thespeed ofthe moving interfaces strongly depends on thepatch frequency andamplitude. The frequency andamplitude ofthe center patchseems tobe optimal in the sense that the moving speed of the patch is highest while the higher frequency patch moves with the slowestspeed.Astimeevolvesthecrystallitesimpingeupononeanotherandformgrainboundaries.

4. Conclusions

In thispaper, we presented thefirst- and second-order operator splitting (OS)methods forsolving the PFC equation.

A coreidea ofthe methodswas todecomposetheoriginal equation intolinearandnonlinearsubequations, inwhichthe

linearsubequationhasaclosed-form solutionintheFourierspace. Weapplied anonlinearNewton-typeiterativemethod tosolve thenonlinearsubequationattheimplicittimelevel.As aresult,only 2–3iterations were involvedinproceeding tothenexttimelevel.Tonumericallydemonstratetheaccuracyandefficiencyoftheproposed OSmethods,wecompared withthe well-known energystable(ES)methods. And we observedthat the OSmethods aremore accurate thanthe ES methods.WealsodemonstratedthefeasibilityoftheOSmethodsforthesimulationofphysicalphenomenasuchascrystal growth.

Acknowledgement

ThisresearchwassupportedbytheBasic ScienceResearchProgramthroughtheNationalResearchFoundationofKorea (NRF)fundedbytheMinistryofEducation(2009-0093827,2012-002298).

References

[1]K.R.Elder,M.Katakowski,M.Haataja,M.Grant,Modelingelasticityincrystalgrowth,Phys.Rev.Lett.88(2002)245701.

[2]K.R.Elder,M.Grant,Modelingelasticandplasticdeformationsinnonequilibriumprocessingusingphasefieldcrystals,Phys.Rev.E70(2004)051605.

[3]J.Swift,P.C.Hohenberg,Hydrodynamicfluctuationsattheconvectiveinstability,Phys.Rev.A15(1977)319.

[4]M.Cheng,J.A.Warren,Anefficientalgorithmforsolvingthephasefieldcrystalmodel,J.Comput.Phys.227(2008)6241–6248.

[5]R.Backofen,A.Rätz,A.Voigt,Nucleationandgrowthbyaphasefieldcrystal(PFC)model,Philos.Mag.Lett.87(2007)813–820.

[6]Z.Hu,S.M. Wise,C.Wang,J.S.Lowengrub, Stableand efficientfinite-differencenonlinear-multigridschemesforthephasefieldcrystalequation, J. Comput.Phys.228(2009)5323–5339.

[7]H.Gomez,X.Nogueira,Anunconditionallyenergy-stablemethodforthephasefieldcrystalequation,Comput.MethodsAppl.Mech.Eng.249–252 (2012)52–61.

[8]Z.Zhang,Y.Ma,Z.Qiao,Anadaptivetime-steppingstrategyforsolvingthephasefieldcrystalmodel,J.Comput.Phys.249(2013)204–215.

[9]G.Strang,Ontheconstructionandcomparisonofdifferenceschemes,SIAMJ.Numer.Anal.5(1968)506–517.

[10]D.Goldman,T.J.Kaper,Nth-orderoperatorsplittingschemesandnonreversiblesystems,SIAMJ.Numer.Anal.33(1996)349–367.

[11]H.G.Lee,J.-Y.Lee,Asemi-analyticalFourierspectralmethodfortheAllen–Cahnequation,Comput.Math.Appl.68(2014)174–184.

[12]H.G.Lee,J.-Y.Lee,AsecondorderoperatorsplittingmethodforAllen–Cahntypeequationswithnonlinearsourceterms,PhysicaA432(2015)24–34.

[13]M.Cheng,A.D.Rutenberg,Maximallyfastcoarseningalgorithms,Phys.Rev.E72(2005)055701(R).