convex splitting for phase-field crystal equation First and second order numerical methods based on a new Journal of Computational Physics

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Journal of Computational Physics

www.elsevier.com/locate/jcp

First and second order numerical methods based on a new convex splitting for phase-field crystal equation

Jaemin Shin

^a

, Hyun Geun Lee

^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:

Received5April2016

Receivedinrevisedform22September 2016

Accepted23September2016 Availableonline28September2016

Keywords:

Swift–Hohenbergfunctional Phase-fieldcrystalequation Convexsplittingmethod Gradientstability Energystability

Thephase-fieldcrystalequationderivedfromtheSwift–Hohenbergenergyfunctionalisa sixthordernonlinearequation. Weproposenumerical methodsbasedonanew convex splittingforthephase-fieldcrystalequation.Thefirstorderconvexsplittingmethodbased ontheproposedsplittingisunconditionallygradientstable,whichmeansthatthediscrete energyis non-increasingfor anytimestep.The secondorderschemeisunconditionally weakly energy stable, which means that the discrete energy is bounded by its initial valueforanytimestep.Weprovemassconservation,uniquesolvability,energystability, and theorderoftruncationerrorforthe proposedmethods.Numerical experimentsare presentedtoshowtheaccuracyandstabilityoftheproposedsplittingmethodscompared totheexistingothersplittingmethods.Numericaltestsindicatethattheproposedconvex splittingisagoodchoicefornumericalmethodsofthephase-fieldcrystalequation.

©^{2016ElsevierInc.Allrightsreserved.}

1. Introduction

Phase-field models have emerged asa powerfulapproach formodeling andpredicting mesoscale morphologicaland microstructuralevolutioninmaterials.Manyofsuchmodels trytominimize anenergyfunctional

E(φ)

^{associatedwitha} phasefieldfunction

φ (

x

,

t

)

.Ingeneral,thephasefieldequationismodeledbygradientflowsfor

E(φ)

^,

∂φ

∂

t

= −

^grad

E (φ),

(1)

where thesymbol “grad” denotes the gradient inthe sense of the Gâteaux derivative. Forexample, theAllen–Cahn and Cahn–Hilliard equationsare gradientflows ofthe Ginzburg–Landau free energy[1,2].It is worth tonote that the energy functional

E(φ)

^isnon-increasingintimesince(1)isofgradienttype.

Themaindifficulty developinganumericalmethodforphasefieldequationsisaseverestabilityrestrictiononthetime stepduetononlinearterms andhighorderdifferentialones. Therehavebeenmanynumericalattemptstoovercomethe stability restriction.Theconvexsplitting(CS)methodshavebeenrevitalized bythework ofEyre[3],whichareoriginally attributed to Elliott and Stuart [4]. Recently, many researchers [5–11] have developed noteworthy schemes by using CS idea where an energy functional

E(φ)

is split into two convex functionals (so called contractive and expansive parts),

*

Correspondingauthor.

E-mailaddress:jyllee@ewha.ac.kr(J.-Y. Lee).

http://dx.doi.org/10.1016/j.jcp.2016.09.053 0021-9991/©^{2016ElsevierInc.Allrightsreserved.}

E(φ)

=

E

^c

(φ)

−

E

^e

(φ)

.Theenergyofthefirstordernumericalsolutionbyaconvexsplittingmethodmonotonicallydecreases when

E

^c

(φ)

isnumericallytreatedimplicitlyand

E

^e

(φ)

explicitly,i.e.,

φ

ⁿ⁺¹

− φ

ⁿ

Δ

t

= −

^grad

E

^c

(φ

ⁿ⁺¹

) +

^grad

E

^e

(φ

ⁿ

).

(2)

Theconvexsplittingmethodsallusivelyindicatethatwemighthavemanykindsofenergysplittings.

Thephase-fieldcrystal(PFC)model,whichisthemainsubjectofthispaper,hasbeensuggestedtostudythemicrostruc- turalevolutionoftwo-phasesystemsonatomiclengthanddiffusivetimescales.Elderet al.[12,13]introducethePFCmodel tominimizetheSwift–Hohenbergfreeenergyfunctional[14],

E (φ) =







1 4

φ

⁴

+

¹

2

φ

 −  + (

1

+ )

²

 φ



dx

,

(3)

where

φ

is thedensity field and



is a positivebifurcation constantwith physicalsignificance.Here,



isthe Laplacian operator and

(

1+ )²=¹+²



+ ^{2.Fortheinterested readers,werefer to[14] foradetailedphysicalmeaning ofthe} functional.Inparticular,

(

1+ )^{2 offreeenergyisfromfittingto an}experimental structurefactor[13].ThePFCequation arisingfrom

E(φ)

^{undertheconstraintofmass}conservationcanbewrittenasfollows:

∂φ

∂

t

=  μ _{= } ^ φ

³

−  φ + (

¹

+ )

²

φ



,

(4)

where

μ

isthechemicalpotential definedas

μ

=^δ_δφ^{E and} _δφ^{δ denotesthe}variationalderivativewithrespectto

φ

.Sinceit isoriginallymodeledtoproducetheperiodicstates[12],weassumethatthedensityfield

φ

isperiodicon



.

Some researchers try to use a convex splitting method basedon the following formof the Swift–Hohenberg energy functional,

E (φ) =







1

4

φ

⁴

+

¹

− 

2

φ

²

− |∇φ|

²

+

¹ 2

(φ)

²



dx (5)

whichisidenticalto(3)andcanbeeasilysplitintotwoconvexfunctionals,

E

_{D F}^c

(φ) =







1

4

φ

⁴

+

¹

− 

2

φ

²

+

¹ 2

(φ)

²



dx

, E

_{D F}^e

(φ) =





|∇φ|

^2dx

,

(6)

with



≤^{1.Here,thediffusion(DF) termisusedfortheexpansivepart.Wiseet al.[5]proposeafirstorderanduncondi-} tionallygradientstableschemebasedontheconvexsplitting(6),

φ

ⁿ⁺¹

− φ

ⁿ

Δ

t

=  μ

ⁿ⁺¹

,

(7)

μ

ⁿ⁺¹

=  φ

ⁿ⁺¹



3

+ (

¹

−  ) φ

ⁿ⁺¹

+ 

²

φ

ⁿ⁺¹

+

²

φ

ⁿ

,

(8)

whichwe aregoingtorefertoasCS_{D F}(1).Huet al.[6]proposeasecondorderconvexsplittingmethod,whichisweakly energystable.Thesecondordermethodcanbewrittenas

φ

ⁿ⁺¹

− φ

ⁿ

Δ

t

=  μ

ⁿ⁺¹²

,

(9)

μ

ⁿ⁺¹²

=  φ

ⁿ⁺¹

 +

¹

− 

2



φ

ⁿ⁺¹

+ φ

ⁿ

 +

¹

2



²



φ

ⁿ⁺¹

+ φ

ⁿ

 +  

3

φ

ⁿ

− φ

ⁿ⁻¹



,

(10)

where



φ

ⁿ⁺¹

=¹₄

φ

ⁿ⁺¹+ φⁿ 

φ

ⁿ⁺¹2

+

φ

ⁿ2

and

φ

⁻¹= φ^{0,whichwe aregoingtorefertoasCS}D F(2).Wecangive an accountforthismethodasamulti-stepimplicit–explicitmethod[15].The implicitpartisdesignedwithasecant-type differencemethodlikeasin[16] andexplicitpartisfromasecondorderAdams–Bashfordmethod.Thesecant-typediffer- enceissecondorderaccurateandplaysanimportantrolefortheproofoftheenergystability.

InordertosolvethePFCequationaccuratelyandefficiently,weproposenewnumericalmethodsbasedonthefollowing convexsplitting

E

_{B F}^c

(φ) =







c

(φ) +

¹

2

φ (

1

+ )

²

φ



dx

, E

^e_{B F}

(φ) =





e

(φ)

dx

,

(11)

where

c

(φ)

= ¹₄

φ

⁴ and

e

(φ)

= ^₂

φ

². Notethat (11)has a differentformto (6)but itis closelyrelatedto theoriginal form(3).Here,thebifurcation(BF)termisusedfortheexpansivepart.Wecaneasilyshowthat(11)isaconvexsplitting andthedetailedproofispresentedinAppendix A.

Theproposed energysplittingisalso applicableto theframeworksoffirstandsecond orderconvexsplittingmethods [5,6]. Applying theseframeworks,we propose thefirst andsecond orderconvexsplitting methods andcompletelyprove massconservation,uniquesolvability,energystability,andtheorderoftruncationerrorfortheproposedmethods.Moreover, wetrytonumericallydemonstratethat(11)isagoodchoiceintheconvexsplittingstrategyforaccuratenumericalmethods.

Forinterestedreaderstothesecondordermethodswiththeenergystability,therehavebeennotablestudiesforsolving the PFC equation. Vignal et al. [7]propose a nonlinear, second order time accurate, and unconditionally gradient stable methodbyusingtheCrank–Nicolsonmethodandtheadditionalstabilizationapproachproposedin[8]fortheCahn–Hilliard equation.Theaccuracyandstabilityoftheirmethoddependonthevalue ofthestabilizationparameter.Thesecondorder method in [7]can be considered asthe method based on the convex splitting (11).Glasner et al. [9]propose a linear, second ordertime accurate, and unconditionallygradient stable method based on a linear convex splitting.In addition, Gomezet al.[17] propose a nonlinear, second ordertime accurate, andunconditionallygradient stablemethodwiththe modifiedCrank–Nicolsonmethod,whichisnotbasedontheconvexsplitting.

This paper is organized as follows. In Sections 2 and 3, we propose the first andsecond convex splitting methods, respectively, forthePFC equationwithdetailedproofs forthemassconservation,unique solvability,andenergystability.

In Section 4, a one-dimensional numerical experiment is presented to show the accuracy andstability of the proposed methods.Inaddition,wedescribenonlineariterativemethodswithnumericalimplementations.InSection5,wenumerically demonstratetheorderofaccuracywithatwo-dimensionalexample.Wealsosimulatetwo- andthree-dimensionalexamples toshowtheapplicabilityofthenumericalmethod.Finally,conclusionsaredrawninSection6.InAppendix B,weshowthat theerrorestimateintimeforthefirstandsecondordermethods.Inaddition,weprovideacomparisonofoursecondorder methodwiththemethodof[7]inAppendix C.

2. Numericalanalysisofthefirstordernumericalmethod,CS_{B F}(1)

Inthissection,we proposea firstordernumericalmethodbasedonthe convexsplitting(11)withdetailedproofsfor themassconservation,unconditionaluniquesolvability,andunconditionalgradientstability.Thefirstorderconvexsplitting method,referredtoasCSB F(1),iswrittenas

φ

ⁿ⁺¹

− φ

ⁿ

Δ

t

=  

φ

ⁿ⁺¹



₃

+ (

¹

+ )

²

φ

ⁿ⁺¹

−  φ

ⁿ



.

(12)

Forsimpledescriptionfortheproof,werewrite(12)as

φ

ⁿ⁺¹

− φ

ⁿ

= Δ

^t

 μ

ⁿ⁺¹

,

(13)

μ

ⁿ⁺¹

₌ ^ φ

ⁿ⁺¹



3

+ (

¹

+ )

²

φ

ⁿ⁺¹

−  φ

ⁿ

.

(14)

Theorem1.Thescheme(13)ismassconserving.

Proof. Themassconservationof(13)followsfrom



φ

ⁿ⁺¹

− φ

ⁿ

,

1



L²

= Δ

^t



 μ

ⁿ⁺¹

,

1



L²

= −Δ

^t



∇ μ

ⁿ⁺¹

, ∇

¹



L²

=

⁰

,

(15)

where

(φ, ψ)

_L2=



φψ

dx denotestheL² innerproduct.Here,theintegrationbypartsformulacanbederivedfor

φ

and

ψ

satisfyingtheperiodicboundaryconditions,

(φ, ψ)

_L2

= − (∇φ, ∇ψ)

L²

.

(16)

Thus,if(13)hasasolution

φ

ⁿ⁺¹,thenitmustbe

φ

ⁿ⁺¹

,

1

L²=

φ

ⁿ

,

1

L².

Weintroducetheusefullemma,whichwillbeusedfortheproofofthesolvabilityforbothofthefirstandsecondorder methods.

Lemma2.WeconsidertheHilbertspaceH0asazeroaveragespace.Forgivenv1

,

v2∈^H0,wedefinetheinnerproductofdualspace by

(

v₁

,

v₂

)

_H−1=

∇

ϕ

v1

,

∇

ϕ

v2



L²,where

ϕ

v1

, ϕ

v2∈^H0arethesolutionsoftheperiodicboundaryvalueproblem−

ϕ

v1=^v1and

−

ϕ

v₂=^v2in



.Fromtheabovedefinition,if

ψ

∈^H0,thenwehavetheidentity

(−φ, ψ)

_H−1

= (φ, ψ)

_L²

.

(17)

Theorem3.Thescheme(13)isuniquelysolvableforanytimestep

Δ

t

>

0.

Proof. WeconsiderthefollowingfunctionalonH=

φ

| (φ,¹

)

_L2=

φ

ⁿ

,

1

L²

:

G

(φ) =

¹

2

φ − φ

ⁿ

²

H⁻¹

+ Δ

^t

E

c

(φ) − Δ

^t

 δ δφ ^E

^e

 φ

ⁿ

 , φ



L²

.

(18)

Itmaybeshownthat

φ

ⁿ⁺¹∈ ^{H istheuniqueminimizerofG ifandonlyifitsolves,forany}

ψ

∈^H0, dG

ds

(φ +

^s

ψ ) 



s=0

= 

φ − φ

ⁿ

, ψ 

H⁻¹

+ Δ

^t

 δ

δφ ^E

^c

(φ) , ψ



L²

− Δ

^t

 δ δφ ^E

^e

 φ

ⁿ

 , ψ



L²

=



φ − φ

ⁿ

− Δ

^t



 δ

δφ ^E

^c

(φ) − δ δφ ^E

^e

 φ

ⁿ



, ψ



H⁻¹

=

⁰

,

(19)

becauseitisclearthatthefunctionalG isstrictlyconvexby

d²G

ds²

(φ +

s

ψ ) 



s=⁰

=

¹

2

ψ

²_H−1

+ Δ

t 2

 δ

²

δφ

²

^E

^c

(φ) ψ, ψ



L²

≥

0

.

(20)

Therefore,(19)istrueforany

ψ

∈^H0ifandonlyifthegivenequationholds

φ

ⁿ⁺¹

− φ

ⁿ

Δ

t

= 

 δ δφ ^E

^c

 φ

ⁿ⁺¹

 − δ δφ ^E

^e

 φ

ⁿ



.

(21)

Hence,minimizingthestrictlyconvexfunctional(18)isequivalenttosolving(12)(or(13)).

Beforeprovingtheenergystability,wepresenttwousefullemmas.Forthesimplicity,wedefinetimedifferenceoperators asJφⁿK = φⁿ⁺¹− φ^{n and}q



φ

ⁿy

= 

φ

ⁿ⁺¹

− 

φ

ⁿ

.

Lemma4.Supposethat

φ

,

ψ

aresufficientlyregularand

(φ)

istwotimescontinuouslydifferentiable.Considerthecanonicalconvex splittingof

(φ)

into

(φ)

= c

(φ)

− e

(φ)

,i.e.,bothfunctions

c

(φ)

and

e

(φ)

areconvexfunctions,then

(φ) − (ψ) ≤ 

_c^

(φ) −

e^

(ψ ) 

(φ − ψ).

⁽²²⁾

Proof. The analogousproof could befoundin [5],howeverweintroduce itforaself-containeddescription.Becauseboth

cand

e areconvex,wehave

c

(φ) −

c

(ψ ) ≤

c^

(φ)(φ − ψ),

⁽²³⁾

e

(φ) −

e

(ψ ) ≥

_e^

(ψ )(φ − ψ).

(24)

Byusingtheseinequalities,wehave

(φ)

c

(φ) −

c

(ψ )) − (

e

(φ) −

e

(ψ ))

≤

c^

(φ)(φ − ψ) −

e^

(ψ )(φ − ψ)

= 

_c^

(φ) −

_e^

(ψ ) 

(φ − ψ).

(25)

As the applicationofLemma 4,we havethe followinglemma forthe proof ofthe energystability for thefirst order method.

Lemma5.Consideranenergyfunction

(φ)

=¹₄

φ

⁴−^₂

φ

²andacanonicalconvexsplittinginto

c

(φ)

=¹₄

φ

⁴and

e

(φ)

=^₂

φ

². Sinceboth

c

(φ)

and

e

(φ)

areconvex,wehave

s

1 4

 φ

ⁿ



4

− 

2

 φ

ⁿ



2

{

≤ 

φ

ⁿ⁺¹



3

−  φ

ⁿ



Jφ

ⁿ

K .

⁽²⁶⁾

Now,weprovetheenergystabilityofthefirstordermethod(13).

Theorem6.Supposethat

φ

ⁿ⁺¹isasolutionto(13).Theconvexsplittingscheme(13)isunconditionallyenergystable,meaningthat foranytimestepsize

Δ

t

>

0,

E (φ

ⁿ⁺¹

) ≤ E (φ

ⁿ

).

(27)

Proof. BasedonLemma 5,theenergydifferenceis

J E (φ

ⁿ

) K =





s

1 4

 φ

ⁿ



₄

− 

2

 φ

ⁿ



₂

{ +

¹

2

r

φ

ⁿ

(

1

+ )

²

φ

ⁿ

z

dx

≤







φ

ⁿ⁺¹



3

−  φ

ⁿ



Jφ

ⁿ

K +

¹ 2

r

φ

ⁿ

(

1

+ )

²

φ

ⁿ

z

dx

.

(28)

By(14),wehave

φ

ⁿ⁺¹3

−

 φ

ⁿ=

μ

ⁿ⁺¹_{− (}1+ )²

φ

ⁿ⁺¹ and(28)followsas

J E (φ

ⁿ

)K ≤







μ

ⁿ⁺¹

− (

¹

+ )

²

φ

ⁿ⁺¹

 Jφ

ⁿ

K +

¹ 2

r

φ

ⁿ

(

1

+ )

²

φ

ⁿ

z

dx

=







μ

ⁿ⁺¹

_Jφ

ⁿ

_{K − Jφ}

ⁿ

_{K (}

1

+ )

²

φ

ⁿ⁺¹

+

¹ 2

r

φ

ⁿ

(

1

+ )

²

φ

ⁿ

z

dx

.

(29)

Wenowproceedtoexpandthedifferenttermsontheright-handsideof(29).Using(13),thefirsttermin(29)ismanipu- latedby





μ

ⁿ⁺¹

Jφ

ⁿ

K

^dx

= Δ

^t





μ

ⁿ⁺¹

 μ

ⁿ⁺¹dx

= −Δ

^t





 ∇ μ

ⁿ⁺¹

^ _

²dx

.

(30)

Next,forthesecondtermin(29),wehave





Jφ

ⁿ

K (

1

+ )

²

φ

ⁿ⁺¹dx

=





Jφ

ⁿ

K (

1

+ )

²

 φ

ⁿ⁺¹

+ φ

ⁿ 2

+ Jφ

ⁿ

K

2



dx

=

¹ 2





r

φ

ⁿ

(

1

+ )

²

φ

ⁿ

z

dx

+

¹ 2





Jφ

ⁿ

K (

¹

+ )

²

Jφ

ⁿ

K

^dx

=

¹ 2





r

φ

ⁿ

(

1

+ )

²

φ

ⁿ

z

dx

+

¹ 2





 (

1

+ ) Jφ

ⁿ

K 

2

dx

.

(31)

Here,forthesecondandthirdlinesin(31),weusetheidentity





φ (

1

+ )

²

ψ

dx

=





(

1

+ )φ(

¹

+ )ψ

^dx

=





ψ (

1

+ )

²

φ

dx

.

(32)

Rearranging(31),wehave







− Jφ

ⁿ

K (

1

+ )

²

φ

ⁿ⁺¹

+

¹ 2

r

φ

ⁿ

(

1

+ )

²

φ

ⁿ

z

dx

= −

¹ 2





 (

1

+ ) Jφ

ⁿ

K 

2

dx

.

₍₃₃₎

Applying(30)and(33)to(29),wehave

J E (φ

ⁿ

)K ≤ −Δ

^t





 ∇ μ

ⁿ⁺¹

^ _

²dx

−

¹ 2





 (

1

+ ) Jφ

ⁿ

K 

2

dx

≤

^{0 (34)}

whichprovestheenergydissipationforanytimestep

Δ

t

>

0.

3. Numericalanalysisofthesecondordermethod,CS_{B F}(2)

Now, we propose a second order numerical method based on the convex splitting (11)with detailed proofs forthe massconservation,unconditionaluniquesolvability,andunconditional(weakly)gradientstability.Thesecondorderconvex splittingmethod,referredtoasCSB F(2),iswrittenas

φ

ⁿ⁺¹

− φ

ⁿ

Δ

t

= 



 φ

ⁿ⁺¹

 +

¹

2

(

1

+ )

²



φ

ⁿ⁺¹

+ φ

ⁿ



− 

2



3

φ

ⁿ

− φ

ⁿ⁻¹



,

(35)

whereasecant-typedifference



φ

ⁿ⁺¹

isdefinedby

 φ

ⁿ⁺¹

 =

c

 φ

ⁿ⁺¹



−

c

 φ

ⁿ

 φ

ⁿ⁺¹

− φ

ⁿ

=

¹

4



φ

ⁿ⁺¹

+ φ

ⁿ

 

φ

ⁿ⁺¹



₂

+ 

φ

ⁿ



2



(36)

forgiven

φ

ⁿ andaninitialsetting

φ

⁻¹= φ^{0.Thefirsttimestepreductiondoesnotaffecttheoverallsecondorderaccuracy} oftheschemeasthenumericalresultswillbeshown. Ifone couldliketoavoidthisreduction,referto [10]andconsider thetheirsuggestion.

Forsimpledescriptionfortheproof,(35)canbewrittenas

φ

ⁿ⁺¹

− φ

ⁿ

= Δ

^t

 μ

ⁿ⁺¹²

,

(37)

μ

ⁿ⁺¹²

₌ ^ φ

ⁿ⁺¹

 +

¹

2

(

1

+ )

²



φ

ⁿ⁺¹

+ φ

ⁿ



− 

2



3

φ

ⁿ

− φ

ⁿ⁻¹



,

(38)

where



φ

ⁿ⁺¹

isdefinedasin(36)and

φ

⁻¹= φ⁰. Theorem7.Thescheme(37)ismassconserving,i.e.,

φ

ⁿ⁺¹

,

1

L²=

φ

ⁿ

,

1

L². Proof. WeskiptheproofsinceitissimilartoTheorem 1.

Theorem8.Thescheme(37)isuniquelysolvableforanytimestep

Δ

t

>

0.

Proof. WeconsiderthefollowingfunctionalonH=

φ

| (φ,¹

)

_L2=

φ

ⁿ

,

1

L²

:

G

(φ) =

¹

2

φ − φ

ⁿ

²

H⁻¹

+ Δ

^{t Q}

(φ) + Δ

^t



1

2

(

1

+ )

²

φ

ⁿ

− 

2



3

φ

ⁿ

− φ

ⁿ⁻¹

 , φ



L²

,

(39)

where

Q

(φ) =

¹ 4

 φ

⁴ 4

+ φ

³

3

φ

ⁿ

+ φ

² 2

 φ

ⁿ



2

+ φ  φ

ⁿ



3

,

1



L²

+

¹

4

(

¹

+ ) φ

²_L2

.

(40)

Thefunctional Q isconvexbecause(¹+ ) φ²_L2 isconvexand

d² d²

φ



1 4

 φ

⁴ 4

+ φ

³

3

φ

ⁿ

+ φ

² 2

 φ

ⁿ



2

+ φ  φ

ⁿ



3



=

¹ 2

φ

²

+

¹

4

 φ + φ

ⁿ



2

≥

⁰

.

(41)

Alsonotethatbyconstruction

d d

φ



1 4

 φ

⁴ 4

+ φ

³

3

φ

ⁿ

+ φ

² 2

 φ

ⁿ



2

+ φ  φ

ⁿ



3



=

¹ 4

 φ + φ

ⁿ

  φ

²

+ 

φ

ⁿ



2



= (φ) .

⁽⁴²⁾

Now,itmaybeshownthat

φ

ⁿ⁺¹∈ H istheuniqueminimizerofG ifandonlyifitsolves,forany

ψ

∈H0, dG

ds

(φ +

^s

ψ ) 



s=0

= 

φ − φ

ⁿ

, ψ 

H⁻¹

+ Δ

^t



(φ) +

¹

2

(

1

+ )

²

φ, ψ



L²

+ Δ

^t



1

2

(

1

+ )

²

φ

ⁿ

− 

2



3

φ

ⁿ

− φ

ⁿ⁻¹

 , ψ



L²

= 

φ − φ

ⁿ

− Δ

t

 μ , ψ 

H⁻¹

=

0

,

(43)

where

μ = (φ) +

¹

2

(

1

+ )

²

 φ + φ

ⁿ



− 

2



3

φ

ⁿ

− φ

ⁿ⁻¹



,

(44)

becauseitisclearthatthefunctionalG isstrictlyconvexby

d²G

ds²

(φ +

^s

ψ ) 



s=0

=

¹

2

ψ

²_H−1

+ Δ

t 2



1 2

φ

²

+

¹

4

 φ + φ

ⁿ



2

 ψ +

¹

4

(

1

+ )

²

ψ, ψ



L²

≥

⁰

.

(45)

Therefore,(43)istrueforany

ψ

∈^{H ifandonlyifthegivenequationholds}

φ

ⁿ⁺¹

− φ

ⁿ

Δ

t

= 



 φ

ⁿ⁺¹

 +

¹

2

(

1

+ )

²



φ

ⁿ⁺¹

+ φ

ⁿ



− 

2



3

φ

ⁿ

− φ

ⁿ⁻¹



.

(46)

Hence,minimizingthestrictlyconvexfunction(39)isequivalenttosolving(35)(or(37)).

Theorem9.Supposethat

φ

ⁿ⁺¹isasolutionto(37).Thesecondorderscheme(37)isunconditionally(weakly)gradientstable,meaning thatforanytimestepsize

Δ

t

>

0 andn≥^1,

E  φ

ⁿ



≤ E  φ

⁰



.

(47)

Proof. Supposethatn≥^{1.Takingtheinnerproductof}

μ

ⁿ⁺¹² with(37)andusingintegrationbyparts,weobtain

−Δ

^t





 ∇ μ

ⁿ⁺¹²

^ _

²dx

= E (φ

ⁿ⁺¹

) − E (φ

ⁿ

)

+ 

2





 φ

ⁿ⁺¹



2

dx

− 

2





 φ

ⁿ



2

dx

− 

2





Jφ

ⁿ

K 

3

φ

ⁿ

− φ

ⁿ⁻¹



dx

.

(48)

Withtheidentity





Jφ

ⁿ

K 

3

φ

ⁿ

− φ

ⁿ⁻¹



dx

=





Jφ

ⁿ

K 

φ

ⁿ⁺¹

− φ

ⁿ



dx

−





Jφ

ⁿ

K 

φ

ⁿ⁺¹

+ φ

ⁿ



dx

+





Jφ

ⁿ

K 

φ

ⁿ

− φ

ⁿ⁻¹



dx

,

(49)

wehave

E 

φ

ⁿ⁺¹

 − E  φ

ⁿ



= −Δ

^t





 ∇ μ

ⁿ⁺¹²

^ _

²dx

− 

2





Jφ

ⁿ

K

^2dx

− 

2





Jφ

ⁿ

K r φ

ⁿ⁻¹

z

dx

.

(50)

UsingCauchy’sinequality,wehavethefollowinginequality

−





Jφ

ⁿ

K r φ

ⁿ⁻¹

z

dx

≤

¹

2





Jφ

ⁿ

K

²dx

+

¹ 2





r φ

ⁿ⁻¹

z

2

dx

.

(51)

Combining(50)and(51),wehave

E 

φ

ⁿ⁺¹

 − E  φ

ⁿ



≤ −Δ

^t





 ∇ μ

ⁿ⁺¹²

^ _

²dx

− 

4





Jφ

ⁿ

K

^2dx

+ 

4





r φ

ⁿ⁻¹

z

2

dx

.

(52)

Now,summing(52),wehave

E 

φ

ⁿ⁺¹

 − E  φ

¹

 =



n k=1

 E (φ

^k+1

) − E (φ

^k

)



≤



n k=¹

⎛

⎝−Δ

^t





 ∇ μ

^k+12

^ _

²dx

− 

4





r φ

^k

z

2

dx

+ 

4





r φ

^k−1

z

2

dx

⎞

⎠

= −Δ

t



n k=1





 ∇ μ

^k+12

^ _

²dx

− 

4





Jφ

ⁿ

K

²dx

+ 

4





r φ

⁰

z

2

dx

.

(53)

Forthecaseofn=^0,using

φ

⁻¹= φ^{0,thelasttermonthe}right-hand-sideof(50)disappearstoyield

E 

φ

¹

 − E  φ

⁰

 = −Δ

^t





 ∇ μ

¹²

^ _

²dx

− 

2





r φ

⁰

z

₂

dx

.

(54)

Adding(54)to(53)yields

E 

φ

ⁿ⁺¹

 − E  φ

⁰

 ≤ −Δ

^t



n k=⁰





 ∇ μ

^k+12

^ _

²dx

− 

4





Jφ

ⁿ

K

^2dx

− 

4





r φ

⁰

z

2

dx

≤

⁰

,

(55)

anditmeansthattheweaklygradientstabilityisproven.

Fig. 1. Time evolution of solution for the PFC equation in 1D.

Remark10.Analternativeapproach tothequestion ofenergystabilityisto introduceamodified energyfunctional ˜E (φ) as

E ˜  φ

ⁿ⁺¹

 = E  φ

ⁿ⁺¹

 + 

4





Jφ

ⁿ

K

^2dx

.

(56)

(52) showsthat thisenergy isnon-increasingfromone time step to thenext. In other words,the second order scheme (37)isstronglyenergystablewithrespectto ˜E (φ)^.Meanwhile, ˜E (φ)^{isconsistentwith}

E (φ)

^as

Δ

t tendtozero.Formore details,pleasereferto[6,10].

4. Convergenceandstabilitytestofthenumericalmethodsin1D

Inthissection,wepresentonedimensionalexamplestonumericallydemonstratetheaccuracy, stability,andefficiency of theproposed methods CSB F(1)andCSB F(2).Becauseproposed numericalmethods arenonlinear, we needa nonlinear iterative method.Wepresentthe detailednumericalimplementations forfirstandsecond order methodsin Sections4.2 and4.4,respectively.Anumericaltolerancetol isdefinedas10⁻⁹unlessmentionedotherwise.

WebeginbyshowingthesolutionevolutionofthePFCequationwiththeperiodicboundaryconditionandthefollowing initialcondition

φ (

x

,

0

) =

⁰

.

07

−

⁰

.

02 cos

 π (

x

−

12

)

16



+

⁰

.

02 cos²

 π (

x

+

10

)

32



−

⁰

.

01 cos²

 π

x 8



(57)

on a domain



= [⁰

,

32]^{. For the numerical} simulations,



=⁰

.

2 is used and the grid size is fixed to

Δ

x=¹

/

2 which providesenoughspatialaccuracy.ThenumericalsolutionisevolvedtotimeT_f=^128.

Fig. 1displaysthetimeevolutionofsolutionwithasmallenoughtimestep

Δ

t=^Tf

/

2¹⁴usingthesecondordermethod CS_{B F}(2).At the earlystage, solutions are rearranged andadjusted to findtheir desiredmode withlow magnitude. After finishingtheadjusting,thesolutiongrowsupandapproachestothesteady-statesolution.Thissolutionwillbeusedasthe referencesolutiontoestimatetheconvergencerate.

Fig. 2displays thetimeevolutionoffreeenergy

E (φ)

corresponding thesolutioninFig. 1.Thepropertyoftheenergy dissipation isvalidandonesteepenergytransitionisobserved.Whenthesolutionisarrangedwithlow magnitudeatthe earlystage,energyisslightlydeclined.Afterthatenergyissteeplydeclinedbecauseofgrowingthesolution.Finally,energy issluggishlydecreasingsincethesolutionisnearatthesteadystate.

Fig. 2. Energy evolution for the PFC equation in 1D.

Fig. 3. Relative l2-errors at t=48 with respect to the grid size for various time steps in 1D.

4.1. SpatialaccuracytestofCS_{B F}(1)andCS_{B F}(2)

Since we assume the periodic boundary condition andfocus only on thetime marching method,we use the Fourier spectralmethodforthespatial discretization.And,thefastFouriertransformwiththeMATLABprogramisappliedforthe wholenumericalsimulations.

Wedemonstratethenumericalconvergenceinspacewiththesameconditionsandparametersinthebeginningofthis section.Toestimatetheconvergenceratewithrespecttoagridsize,simulationsareperformedbyvaryingthegridpoints 16

,

24

,

32

, . . . ,

128.Furthermore,wecalculatethemwithvarioustimesteps.

Fig. 3showstherelativel2-errorswithrespecttovariousgridsize.Here,theerrorsarecomputedbycomparisonwith thereferencenumericalsolutionobtainedusingthesecondordermethodCS_{B F}(2)withatimestep

Δ

t=^Tf

/

2¹⁴and256 grid points.As can be seen,the spatial convergenceofthe resultsunderthe gridrefinements isevident. Furthermore,it showsthat64 gridpoints(

Δ

x=¹

/

2)providethesufficientspatialaccuracy.

4.2. Numericalimplementationofthefirstordermethod,CS_{B F}(1)

Forcompletedescriptionofthenumericalsolver,letusrecallthefirstordernumericalmethodCSB F(1)

φ

ⁿ⁺¹

− φ

ⁿ

Δ

t

=  

φ

ⁿ⁺¹



3

+ (

¹

+ )

²

φ

ⁿ⁺¹

−  φ

ⁿ



.

(58)

Since(58)isnonlinear,weintroducetheNewton-typeiterativemethod,whichisawell-knownnonlinearsolver.Usingthe linearizationofthenonlinearterm

 φ

^n,m+1



3

=

3

 φ

^n,m



2

φ

^n,m+1

−

2

 φ

^n,m



3

,

(59)

wedevelopaNewton-typeiterativemethodas



I

− Δ

^t

 

3

φ

^n,m



2

+ (

¹

+ )

²



φ

^n,m+1

= φ

ⁿ

−  Δ

t

φ

ⁿ

−

²

Δ

t

  φ

^n,m



3

.

(60)

Fig. 4. Number of nonlinear and BICG iterations for the first order method.

The initial guessissetas

φ

^n,0= φ^{n andthenonlineariteration (60)is}recursively applieduntil arelative l2-norm ofthe consecutiveerrorof

φ

^n,m+1islessthantolerancetol,i.e., ^φn,m+1−φ

n,m ₂

φ^n,m2

<

tol.Thenlet

φ

ⁿ⁺¹be

φ

^n,m+1.

Inaddition,weneedtosolvethelinearsystem(60)foreachm-step.Since(60)isnotsymmetry,weusethebi-conjugate gradient(BICG)methodinthisstudy.ThestoppingcriterionfortheBICGiterationisthatarelativel2-normoftheresidual errorislessthanatol.Moreover,toacceleratetheconvergencespeedoftheBICGalgorithm,wesuggestapre-conditioner

M

=

^I

− Δ

^t





A I

¯ + (

¹

+ )

²



,

(61)

where A is¯ theaveragevalueof3

φ

^n,m2

.

We note that the above description for the numerical implementation is not specified in one dimensional case. We also remarkthatspectral methodsare usedinall ofnumericalcomputations.Simulationsare executedby usingMATLAB programwithafastFouriertransform.Evenintwoandthreedimensions,thecomputationalspeedisreasonablyfast.

Toshowtherobustnessofnonlinearsolverandthenecessityofpre-conditioner,wecountthenumberofbothiterations anddisplaytheaveragenumberwithrespecttothevarioustimestep

Δ

t=^Tf

/

2¹²

,

Tf

/

2¹¹

, . . . ,

Tf

/

2³.Theinitialstateand otherparametersareidenticaltothoseinthebeginningofthissectionexceptthetolerancetol=^{10−8.Therelativelysmall} toleranceisbecauseoftheconvergenceissueofBICGwhenweusethelargetimestepwithoutthepre-conditioner.

Thenumberofnonlineariterationsaveragedoverthesimulationtime0

<

t=ⁿ

Δ

t≤^Tf isshownasafunctionoftime step

Δ

t in Fig. 4 (a).On average,2–4iterations were involved inproceedingto thenext time step.We canexplain that suchafastiterativeconvergencecanbeachievedsincethesuccessiveiterationsareNewton-typeiterativemethods.Onthe other hand,thenumber ofBICGiterations averaged over thesimulation time isshownasa function oftime step

Δ

t in Fig. 4(b).Inthiscase,weregard thenumberofBICGiterationofonetime stepastheaveragednumberofBICGiteration inthenonlineariterations foronetime step.As showninthefigure,theBICGiterations areremarkablyreducedwiththe pre-conditioner.

4.3. ComparisonbetweenCSB F(1)andCSD F(1)

Now,wecomparetheproposedfirstorderschemeCS_{B F}(1)tootherwell-knownconvexsplittingmethodCS_{D F}(1)using the numericalconvergence.We demonstratethenumericalconvergence withthesameconditions andparameters in the beginning ofthissection.Toestimate theconvergenceratewithrespect toatime step

Δ

t, simulationsareperformedby varyingthetimestep

Δ

t=^Tf

/

2¹²

,

Tf

/

2¹¹

, . . . ,

Tf

/

2³.

Fig. 5showstherelativel₂-errorswithrespecttovarioustimesteps.Here,theerrorsarecomputedbycomparisonwith aquadruplyover-resolvedreferencenumericalsolutionobtainedusingthesecondordermethodCS_{B F}(2).Itisobservedthat bothschemesgivedesiredfirstorderaccuracyintime.

Fig. 6 shows the time evolutions of the free energy functional with

Δ

t=¹

/

2, 1

/

2³, and 1

/

2⁵. Using the proposed method, CSB F(1), the free energy functional at the same time t is almost similar for the different time steps, whereas significantdifferenceemergeswithlargertimestepsforCS_{D F}(1).

4.4. Numericalimplementationofthesecondordermethod,CSB F(2)

Numericalsolvers forCS_{B F}(2)aresimilartothoseforCS_{B F}(1)inSection4.2,sothat wesimplydescribethederivation oftheiterativemethodforthesecondordermethodCSB F(2)

φ

ⁿ⁺¹

− φ

ⁿ

Δ

t

= 



 φ

ⁿ⁺¹

 +

¹

2

(

1

+ )

²



φ

ⁿ⁺¹

+ φ

ⁿ



− 

2



3

φ

ⁿ

− φ

ⁿ⁻¹



,

(62)