Unconditionally stable methods for gradient flow using Convex Splitting Runge–Kutta scheme

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Journal of Computational Physics

www.elsevier.com/locate/jcp

Unconditionally stable methods for gradient flow using Convex Splitting Runge–Kutta scheme

Jaemin Shin

^a

, Hyun Geun Lee

^b

, June-Yub Lee

^c

^,∗

aInstituteofMathematicalSciences,EwhaWomansUniversity,Seoul 03760,RepublicofKorea bDepartmentofMathematics,KwangwoonUniversity,Seoul 01897,RepublicofKorea cDepartmentofMathematics,EwhaWomansUniversity,Seoul 03760,RepublicofKorea

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

Articlehistory:

Received3November2016

Receivedinrevisedform30June2017 Accepted3July2017

Availableonline10July2017

Keywords:

Gradientflow Convexsplitting Gradientstability Energystability Phase-fieldmodel Cahn–Hilliardequation

We propose a Convex Splitting Runge–Kutta (CSRK) scheme which provides a simple unifiedframeworktosolveagradientflowinanunconditionallygradientstablemanner.

The key feature of the scheme is a combination of a convex splitting method and a specially designedmulti-stagetwo-additiveRunge–Kuttamethod. Ourmethodsare high orderaccurateintimeand assurethegradient (energy)stability foranytimestepsize.

We providedetailed proof of the unconditional energy stability and present issues on thepracticalimplementations.Wedemonstratetheaccuracyandstabilityoftheproposed methodsusingnumericalexperimentsoftheCahn–Hilliardequation.

©^{2017ElsevierInc.Allrightsreserved.}

1. Introduction

Gradientsystemshavebeenfundamentalinthedevelopmentofmanyimportantconceptsindynamicalsystems[1].In particular, gradientsystemsare importantin manyphenomenologicalmodels ofphasetransitionsuch asthe Allen–Cahn andCahn–Hilliardequations[2,3].Weconcentrateonanumericalmethodforsolvingtheinitialvalueproblem,whichcan berepresentedasthegradientflowforF

: R

^N

→ R

^,

∂φ

∂

t

= −

grad F

(φ) ,

(1)

wheregrad F

(φ)

isagradient of F ,

φ (

t

) ∈ R

^{N,andN isthe dimensionofdataspace. It isworth notingthat theenergy} function F

(φ)

isnon-increasingintimesince(1)isofgradienttype.

d dtF

(φ) =



grad F

(φ) , ∂φ

∂

t



= − 

^{grad F}

(φ)

²

≤

⁰

,

(2)

where

·, ·

^{is an inner product on}

R

^{N withthe} corresponding norm

φ

²

= φ, φ

^{. Here, the gradient operator “grad”}

dependson the giveninner product space. Forexample,Allen–Cahn and Cahn–Hilliard equationscan be represented by gradientflowswiththeGinzburg–LandaufreeenergyfunctionalundertheL² andH⁻¹ innerproducts,respectively.

*

Correspondingauthor.

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

http://dx.doi.org/10.1016/j.jcp.2017.07.006 0021-9991/©^{2017ElsevierInc.Allrightsreserved.}

Denoting

φ

ⁿ asa numericalapproximationof

φ 

tⁿ



,weaimtodevelop anumericalmethodforonestepevolution to obtain

φ

ⁿ⁺¹attⁿ

+ Δ

^t.Anumericalintegrationmethodfor(1)issaidtobegradientstable[1,4]ifacriticaltimestepsize

Δ

tc andafunction

F : R

^N

→ R

^{existsuchthat:}

F (φ) ≥

^{0 for all}

φ ∈ R

^N

,

(3)

F (φ) → ∞

^as

φ → ∞,

⁽⁴⁾

F  φ

ⁿ⁺¹

 ≤ F  φ

ⁿ



for

φ

ⁿ

∈ R

^N

,

(5)

forall0

< Δ

t

≤ Δ

^tc.Weareparticularlyinterestedinunconditionalstabilitywhichmeansthatamethodisstablenomatter whatatimestepistaken.A methodissaidtobeenergystableifitsatisfiestheenergydissipationproperty(5)and

F

^may ormaynotbeidenticaltotheoriginalenergyfunction F .

Itisnumericallyveryimportanttousean unconditionallygradient(energy)stablemethodandthishasbeenaserious research objectiveinthefields.Fora convexF , findingan unconditionallygradientstablemethodisrelatively easysince (1) isacontractive problemandthe setofequilibriadefinesaconvexset.However, manyproblemswithanonconvex F canhavemultipleisolatedequilibria[4],anddevelopinganunconditionallygradient(energy)stablemethodisachallenging task.

TosolvethegradientflowwiththenonconvexF ,aconvexsplitting(CS)methodwasdevelopedbyElliottandStuart[5]

and revitalized by the work of Eyre [6]. An energy function F

(φ)

is split into contractive and expansiveparts, F

(φ) =

Fc

(φ) −

^Fe

(φ)

,whereboth Fc

(φ)

and Fe

(φ)

are convex.Bytreating Fc

(φ)

implicitlyand Fe

(φ)

explicitly,thefirstorder CSmethodisobtainedby

φ

ⁿ⁺¹

− φ

ⁿ

Δ

t

= −

^{grad F}c

 φ

ⁿ⁺¹

 +

^{grad F}e

 φ

ⁿ



.

(6)

It iswellknownthattheenergyofthenumericalsolutionof(6)monotonicallydecreasesregardlessofthetime stepsize

Δ

t

>

0.Thus,(6)issaidtobeunconditionallygradient(energy)stable.

Ingeneral,manyphasefieldequationscanbemodeledbygradientflowswithaparticularenergyfunctional.Duetothe nonlinearityoftheenergyfunctionalaswellasthehighorderdifferentialterms,aseverestability restrictionexistsonthe timestepwhenapplyingtypicalnumericalmethodstothephasefieldequations.Thefirstorderandunconditionallygradi- entstableCSmethod,proposedforsolvingtheCahn–Hilliardequation [6],hasaffectedmanytypesofnumericalmethods for phase field equations.Semi-implicit methodssuch as theCS methodimprovethe stability,however all semi-implicit methodsdonotguaranteetheunconditionalgradientstabilityasreferredin[7].

While manyresearchers[8–11]haveproposed higherorderaccuratemethods,their methodsdonot guaranteeuncon- ditionalenergystability.Thisisnot areview paperbutwe wouldliketomentionsome ofnoteworthyresearchesrelated tosecondorderenergystablemethods.Recently,anumberofsignificantsecondorderconvexsplittingmethodshavebeen proposed consideringtheenergystability forawideclassofphase fieldmodel,suchasCahn–Hilliard[12,13],phase-field crystal[14–16],modifiedphase-fieldcrystal[17],andepitaxialthinfilmgrowth[18,19]equations.Furthermore,theexten- siveworksshowthatCSisavailabletothephasefieldmodelsforthetumorgrowth[12]andpolymerblend[20].Weshould note thattherehavebeenanotherdirectionofthesecondorderenergystablemethods forvariousphase fieldmodel[21, 22],whichisnotextensionoftheCSidea.

Recently, we proposed a Convex Splitting Runge–Kutta (CSRK) scheme in [23], which combines the convex splitting strategy with theimplicit–explicit (IMEX) Runge–Kutta(RK) method[24,25]. Since IMEX-RK methodshave manychoices for thecoefficientsof theRK tables, we were motivatedto develop ahighorder accurate methodwithsufficient energy stability.Wenumericallydemonstratedtheimprovementoftheenergystabilityforsolvingthephasefieldequationsusing various typesofCSRKmethods.Whilenumericalresultsindicatethenotable improvementoftheenergystability,we did notdisclosethefundamentalreasonswhythestability wasimproved.Nonetheless,thepreviousstudyempiricallysuggests thelikelyexistenceofhighordermethodswiththeunconditionalenergystability.

The objectofthiscurrentpaperistoshow theexistenceofhigherorderunconditionallyenergystableCSRKmethods.

In Section2,we propose atwo-additiveRKmethodcombined witha convexsplittingstrategyanda resemblecondition, referred to asthe CSRK-Rmethod. InSection 3,we provide asufficient condition to ensurethe proposed methodisun- conditionally energystablewithdetailedproof.InSection 4,forthe practicalimplementations,we considerthe IMEX-RK methodswhileguaranteeingtheuniquesolvability.Wealsopresentanumberofnumericalmethodsthatachievethehigher order ofaccuracy intime.InSection 5,weapply theproposed methodstotheCahn–Hilliard equation,whichisa typical gradientflowinthephasefieldmodel.Finally,conclusionsaredrawninSection6.IntheAppendix A,toensurethisarticle isself-contained,webrieflyintroducetheorderconditionsforthetwo-additiveRKmethods.

2. ConvexSplittingRunge–Kuttamethodswitharesemblecondition(CSRK-R)

First,webrieflyreviewatwo-additiveRKmethod.Inmanyapplications,initialvalueproblemscanbegivenasadditively partitionedsystems,

∂φ

∂

t

=

f

(φ) +

g

(φ)

and

φ (

0

) = φ

⁰

,

(7)

wheretheright-handsideissplitintotwodifferentpartswithrespecttostiffness,nonlinearity,dynamicalbehavior,evalu- ationcost,etc.Wereferto[25,26]forinformationonthecaseofmorethantwosplittingsystems.Anadditiveschemefor (7)appliestwodifferentRKmethodsforeach f

(φ)

andg

(φ)

.Wedenotetwos-stageRKtablesintheButchernotation

c A

b^T

=

0 0 0 0

· · ·

⁰

c₁ a₁₀ a₁₁ a₁₂

· · ·

^a1s

c₂ a₂₀ a₂₁ a₂₂

· · ·

^a2s

.. . .. . .. . .. . . . . .. .

cs as0 as1 as2

· · ·

^ass

b₀ b₁ b₂

· · ·

^bs

,

^c

ˆ

A

ˆ

b

ˆ

^T

=

0 0 0 0

· · ·

⁰

ˆ

c₁ a

ˆ

₁₀ a

ˆ

₁₁ a

ˆ

₁₂

· · · ˆ

^a1s

ˆ

c₂ ^a

ˆ

20 ^a

ˆ

21 ^a

ˆ

22

· · · ˆ

^a2s

.. . .. . .. . .. . . . . .. . ˆ

cs a

ˆ

s0 a

ˆ

s1 a

ˆ

s2

· · · ˆ

^ass

b

ˆ

_{0 b}

ˆ

_{1 b}

ˆ

₂

· · ·

_b

ˆ

_s

.

(8)

where A

,

A

ˆ ∈ R

^{(s+1)×(s+1),b}

,

_b

ˆ ∈ R

^{s+1, andthecoefficients c and}

ˆ

c areusually givenby c

=

^{A1 and} c

ˆ = ˆ

^A1,respectively, with1

= (

¹

,

1

, . . . ,

1

)

^T

∈ R

^s+1.

WiththeButchertables(8),thetimemarchingalgorithmcanbewrittenasfollows:Wedenote

φ

ⁿ asanapproximation of

φ 

tⁿ



. Witha time stepsize

Δ

t, one step evolutionfromtⁿ to tⁿ⁺¹

=

^tn

+ Δ

^{t ofthe additive RK methodisgiven as} follows.Given

φ

ⁿ,wewillcalculatethenexttimeapproximation

φ

ⁿ⁺¹ withtheintermediatestages,

φ

0

, φ

1

, · · · , φ

s.Weset azero-stageask₀

=

^f

(φ

0

)

andk

ˆ

₀

=

^g

(φ

0

)

,where

φ

0

= φ

^{n.Foreachstagei}

=

¹

,

2

, . . . ,

s,wecalculate

φ

_i

= φ

0

+ Δ

^t

⎛

⎝

^s

j=⁰ a_{i j}k_j

+

s

j=⁰

ˆ

a_{i jk}

ˆ

_j

⎞

⎠ ,

(9)

whereki

=

^f

(φ

i

)

andk

ˆ

i

=

^g

(φ

i

)

.Here,

φ

i istheapproximation of

φ 

tⁿ

+

^ci

Δ

t



onthei-thstage.Inaddition,weevaluate thenexttimeapproximation

φ

ⁿ⁺¹ as

φ

ⁿ⁺¹

= φ

0

+ Δ

^t

s

j=⁰



b_jk_j

+ ˆ

^bjk

ˆ

_j



.

(10)

IntheAppendix A,wedescribetheorderconditionsofthetwo-additiveRKmethoduptothirdorderaccuracy.

WenowintroduceanewclassofCSRKmethodwiththeadditionaldesigncriterioncalledtheresemblecondition, which playsakeyroleinthedevelopmentofenergystablemethods.Westartbyconsideringtwoconditionsasfollows.

Condition1(Resemblecondition).FortheButchernotationsin(8),A andA are

ˆ

saidtobesatisfyingtheresembleconditionifA

ˆ =

^AS, whereS_{i j}

= δ

i,j+¹isashiftmatrixand

δ

i jistheKroneckerdeltasymbol.

Condition2(Stifflyaccuratecondition).FortheButchernotationsin(8),A,b andA,

ˆ

b are

ˆ

saidtobesatisfyingthestifflyaccurate conditionifbj

=

^asjand_b

ˆ

_j

= ˆ

^asjforj

=

⁰

,

1

, . . . ,

s.

Tobesatisfiedthecondition1,A andA should

ˆ

beconstructedas A

=

0 0^T 0 R

 ,

A

ˆ =

0^T 0 R 0



,

(11)

wherewedenoteasaresemblebasematrixR,

R

=

⎛

⎜ ⎝

r11

· · ·

^r1s

.. . . . . .. .

rs1

· · ·

^rss

⎞

⎟ ⎠ .

⁽¹²⁾

Therefore, the condition 1 automatically impliesthat the (canopy node)condition c

= ˆ

^{c. In addition, the condition 2 is} motivatedbecause it isknown that an A-stable RK methodis L-stable ifb_j

=

^asj [27].With theconditions 1 and 2,we considerthes-stageresembleRKtableonlywitharesemblebasematrixR

c A b^T

=

0 0 0 0

· · ·

⁰

c1 0 r11 r12

· · ·

r1s

c₂ 0 r₂₁ r₂₂

· · ·

^r2s

.. . .. . .. . .. . . . . .. .

c_s 0 r_s1 r_s2

· · ·

^rss

0 rs1 rs2

· · ·

^rss

,

c

ˆ

A

ˆ

b

ˆ

^T

=

0 0 0 0

· · ·

⁰

c1 r11 r12 r13

· · ·

0 c₂ r₂₁ r₂₂ r₂₃

· · ·

⁰

.. . .. . .. . .. . . . . .. .

c_s r_s1 r_s2

· · ·

^rss 0 rs1 rs2

· · ·

^rss 0

.

(13)

Using the conditions 1 and 2, we can describe the resemble RK method (13) only using a resemble base matrix R.

Furthermore,bycondition2,thetimeevaluation(10)isreducedto

φ

ⁿ⁺¹

= φ

s.

We nowproposethe s-stageCSRKmethodswiththeresemble condition(CSRK-R). First,werecalltheconvexsplitting (CS)method(6)for

∂φ

∂

t

= −

grad

(

Fc

(φ) −

Fe

(φ))

(14)

canbewrittenas

φ

1

= φ

0

− Δ

^{t grad}

(

F_c

(φ

1

) −

^Fe

(φ

0

)) ,

(15)

where

φ

0

= φ

^nand

φ

1

= φ

^{n+1.Wecanrepresent(15)astheframeworkoftheresembleRKmethodwithatrivialresemble} basematrixR

= (

¹

)

bydefining f

(φ) = −

^{grad F}c

(φ)

andg

(φ) =

^{grad F}e

(φ)

.

We now define the p-thorder s-stageCSRK-R

(

p

)

methods by combiningthe CS scheme andthe p-th order accurate resemble RKmethod(13).Letusconsidera generalresemblebasematrixR with s-stages.Toevaluate onetimestep,we set

φ

0

= φ

^nandestimate

φ

i

= φ

0

− Δ

^t

s

j=1

ri jgrad



Fc

 φ

j

 −

^Fe

 φ

j−¹



,

(16)

fori

=

¹

,

2

, . . . ,

s andwehave

φ

ⁿ⁺¹

= φ

s.

3. UnconditionalenergystabilityrequirementofCSRK-R

WegiveasinglelemmabeforewepresentaconditiontomaketheproposedCSRK-Rscheme(16)unconditionallyenergy stable.

Lemma1.Supposethat

φ

and

ψ

aresufficientlyregularandF

(φ)

istwotimescontinuouslydifferentiable.Considerthecanonical convexsplittingofF

(φ)

intoF

(φ) =

^Fc

(φ) −

^Fe

(φ)

;i.e.bothF_c

(φ)

andF_e

(φ)

areconvex,then

F

(φ) −

^F

(ψ ) ≤ 

^gradFc

(φ) −

^{grad F}e

(ψ ) , φ − ψ .

⁽¹⁷⁾

Proof. The analogousproof could befoundin [14],howeverwe introduceitforaself-containeddescription.Since F_c

(φ)

and Fe

(φ)

areconvex,

F_c

(φ) −

^Fc

(ψ ) ≤ 

^{grad F}c

(φ) , φ − ψ ,

⁽¹⁸⁾

F_e

(φ) −

^Fe

(ψ ) ≥ 

^{grad F}e

(ψ ) , φ − ψ .

⁽¹⁹⁾

Usingtwoinequalities,

F

(φ) −

^F

(ψ ) = (

^Fc

(φ) −

^Fe

(φ)) − (

^Fc

(ψ ) −

^Fe

(ψ ))

≤ 

^{grad F}c

(φ) , φ − ψ − 

^{grad F}e

(ψ ) , φ − ψ

= 

^{grad F}c

(φ) −

^{grad F}e

(ψ ) , φ − ψ . 2

(20)

Condition3(Positivedefinitecondition).TheCSRK-Rmethoddefinedin(16)issaidtobesatisfyingthepositivedefiniteconditionif arowdifferencematrix



R ispositivedefiniteafterasymmetrictransformation,where



R

=

^R

−

^{SR isdefinedasfollowswithashift} matrixSi j

= δ

i,j+¹,



R

=

⎛

⎜ ⎜

⎜ ⎝

˜

r11 r

˜

12

· · · ˜

^r1s

˜

r21 r

˜

22

· · · ˜

r2s

.. . .. . . . . .. .

˜

r_s1 ^r

˜

s2

· · · ˜

^rss

⎞

⎟ ⎟

⎟ ⎠ =

⎛

⎜ ⎜

⎜ ⎝

r11 r12

· · ·

^r1s

r21 r22

· · ·

r2s

.. . .. . . . . .. .

r_s1 r_s2

· · ·

^rss

⎞

⎟ ⎟

⎟ ⎠ −

⎛

⎜ ⎜

⎜ ⎝

0 0

· · ·

⁰

r11 r12

· · ·

r1s

.. . .. . . . . .. .

r_s−1,1 r_s−1,2

· · ·

^rs−¹,s

⎞

⎟ ⎟

⎟ ⎠ .

(21)

Theorem2(Unconditionalenergystability).Supposethattherowdifferencematrix



R ispositivedefinite.TheproposedCSRK-Rscheme (16)isthenunconditionallyenergystable,meaningthatforanytimestepsize

Δ

t

>

0,

F

 φ

ⁿ⁺¹

 ≤

^F

 φ

ⁿ



.

(22)

Proof. For simple description, we define the difference operator by

JφK

ⁿm

= φ

n

− φ

m and the auxiliary variable by G_j

=

Fc

 φ

j

 −

^Fe

 φ

j−¹



.Then,wecanrewrite(16)as

JφK

ⁱ0

= −Δ

^t

s

j=1

ri jgrad Gj

,

(23)

fori

=

¹

,

2

, . . . ,

s.Thedifferencebetweenthetwoadjacentstagescanbecalculatedas

JφK

ⁱ_i−1

= JφK

ⁱ₀

− JφK

ⁱ₀⁻¹

= −Δ

t

s

j=1

˜

ri jgrad Gj

,

(24)

foralli.ByLemma 1,theenergydifferencebetweentwoadjacentstagesis

J

^F

(φ)K

ⁱ_i−1

≤ 

grad F_c

(φ

i

) −

^{grad F}e

(φ

i−1

) ,JφK

ⁱ_i−1



= 

grad G_i

, JφK

ⁱ_i−1



.

(25)

Aftersummingup,wehave

J

^F

(φ) K

^s0

=

s

i=1

J

^F

(φ) K

ⁱi−¹

≤

s

i=1



grad G_i

, JφK

ⁱi−¹

 = −Δ

^t



grad G

,

R grad G



s

,

(26)

where grad G

= (

^{grad G}1

, · · · ,

^{grad G}s

)

^T and we define an s-dimensional inner product as

φ, ψ

s

=

s

i=¹

φ

i

, ψ

i



^for φ

= (φ

1

, · · · , φ

s

)

^T and ψ

= (ψ

1

, · · · , ψ

s

)

^T. Since



R is positive definite,

J

^F

(φ)K

^s₀

≤

^{0. It follows that the energy} dissipation F



φ

ⁿ⁺¹



=

^F

(φ

s

) ≤

^F

(φ

0

) =

^F

 φ

ⁿ



.

2

Finally,wecoulddesignanenergystablemethodfortheCSRK-Rmethodsusingonlythepositivedefinitenesscondition of



R.ThethreesufficientconditionsplayanimportantroleintheCSRKmethodinbeingunconditionallyenergystable.Note thatthisdevelopmentprovidesthepossibilityofexistingconditionsordesignstoguaranteetheenergystability.

4. ImplementationsofunconditionallyenergystableCSRK-Rmethod

Asimple choice forCSRK-R is toconsider the type ofimplicit–explicit Runge–Kutta(IMEX-RK) method whichapplies animplicitdiscretizationtoone termandan explicitdiscretizationtotheotherterm.Amethodwithafullmatrixin(13) requiresthe solutionof s simultaneousimplicit (ingeneral nonlinear) equationsper step.Tocircumvent thisdifficulty,a lowertriangularmatrixcouldbe suggested.Thenwe justneedtosolvea singleimplicitequationforeachstage,whichis motivatedby “diagonallyimplicit”RK method[28].Thismeans that,in CSRK-R,we selectalower triangulartype forthe resemblebasematrix

R

=

⎛

⎜ ⎜

⎜ ⎝

r11 0

· · ·

0

r₂₁ r₂₂

· · ·

⁰

.. . .. . . . . .. .

r_s1 r_s2

· · ·

^rss

⎞

⎟ ⎟

⎟ ⎠ .

(27)

Withtheresemblebasematrix(27),theresembleRKmethod(13)isthenreducedasfollows:

c A

b^T

=

0 0 0 0

· · ·

⁰

c₁ 0 r₁₁ 0

· · ·

⁰

c₂ 0 r₂₁ r₂₂

· · ·

⁰

.. . .. . .. . .. . . . . .. .

c_s 0 r_s1 r_s2

· · ·

^rss

0 rs1 rs2

· · ·

^rss

, ˆ

c A

ˆ

b

ˆ

^T

=

0 0 0 0

· · ·

⁰

c₁ r₁₁ 0 0

· · ·

⁰

c₂ r₂₁ r₂₂ 0

· · ·

⁰

.. . .. . .. . .. . . . . .. .

c_s r_s1 r_s2

· · ·

^rss 0 rs1 rs2

· · ·

^rss 0

.

(28)

As(16)comesfrom(13),wecanderivethefollowingschemefrom(28).First,wesetazero-stageask0

= −

^{grad F}c

(φ

0

)

andk

ˆ

0

=

^{grad F}e

(φ

0

)

,where

φ

0

= φ

^{n.Foreachstagei}

=

¹

,

2

, . . . ,

s,wecalculate

Table 1

Orderconditionsoftwo-additiveRKmethodsuptothethirdorder accuracy.

Order Stand-alone conditions Coupling conditions

1 b·1=1 bˆ·1=1 –

2 b·^c=¹/2 bˆ·^c=¹/2 –

3 b·^c2=¹/3 bˆ·^c2=¹/3 b· ˆ^Ac+ ˆ^b·^Ac=¹/3 b·^Ac=¹/6 bˆ· ˆ^Ac=¹/6

φ

i

= φ

0

+ Δ

t

⎛

⎝

ⁱ

j=1

ri jkj

+

i−¹

j=0

ri,j+¹_k

ˆ

_j

⎞

⎠ ,

(29)

whereki

= −

^{grad F}c

(φ

i

)

and_k

ˆ

_i

=

^{grad F}e

(φ

i

)

.Finally,weevaluatethenexttimeapproximationas

φ

ⁿ⁺¹

= φ

s.

Theorem3(Uniquesolvability).TheproposedCSRK-Rscheme(29)isuniquelysolvableforanytimestepsize

Δ

t

>

0,providedthat r_ii

≥

^{0 foralli.}

Proof. Foreachstagei

=

¹

,

2

, · · · ,

^{s of(29),weneedtosolve}

φ

i

+

rii

Δ

t grad Fc

(φ

i

) =

Si

,

(30)

where

Si

= φ

0

− Δ

t

⎛

⎝

ⁱ⁻¹

j=1

ri jgrad Fc

 φ

j

 −

i−¹

j=0

ri,j+¹grad Fe

 φ

j

 ⎞

⎠ .

(31)

Since Fc isconvex,thefunction Q

(φ) =

¹

2

φ

²

+

^rii

Δ

t F_c

(φ) − φ,

^Si



⁽³²⁾

hasauniqueminimizationforanytime stepsize

Δ

t

>

0.Attheuniqueminimizerforeachstage,grad Q

(φ) =

^{0 and(30)} hasauniquesolution

φ

i.Therefore,theproposedscheme(29)isuniquelysolvableforanytimestepsize

Δ

t

>

0.

2

Remark1.Thes-stageCSRK-Rdefinedin(29)requiresthes inversionspersteplike(30),whileoneinversionisneededto solvesomeofexistingsecondorderconvexsplittingmethods.

WenowintroducesomeexamplesfortheresemblebasematrixR sothat



R ispositivedefinite.Ofcourse,A andA made

ˆ

fromagivenresemblebasematrixR shouldsatisfytheorderconditions.Wehavec and^{c by}

ˆ

^{theusualrelationsc}

=

^{A1 and}

ˆ

c

= ˆ

^A1,respectively.Bycondition 2,wereconstructb and_{b from}

ˆ

_{A and}A,

ˆ

respectively.Condition1automaticallyimplies thatc

= ˆ

^{c,anditreducesthemassiveorderconditionslistedintheAppendix A(Table 2);thereducedconditionsarelisted} inTable 1.Obviously,theconditions1 and2implythatthetwo orderconditionsb

·

¹

=

^{1 and}_b

ˆ ·

¹

=

^{1 areidentical.We} have 1 condition forthe first orderaccuracy, 3 conditionsforthe second order accuracy, and8 conditions forthe third orderaccuracy.

Inaddition,forconvenientdesignofthetables,we considerasinglydiagonalmatrix,i.e.rii

= γ

,fori

=

¹

,

2

, . . . ,

s.For thefirstorderaccuracy,aresemblebasematrixR is

R

= 

1



.

(33)

Itiseasytoshowthat(33)constructsawell-knownCSmethod.Forthesecondorderaccuracy,CSRK-R(2),apossiblechoice ofresemblebasematrixR is

R

=

⎛

⎜ ⎝

2

3 0 0

−

₁₂^{7 2}₃ ⁰

−

¹₃ ²₃ ²₃

⎞

⎟ ⎠ .

⁽³⁴⁾

The positive definiteness of



R iseasily shown by evaluating theall eigenvalues of



R

+

^RT



/

2 are positive.For (34), the eigenvaluesof



R

+

^RT



/

2 are0

.

0293,0

.

6667,and1

.

3040.Forthethirdorderaccuracy,apossiblechoiceofresemblebase matrixR is

R

=

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎝

1

2 0 0 0 0 0

1 2

1

2 0 0 0 0

−

₁₀¹ ₁₀^{1 1}₂ ^{0 0 0}

a41 a42 a43 1

2 0 0

a₅₁ a₅₂ a_{53 20}^{1 1}₂ 0 a₆₁ a₆₂ a₆₃ a₆₄ a₆₅ ¹₂

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎠

,

(35)

where

a₄₁

=

^13252051_50981620

,

a₄₂

= −

^100507933_407852960

,

a₄₃

=

^19290953_81570592

,

a₅₁

=

_5098162000^401851541

,

a₅₂

= −

_637270250^20327867

,

a₅₃

= −

_1019632400^200790581

,

a₆₁

=

₁₄₃₀₀³²¹⁷

,

a₆₂

= −

₇₁₅₀⁷⁰³

,

a₆₃

= −

₄₂₉₀₀⁶³⁵⁹

,

a₆₄

= −

₁₀₇₂₅⁴⁵⁵⁶

,

a₆₅

=

⁴⁰⁶₄₂₉

.

For(35),theeigenvaluesof



R

+

^RT



/

2 are0

.

0063,0

.

1105,0

.

3582,0

.

5722,0

.

9225,and1

.

0303.Therefore,wecansaythat ourmethodsmadeby (34)and(35)satisfy thecondition 3anditiseasy tocheck thatthecoefficientsinduced bythose matricessatisfytheorderconditionsinTable 1.

Remark2.The examples(34) and(35) showthat thedesiredmethods doindeedexist,which aretheIMEX-RK methods withsome special conditions.Forinterested readers onthe construction oftables, we note that no 2-stage resemble RK method exists for the second order accuracy. Furthermore, the finding of the resemble RK methods for the third order accuracywithlessthan6stagesisopen.

Remark3.Weonlyconsideruptothethirdorder,becausetheobjectofthispaperistoshowtheexistenceoftheuncon- ditionallygradientstablemethodswiththesufficienthighorderaccuracyintime.Findinghigherordermethodsappearsto berathertediousandcouldbesomewhatdifficultwithoutsometypeofsimplifyingassumptions.

5. NumericalexampleswithCahn–Hilliardequation

Inthissection,we applythe proposedCSRK-R methodtotheCahn–Hilliard(CH) equationtonumericallydemonstrate thatthe methodachieveshighorderaccuracyandenergystability.Becauseofthehighderivative termandthenonlinear term, whichintroduce asevere time steprestriction forstability,theCH equation isa goodexample todemonstratethe applicabilityoftheproposedmethods.Tothebestofourknowledge,nothirdordermethodguaranteeingtheunconditional gradientstabilityhasbeenproposedtodate.

Fortheorderparameter

φ

,theGinzburg–Landaufreeenergyfunctionalisgivenby

E (φ) =





(φ) +

²

2

|∇φ|

²



dx

,

(36)

where



is adomain in

R

^d

(

d

=

¹

,

2

,

3

)

,

(φ)

isabulk free energywithtwo minimacorresponding tothetwo phases, and

>

0 isaparameterrelatedtotheinterfacialthickness.Forsimplicity,weconsiderthepolynomialfreeenergy

(φ) =

1 4

 φ

²

−

¹



2

.Wenotethattheenergyfunctional(3)satisfiestheconditions(3)and(4),sothatcombiningwiththeCSRK-R methodguaranteeingenergystability(5)providesagradientstablemethod.

We considerthe Hilbertspace H0 asa zeroaverage space. Forgiven v1

,

v2

∈

^H0, wedefine the inner productofthe dualspaceby

(

v1

,

v2

)

_H−1

= 

∇ ϕ

v1

, ∇ ϕ

v2



L²,where

ϕ

v1

, ϕ

v2

∈

^H0arethesolutionsofthehomogeneousNeumannboundary valueproblems

− ϕ

v1

=

^v1and

− ϕ

v2

=

^v2 in



.Fromtheabovedefinition,if

ψ ∈

^H0,thenwehavetheidentity

(−φ, ψ)

_H−1

= (φ, ψ)

_L²

.

(37)

TheCHequationisderivedbythegradientflowwiththeGinzburg–Landaufreeenergyfunctional(36)undertheH⁻¹inner product

∂φ

∂

t

= −

^gradH⁻¹

E (φ) =  



(φ) −

²

φ



,

(38)

where



grad_H−1

E (φ) ,

v



H⁻¹

=

^d

d

θ ^E (φ + θ

v

) 



_θ=

0

= 



(φ) −

²

φ,

v



L2

= 

− 



(φ) −

²

φ

 ,

v



H⁻¹

.

(39)

Toobtaintheenergystablemethod,Eyre[29]developedthefirstordernonlinearCSmethodfortheCHequationas

φ

ⁿ⁺¹

− φ

ⁿ



t

=  

φ

ⁿ⁺¹



3

−

²

φ

ⁿ⁺¹

− φ

ⁿ



.

(40)

Inthismethod,

E(φ)

issplitintoacontractivepartandanexpansivepart:

E(φ) = E

c

(φ) − E

e

(φ)

where

E

c

(φ) =





1 4

φ

⁴

+

¹

4

+

²

2

|∇φ|

²



dx

, E

e

(φ) =



 1

2

φ

²dx

.

(41)

And

E

c

(φ)

istreatedimplicitly,whereas

E

e

(φ)

istreatedexplicitly.Itiseasytoshowthatboth

E

c

(φ)

and

E

e

(φ)

areconvex.

Now, we can constructthe numerical method using the splitting(41) and s-stage CSRK-R(p) method (29) in Section 4.

Giventhe resemblebasematrixR with s-stagesanddesiredorderofaccuracy,we evaluateone timestepasfollows:we set

φ

0

= φ

^nandevaluate

φ

_i

− φ

0

Δ

t

=

i

j=1

ri j

 

φ

j



3

−

²

φ

j

− φ

j−¹



,

(42)

fori

=

¹

,

2

, . . . ,

s andwehave

φ

ⁿ⁺¹

= φ

s.Tosolvethei-thstageevaluation(42),weuseanonlineariterativemethodwith aNewton-typelinearizationof

(φ

i

)

³.Wereferto[16,30]fordetailsofthenonlineariterativesolver.

Remark4. As we introduce in (38), the CH equation is one of gradient flows under the H⁻¹ inner product. Intuitively, thegradientin H⁻¹ canbe consideredasgrad_H−1

= −

_δφ^{δ ,where} _δφ^{δ isthefunctional}derivative. Therefore,theuncondi- tionalenergystabilityanduniquesolvabilityof(42)areguaranteedbyTheorem 2 and 3withtheresemble basematrices introducedinSection4.

Remark5. We apply anonlinear convexsplitting(41)to constructa CSRK-Rmethod,so that thenonlinear systems(42) are obtained. However, itisjustan exampletoshow theapplicability ofCSRK-R methods.We canmake alinearCSRK-R methodbyconsideringthelinearconvexsplittingfor(36).Wereferto[23]forinterestedreaders.

We nowpresentexamples tonumericallydemonstratethe accuracyandstability oftheproposed method,CSRK-R

(

p

)

, correspondingtothefirstordermethod(33),secondordermethod(34),andthirdordermethod(35).

5.1. Solutionevolutionwithrandominitialdatain1D

We beginby showingthe solutionevolution oftheCH equation withthe zeroNeumannboundary condition andthe randomlyperturbedinitialcondition

φ (

x

,

0

) =

⁰

.

01

·

^rand

(

x

)

(43)

onadomain

 = [

⁰

,

1

]

^{,wheretherandomnumberrand}

(

x

)

isuniformlydistributedbetween

−

^{1 and 1.Forthenumerical} simulations,

=

⁰

.

02 is used and the grid size is fixed at



x

=

¹

/

128 which provides sufficient spatial accuracy. The numerical solutionisevolvedtotime Tf

=

⁰

.

08.Weremarkthat spectralmethods areused forspatialderivatives inthe numericalcomputations.SimulationsareexecutedbyusingtheMATLABprogramwithafastFouriertransform.

Fig. 1 showsthe time evolutionofthe solutionwitha sufficientlysmalltime step

Δ

t

=

^Tf

/

2¹⁷ usingthethird order methodCSRK-R(3).Attheearlystage,solutionsarerearrangedandadjustedtofindtheirdesiredmodewithlowmagnitude.

Afterfinishingtheadjustment,thesolutionmaturesandapproachesthesteady-statesolution.Thesesolutionswillbeused asthereferencesolutiontoestimatetheconvergencerateinSection5.2.

Fig. 2 showsthe time evolution offree energy functional

E (φ)

corresponding tothe solution in Fig. 1. The property of theenergydissipation isvalidatedandenergytransitionsareobserved.The evolutionwill beused asthe referenceto comparewiththeenergyevolutioninSection5.3.

5.2. Numericalconvergencetestwithrandominitialdatain1D

We demonstratethe numericalconvergence usingthe sameconditions andparameters asthoseused inthe previous section. Toestimate theconvergenceratewithrespectto atime step

Δ

t, simulationsare performedbyvarying thetime steps

Δ

t

=

Tf

/

2¹⁵

,

Tf

/

2¹⁴

, . . . ,

Tf

/

2³.

Fig. 3showstherelativel₂-errorsof

φ (

x

,

t

)

withvarioustimesteps.Here,theerrorsarecomputedbycomparisonwith thereferencesolution.ItisobservedthattheCSRK-R(p)methodsgivethedesiredp-thorderaccuracyintime.

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

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

5.3. Energystabilityperformancewithrandominitialdatain1D

We display the energy evolutions with the same conditions and parameters as those used in the previous section.

Simulationsareperformedbyrelativelylargetimesteps

Δ

t

=

^Tf

/

2⁷

,

Tf

/

2⁵

,

Tf

/

2³.

Fig. 4showstheenergyevolutionwithlargetimestepsandtheenergydissipationsareobservedforallsimulations.

5.4. Solutionevolutionwithrandominitialdatain2D

Next,wedisplaythesolutionevolutionoftheCHequationwiththezeroNeumannboundaryconditionandtherandomly perturbedinitialcondition

φ (

x

,

y

,

0

) =

⁰

.

001

·

^rand

(

x

,

y

)

(44)

ona domain

 = [

⁰

,

1

] × [

⁰

,

1

]

^{.Forthenumerical}simulations,

=

⁰

.

02 isusedandthe gridsize isfixedat



x

= 

^y

=

1

/

512.ThenumericalsolutionisevolvedtotimeT_f

=

⁰

.

08.

Fig. 3. Relative l2-errors for various time stepst in 1D.

Fig. 4. Energy evolution for the CH equation in 1D.

Fig. 5. TimeevolutionofsolutionfortheCHequationin2D.(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtotheweb versionofthisarticle.)

Fig. 6. Energy evolution for the CH equation in 2D.

Fig. 5 showsthetime evolution ofthesolution withasufficiently smalltime step

Δ

t

=

Tf

/

2¹⁷ usingthe third order methodCSRK-R(3). Ineach figure,thered, green,andblueregions indicate

φ =

^1,0,

−

^1,respectively. Atthe earlystage, solutionsare rearrangedandadjustedtofindtheir desiredmode withlow magnitude.Afterfinishingtheadjustment,the solutionmatures andmerges. Thesesolutions willbe used asthereferencesolution toestimate the convergenceratein Section5.6.

Fig. 6shows thetime evolutionof free energyfunctional

E (φ)

corresponding to the solutionin Fig. 5.The property oftheenergydissipation isvalidatedandenergytransitionsare observed.The evolutionwillbe usedasthereferenceto comparewiththeenergyevolutioninSection5.6.

5.5. Numericalconvergencetestwithrandominitialdatain2D

We demonstratethe numericalconvergenceusing thesame conditionsandparameters asthose usedin theprevious section. Toestimatethe convergenceratewithrespect toa timestep

Δ

t,simulations areperformedby varying thetime steps

Δ

t

=

^Tf

/

2¹⁵

,

T_f

/

2¹⁴

, . . . ,

T_f

/

2³.

Fig. 7 showstherelative l2-errors of

φ (

x

,

y

,

t

)

withvarious time steps.Here, the errorsare computedby comparison withthereferencesolution.ItisobservedthattheCSRK-R(p)methodsgivethedesired p-thorderaccuracyintime.

5.6. Energystabilityperformancewithrandominitialdatain2D

We display the energy evolutions with the same conditions and parameters as those used in the previous section.

Simulationsareperformedbyrelativelylargetimesteps

Δ

t

=

Tf

/

2⁷

,

Tf

/

2⁵

,

Tf

/

2³.

Fig. 8showstheenergyevolutionwithlargetimestepsandtheenergydissipationsareobservedforallsimulations.

5.7. Long-timeevolutionwithrandominitialdatain2D

We display the long-time evolution usingthe same conditions andparameters asthose used in theprevious section except T_f

=

^{16 and}

Δ

t

=

^{2−12.Here,we justprovide aresolved} computation forthe long-timeevolution withrelatively

Fig. 7. Relative l2-errors for various time stepst in 2D.

Fig. 8. Energy evolution for the CH equation in 2D.

Fig. 9. TimeevolutionofsolutionfortheCHequationin2D.(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtotheweb versionofthisarticle.)

Fig. 10. Energy evolution for the CH equation in 2D.

smallertimestepsizeandtheinterestedreaderscanconsideranalgorithmsuchasanadaptivetimesteppingprocedureto chooseabiggertimestepsize.

Fig. 9showsthe timeevolutionofthesolutionusingthethirdordermethodCSRK-R(3).Ineachfigure,thered, green, and blue regions indicate

φ =

^{1, 0,}

−

^1, respectively. The coarsening is observed for long time and the solution finally approachesthesteadystate.

Fig. 10showsthetimeevolutionoffreeenergyfunctional

E (φ)

correspondingtothesolutioninFig. 9.

6. Conclusions

Weproposedunconditionallygradientstablemethodstosolveagradientflowwhichisafundamentalconceptinvarious dynamical systems.Themainfeature oftheschemeisa combinationofthe convexsplitting methodandthe multi-stage two-additiveRunge–Kuttamethod withtheresemble design.The proposed methods arehighorder accuratein time and assureunconditionalgradientstability.Inpracticalterms,weprovidedsomeexamplesofimplicit–explicitRunge–Kuttatype methods,whicharenotonlyunconditionallygradientstablebutalsouniquelysolvable.Applyingtheproposedmethodsto theCahn–Hilliardequation, wepresentednumericalexperimentstoshow highorderaccuracy andunconditionalgradient stability.Wedemonstratedtheexistenceofthethirdorderandenergystablemethodbypresentingthespecificexamples.In future,weneedtostudywhatareoptimalchoicesamongtheproposedmethodsandcomparewiththeexistingnumerical methodsintermsofnumericalconvergenceerror.

Acknowledgements

Thisresearch was supported by the BasicScience ResearchProgramthrough the NationalResearchFoundation ofKo- rea (NRF), funded by the Korean Government MOE (2009-0093827) and MSIP (2015-003037, 2017R1D1A1B0-3032422, 2017R1D1A1B0-3034619).

Appendix A. Orderconditionsfortwo-additiveRunge–Kuttamethod

Althoughcouldreferto[23,25,26,31]forinformationonthederivationandlistoftheorderconditions,inthisappendix, webrieflyintroducetheorderconditionsofatwo-additiveRunge–Kuttamethodtoprovidedefinitenotationsandinforma- tion.Thetwo-additivelypartitionedsystem

∂φ

∂

t

=

^f

(φ) +

^g

(φ)

and

φ (

0

) = φ

⁰

,

(45)