Structure preserving schemes for the continuum Kuramoto model: Phase transitions

(1)

Contents lists available atScienceDirect

Journal

of

Computational

Physics

www.elsevier.com/locate/jcp

Structure

preserving

schemes

for

the

continuum

Kuramoto

model:

Phase

transitions

José

A. Carrillo

a

,

∗

,

Young-Pil Choi

b

,

Lorenzo Pareschi

c a_Department_of_Mathematics,_Imperial_College_London,_London_SW7_2AZ,_United_Kingdom

b_Department_of_Mathematics_and_Institute_of_Applied_Mathematics,_Inha_University,_Incheon,_402-751,_Republic_of_Korea c_Department_of_Mathematics_and_Computer_Science,_University_of_Ferrara,_Via_N._Machiavelli_35,_44121,_Ferrara,_Italy

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received10March2018

Receivedinrevisedform6September2018 Accepted27September2018

Availableonline2October2018

Keywords:

Structurepreserving Kuramotooscillators Kineticequations Phasetransitions

TheconstructionofnumericalschemesfortheKuramotomodelischallengingduetothe structural properties of the system whichare essentialin orderto capture the correct physicalbehavior,likethedescriptionofstationarystatesandphasetransitions.Additional diﬃculties are represented by the high dimensionality of the problem in presence of multiplefrequencies.Inthispaper, wedevelopnumericalmethods whichare capableto preservethesestructuralpropertiesoftheKuramotoequationinthepresenceofdiffusion and tosolve eﬃciently themultiple frequenciescase.The novelschemes are thenused tonumericallyinvestigatethephasetransitionsinthecaseofidenticalandnonidentical oscillators.

1. Introduction

Synchronization phenomena of large populations of weakly coupled oscillators are very common in naturalsystems, andit has beenextensively studied invarious scientiﬁc communities such asphysics, biology,sociology, etc. [2,4,29,52]. Synchronization arises due to the adjustment of rhythms of self-sustained periodic oscillators weakly connected [2,47], anditsrigorousmathematicaltreatmentispioneeredbyWinfree[53] andKuramoto[4].In[4,53],phasemodelsforlarge weaklycoupledoscillatorsystemswereintroduced,andthesynchronizedbehaviorofcomplexbiologicalsystemswasshown to emerge from the competingmechanisms of intrinsicrandomness andsinusoidal couplings. Since then, the Kuramoto modelbecomesaprototypemodelforsynchronizationphenomenaandvariousextensionshavebeenextensivelyexplored invariousscientiﬁc communitiessuch asapplied mathematics,engineering, controltheory,physics,neuroscience,biology, andsoon[28,47,52].

Givenan ensembleof sinusoidallycoupled nonlinear oscillators, whichcan be visualized as active rotorson the unit circle

S

1_,_let _z

j

=

eiϑj be thepositionofthe j-throtor.Then, thedynamicsof zj iscompletelydetermined bythat ofits phase

ϑ

j.Letusdenotethephaseandfrequencyofthe j-thoscillatorby

ϑ

jand

˙ϑ

j,respectively.Then,thephasedynamics ofKuramotooscillatorsaregovernedbythefollowingﬁrst-orderODEsystem[4]:

˙ϑ

i

=

ω

i

−

K N N

j=1 sin

(ϑ

i

− ϑ

j

),

i

=

1

,

· · · ,

N

,

t

>

0

,

(1.1)

*

Correspondingauthor.

E-mailaddresses:carrillo@imperial.ac.uk(J.A. Carrillo),ypchoi@inha.ac.kr(Y.-P. Choi),lorenzo.pareschi@unife.it(L. Pareschi).

https://doi.org/10.1016/j.jcp.2018.09.049

(2)

subject to initial data

ϑ

i

(

0

)

=: ϑ

i0, i

=

1

,

· · · ,

N, where K is the uniform positive couplingstrength, and

ω

i denotes the naturalphase-velocity(frequency)andisassumedtobearandomvariableextractedfromthegivendistribution g

=

g

(

ω

)

satisfying

R

g

(

ω

)

d

ω

=

1

.

Wenoticethattheﬁrsttermandsecondtermintherighthandsideoftheequation(1.1) representtheintrinsicrandomness andthenonlinearattraction–repulsioncoupling,respectively.

Thesystem(1.1) hasbeenextensivelystudied,anditstillremainsapopularsubjectinnonlineardynamicsandstatistical physics.Wereferthereaderto[2] andreferencesthereinforgeneralsurveyoftheKuramotomodelanditsvariants.In[4], Kuramoto ﬁrstobserveda continuousphase transitioninthecontinuum Kuramotomodel(N

→ ∞

,see(1.2) below)with a symmetric distribution function g

(

ω

)

by introducing an orderparameter r∞ whichmeasures thedegree of the phase synchronizationinthemean-ﬁeldlimit.Moreprecisely,theorderparameterisgivenby

rN

(

t

)

eiϕN(t)

:=

1 N N

j=1 eiϑj(t)

_,

_r∞

:=

_lim t→∞Nlim→∞r N

₍

_t

_).

In fact,Kuramoto showedthat theorderparameter r∞ asafunction ofthecoupling strength K changesfromzero (dis-ordered state)toanon-zerovalue(orderedstate)whenthecouplingstrength K exceedsacriticalvalue Kc

:=

2

/(

π

g

(

0

))

, i.e., r∞

(

K

)

=

0 (incoherentstate)for K

∈ [

0

,

Kc

)

,1

≥

r∞

(

K

)

>

0 (coherentstate)forK

>

Kc,andr∞

(

K

)

increaseswith K . Later,itisalsoobservedthatthecontinuumKuramotomodelcanexhibitcontinuous ordiscontinuous phasetransitionby takingintoaccountdifferenttypesofnaturalfrequencydistributionfunctions[11,12,45].Here,continuousphasetransitions refertothecontinuityatK

=

Kc oftheorderparameterr∞

(

K

)

.ItisknowntobecontinuousfortheGaussiandistributed infrequencyoscillators[52] anddiscontinuousfortheuniformlydistributedinfrequencyoscillators[45].

As the numberofoscillatorsgoes toinﬁnity

(

N

→ ∞)

,a continuum descriptionofthe system(1.1) canbe rigorously derivedbyemployingbynowstandardmean-ﬁeldlimittechniquesfortheVlasovequation[19,21,40,42].Let

ρ

=

ρ

(ϑ,

ω

,

t

)

betheprobabilitydensityfunctionofKuramotooscillatorsin

ϑ

∈ T

:= R/(

2

π

Z)

withanaturalfrequency

ω

extractedfrom adistributionfunctiong

=

g

(

ω

)

attimet.ThenthecontinuumKuramotoequationisgivenby

∂

t

ρ

+ ∂

ϑ

(

u

[

ρ

]

ρ

)

=

0

,

(ϑ,

ω

)

∈ T × R,

t

>

0

,

u

[

ρ

](ϑ,

ω

,

t

)

=

ω

−

K

T×R

sin

(ϑ

− ϑ

_∗

)

ρ

(ϑ

_∗

,

ω

,

t

)

g

(

ω

)

d

ϑ

_∗d

ω

,

(1.2)

subjecttotheinitialdata:

ρ

(ϑ,

ω

,

0

)

=:

ρ0

(ϑ,

ω

),

T

ρ0

(ϑ,

ω

)

d

ϑ

=

1

.

(1.3)

Theorderparameterr andtheaveragephase

ϕ

associatedto(1.2) aregivenas

r

(

t

)

eiϕ(t)

=

T×R eiϑ

ρ

(ϑ,

ω

,

t

)

g

(

ω

)

d

ϑ

d

ω

,

leadingto r

(

t

)

=

T×R cos

(ϑ

−

ϕ

(

t

))

ρ

(ϑ,

ω

,

t

)

g

(

ω

)

d

ϑ

d

ω

.

(1.4)

Forthecontinuum equation(1.2), globalexistence anduniqueness ofmeasure-valued solutionsarestudiedin[21,40] and important qualitative properties such as theLandau damping towards the incoherentstationary stateswere analyzed in [30].

Averyrelevantissuefromtheapplicationviewpointishowstablethesestationarystatesandphasetransitionsareby addingnoisetothesystem[4,50].Themean-ﬁeldlimitequationassociatedto(1.1) withstandardGaussiannoiseofstrength

√

2D,withD

>

0,is

∂

t

ρ

+ ∂

ϑ

(

u

[

ρ

]

ρ

)

=

D

∂

ϑ2

ρ

,

(ϑ,

ω

)

∈ T × R,

t

>

0

,

u

[

ρ

](ϑ,

ω

,

t

)

=

ω

−

K

T×R sin

(ϑ

− ϑ

_∗

)

ρ

(ϑ

_∗

,

ω

,

t

)

g

(

ω

)

d

ϑ

_∗d

ω

,

(1.5)

(3)

subjecttotheinitial data(1.3). Phasetransitionshavealsobeenfound intermsoftheorder parameterr∞ asafunction of K for D

>

0. In fact, it was proven in [50] that foridentical oscillators there is a continuous phase transition. This phasetransitionisalsocontinuousforGaussiananduniformlycouplednoisy oscillators[2] anddiscontinuous forbimodal distributions[16,25].StabilityofthecoherentandincoherentstatesandtheasymptoticdynamicsfortheKuramotomodel withnoise(1.5) wereanalyzed in[31,32,37,38].Noise-drivenphasetransitionsininteractingparticlesystemsareahottopic ofresearchfromthenumericalandtheoreticalviewpoints,see[5–10,25,33] andthereferencestherein.

Inthepresentwork,wefocusontheconstructionofeffectivenumericalschemesforthecontinuumKuramotosystem withdiffusionobtainedinthelimit ofaverylarge numberofoscillatorsandinpresenceofnoise.Thenumericalsolution of such system is challenging due to the high dimensionality of the problem and the intrinsic structural properties of the system which are essential in order to capture the correct physical behavior. The literature dealing with the study ofnumericalschemes to thecontinuum Kuramoto model(1.2) ismainly focused ondeterministic spectral andstochastic methods.Wereferto[2,SectionVI.B] andthereferencesthereinforgeneraldiscussionsonthestochasticanddeterministic numerical methods used in Kuramoto models, see also [10,35,36,47] for Monte Carlo related methods and [1,3,46] for spectralmethods.

Structurepreservingnumericalschemesformean-ﬁeldtypekineticequationshavebeenpreviouslyconstructedin[2,14,

17,18,20,22,44,51].Wereferalsoto[34] forrelatedapproachesforgeneralsystemsofbalancelawsandto[27] foran intro-ductiontonumericalmethodsforkineticequations.Here,followingtheapproachin[18,22,44] weintroduceadiscretization ofthephasevariablesuch thatnonnegativity of thesolution,physicalconservations,asymptoticbehavior andfree energy dissipationarepreservedatadiscretelevelforidenticaloscillators.Thisapproachisthencoupledwithasuitablecollocation methodforthefrequencyvariablebasedonorthogonalpolynomialswithrespecttothegivenfrequencydistribution.Similar collocationstrategieshavebeenusedforkineticequationsinthecaseoflineartransportandsemiconductors[39,41,49].

Therestofthemanuscriptisorganizedasfollows.First,inSection2werecallsomebasicpropertiesofthecontinuum Kuramoto model, in particular concerning some relevant existence theory and some useful estimates underpinning our numericalstrategy.WealsodiscusstheasymptoticbehavioroftheKuramotomodelwithnoise,stationarystatesandrelated freeenergyestimatesinthecaseofidenticaloscillators.ThediscretizationofKuramotosystemsisdiscussedinSection 3. Structure preserving schemes are developed in combination with a collocation method for the frequency variable. The asymptoticbehavioroftheschemesaswellastheotherrelevantphysicalpropertiesarediscussed.InSection4wepresent severalnumericaltests,witha particularemphasisonthestudyofphasetransitions.We willalsocompareourstructure preserving methodsto MonteCarloandspectral methods.Futureresearch directionsandconclusions are reportedinthe lastSection.

2. The continuum Kuramoto model

2.1. StabilityoftheKuramotomodelwithrespecttothefrequencydistribution

Inthissubsection,webrieﬂy reviewsome relevantissuesrelatedtothewell-posedness theoryandseveraluseful esti-matesforthecontinuumKuramotoequation(1.2) anditsversionwithnoise(1.5).Wereferto[19,21,40] forfurtherdetails. Weﬁrstprovidea priori estimatesfortheequation(1.2).

Lemma 2.1. Let

ρ

beasmoothsolutionto(1.2)–(1.3).Thenwehave

(

i

)

d dt

T

ρ

(ϑ,

ω

,

t

)

d

ϑ

=

0 i.e.,

T

ρ

(ϑ,

ω

,

t

)

d

ϑ

=

T

ρ0

(ϑ,

ω

)

d

ϑ

=

1 for w

∈ R,

(

ii

)

d dt

T×R

ρ

(ϑ,

ω

,

t

)

log

(

ρ

(ϑ,

ω

,

t

))

g

(

ω

)

d

ϑ

d

ω

=

K r2

(

t

),

fort

≥

0,wherer isgivenin(1.4).

Proof. The proofof

(

i

)

isstraightforward.Fortheidentity

(

ii

)

,byusingthedeﬁnitionofr in(1.4) werewriteu

[

ρ

]

as

u

[

ρ

](ϑ,

ω

,

t

)

=

ω

−

K r sin

(ϑ

−

ϕ

(

t

)).

(2.1)

Directcomputationsyield

d dt

T×R

ρ

(ϑ,

ω

,

t

)

log

(

ρ

(ϑ,

ω

,

t

))

g

(

ω

)

d

ϑ

d

ω

= −

T×R g

(

ω

)∂

ϑ

(

u

[

ρ

]

ρ

)

log

ρ

d

ϑ

d

ω

=

T×R g

(

ω

)

u

[

ρ

]∂

ϑ

ρ

d

ϑ

d

ω

(4)

= −

T×R

(∂

ϑu

[

ρ

])

ρ

(ϑ,

ω

)

g

(

ω

)

d

ϑ

d

ω

=

K r

T cos

(ϑ

−

ϕ

(

t

))

ρ

(ϑ,

ω

)

g

(

ω

)

d

ϑ

d

ω

=

K r2

,

completingtheproof.

2

We next discussglobalexistence anduniqueness ofmeasure-valuedsolutions. Letusdenoteby

M(T

× R)

the setof nonnegative Radonmeasures on

T

× R

,which canbe regardedasnonnegativeboundedlinearfunctionals on

C

0

(

T

× R)

. Thenournotionofmeasure-valuedsolutionsto(1.2) isdeﬁnedasfollows.

Deﬁnition 2.1. For T

∈ [

0

,

∞)

,let

μ

∈

C

w

(

0

,

T

;

M(T

× R))

be ameasure-valuedsolutionto (1.2) withan initial measure

μ

0

∈

M(T

× R)

ifandonlyif

μ

satisﬁesthefollowingconditions:

• μ

isweaklycontinuous,i.e.,

μ

t

,

h

is a continuous as a function of t

,

∀

h

∈

C

0

(

T × R),

where

μ

t

,

h

:=

T×R h

(ϑ,

ω

)

μ

t

(

d

ϑ,

d

ω

).

•

Forh

∈

C

1₀

(

T

× R

× [

0

,

T

))

,

μ

satisﬁesthefollowingintegralequation:

μ

t

,

h

(

·, ·,

t

)

−

μ0

,

h

(

·, ·,

0

)

=

t

0

μ

s

, ∂

sh

+

u

[

μ

]∂

ϑh

ds

,

whereu

[

μ

]

isgivenby u

[

μ

](ϑ,

ω

,

t

)

:=

ω

−

K

T×R sin

(ϑ

− ϑ

∗

)

g

(

ω

)

μ

t

(

d

ϑ

∗

,

d

ω

).

WealsointroducethedeﬁnitionoftheboundedLipschitzdistance.

Deﬁnition 2.2. Let

μ

,

ν

∈

M(T

× R)

betwoRadonmeasures.ThentheboundedLipschitzdistanced

(

μ

,

ν

)

between

μ

and

ν

isgivenby

d

(

μ

,

ν

)

:=

sup

h∈S

|

μ

,

h

−

ν

,

h

| ,

wheretheadmissibleset

S

oftestfunctionsisdeﬁned as

S

:= {

h

: T × R → R :

max

{

h

L∞,Lip

(

h

)

} ≤

1

}

with Lip

(

h

)

:=

sup (ϑ,ω) =(ϑ∗,ω∗)

|

h

((ϑ,

ω

))

−

h

((ϑ

_∗

,

ω

_∗

))

|

|(ϑ,

ω

)

− (ϑ

_∗

,

ω

_∗

)

|

.

Wenowpresenttheresultsontheglobalexistenceandstabilityofmeasure-valuedsolutionsto(1.2).Wereferthereader to[19,21,40] fordetailsoftheproof.

Proposition 2.1. Forany

μ

0

∈

M(T

×R)

,(1.2) hasauniquesolution

μ

∈

C

w

(

0

,

T

;

M(T

×R))

withtheinitialdata

μ

0.Furthermore,

if

μ

and

ν

aretwosuchsolutionsto(1.2),thenthereexistsaconstantC

>

0 dependingonT suchthat

(5)

Remark 2.1. Onecanalsocharacterizethesolution

μ

oftheinitial-valueproblemfor(1.2) asthepush-forwardoftheinitial data

μ

0throughtheﬂowmapgeneratedbyu

[

μ

]

,i.e.,foranyh

∈

C

c1

(

T

× R)

andt

,

s

≥

0

T×R h

(ϑ,

ω

)

μ

t

(

d

ϑ,

d

ω

)

=

T×R h

( (

0

;

t

, ϑ,

ω

),

ω

)

μ0

(

d

ϑ,

d

ω

),

where

satisﬁes d dt

(

t

;

s

, ϑ,

ω

)

=

u

[

μ

]( (

t

;

s

, ϑ,

ω

),

ω

,

t

)

with

(

s

;

s

, ϑ,

ω

)

= ϑ.

(2.2)

Note that the characteristic system (2.2) is well deﬁnedsince the velocity ﬁeld u

[

μ

]

is globally Lipschitz in

(ϑ,

ω

)

and continuousont evenforgeneralmeasures

μ

∈

C

w

(

0

,

T

;

M(T

× R))

.

Remark 2.2. AsasimpleextensionofProposition2.1,see[21],wealsoobtainthatthesolution

μ

t canbeapproximatedas asumofDiracmeasuresofthefollowingform

μ

N t

=

1 N N

i=1

δ

ϑi(t)

⊗ δ

ωi

,

i.e., d

(

μ

t

,

μ

Nt

)

→

0 as N

→ ∞

. This can be accomplished by choosing a particle approximation of the initial data. One proceduretoconstructparticleapproximationsisthefollowing.Letuschoosethesmallestsquarecontainingthesupportof theinitialdata

μ

0,i.e.,supp

(

μ

0

)

⊂

R.Foragivenn,wedividethesquare R inton2subsquares Ri,i.e.,

R

=

n2

i=1

Ri

.

Let

ϑ

_i0

,

ω

i bethecenterofRi fori

=

1

,

· · · ,

n2.Thenweconstructtheinitialapproximation

μ

n0 as

μ

n₀

=

n2

i=1 mi

δ

ϑ0 i

⊗ δ

ωi with mi

=

Ri

μ0

(ϑ,

ω

)

d

ϑ

d

ω

.

Thenwecaneasilyshowthatd

(

μ

n₀

,

μ

0

)

≤

C

/

n

→

0 asn

→ ∞

.

Fromnow on,weassumethat theinitialmeasureisa smoothabsolutely continuous withrespecttoLebesguedensity withconnectedsupport.ThisassumptioncanberemovedbythestabilitypropertyinProposition2.1.Wenextshowstability oftheorderparameterr withrespecttoapproximatedfrequencydistributionsandinitialdataby againusingthestability estimatepresentedinProposition2.1.

Proposition 2.2. Let

μ

bethemeasure-valuedsolutiontotheequation(1.2)–(1.3) inthetimeinterval

[

0

,

T

]

withg

∈

S

andtheinitial data

μ

0

∈

M(T

× R)

.Letusdeﬁnernas

rn

(

t

)

eiϕn(t)

=

T×R

eiϑ

ρ

n

(ϑ,

ω

,

t

)

gn

(

ω

)

d

ϑ

d

ω

,

where

ρ

n_is_the_{measure-valued}_solution_to₍_1.2_{) with}_the_initial_data

_ρ

n

0

∈

S

satisfyingd

(

ρ

n

0

,

μ

0

)

→

0.Thenforany g

n

_∈

_M(R)

satisfyingd

(

gn

,

g

)

→

0 asn

→ ∞

,wehave

lim

n→∞r

n

₍

_t

₎

₌

_r

₍

_t

₎

_{for t}

_{∈ [}

₀

_,

_T

_].

Proof. We ﬁrstestimate the1-Wassersteindistancebetween

ρ

n _and

_μ

_._Observe_that

_ρ

n

₍

_t

₎

_∈

_S

_for_all _t

_≥

_{0 by}_the push-forwardcharacterizationinRemark2.1.Moreover,italsofollowsfromRemark2.1that

T×R h

(ϑ,

ω

)

ρ

n

(ϑ,

ω

,

t

)

d

ϑ

d

ω

−

T×R h

(ϑ,

ω

)

μ

t

(

d

ϑ,

d

ω

)

=

T×R h

(

n

(

0

;

t

, ϑ,

ω

),

ω

)

ρ

n 0

(ϑ,

ω

)

d

ϑ

d

ω

−

T×R h

( (

0

;

t

, ϑ,

ω

),

ω

)

μ0

(

d

ϑ,

d

ω

)

,

(6)

foranyh

∈

C

c1

(

T

× R)

andt

,

s

≥

0,where

n satisﬁes d dt

n

₍

_t

_;

_s

_{, ϑ,}

_ω

₎

₌

_un

_[

_ρ

n

_](

n

₍

_t

_;

_s

_{, ϑ,}

_ω

_),

_ω

_,

_t

₎

=

ω

−

K

T×R sin

(

n

(

t

;

s

, ϑ,

ω

)

− ϑ

_∗

)

ρ

n

(ϑ

_∗

,

ω

,

t

)

gn

(

ω

)

d

ϑ

_∗d

ω

,

with

n

(

s

;

s

,

ϑ,

ω

)

= ϑ

forall n

≥

1 and

satisﬁes (2.2). Fornotationalsimplicity, we denote

n

(

t

;

0

,

ϑ,

ω

)

=

n

(

t

)

and

(

t

;

0

,

ϑ,

ω

)

= (

t

)

.Thenastraightforwardcomputationyields

d dt

|

n

₍

_t

₎

_{− (}

_t

₎

_{| ≤}

_K

T×R sin

(

n

(

t

)

− ϑ

_∗

)

ρ

n

(ϑ

_∗

,

ω

,

t

)

d

ϑ

_∗

(

gn

−

g

)(

dw

)

+

K

T×R sin

(

n

(

t

)

− ϑ

_∗

)

g

(

w

) (

ρ

n t

−

μ

t

)(

d

ϑ

∗

,

d

ω

)

+

K

T×R

sin

(

n

(

t

)

− ϑ

∗

)

−

sin

( (

t

)

− ϑ

∗

)

g

(

w

)

μ

t

(

d

ϑ

∗

,

d

ω

)

≤

K d

(

gn

,

g

)

+

K d

(

ρ

tn

,

μ

t

)

+

K

|

n

(

t

)

− (

t

)

|.

Thusweconclude

|

n

₍

_t

₎

_{− (}

_t

₎

_{| ≤}

_C

₍

_K

_,

_T

₎

⎛

⎝

d

(

gn

,

g

)

+

t

0 d

(

ρ

_sn

,

μ

s

)

ds

⎞

⎠ ,

whereC

>

0 isindependentofn.Thistogetherwiththeboundednessof

n and

in

[

0

,

T

]

gives

T×R h

(

n

(

0

;

t

, ϑ,

ω

),

ω

)

ρ

₀n

(ϑ,

ω

)

d

ϑ

d

ω

−

T×R h

( (

0

;

t

, ϑ,

ω

),

ω

)

μ0

(

d

ϑ

d

ω

)

≤

C

⎛

⎝

d

(

gn

,

g

)

+

t

0 d

(

ρ

n s

,

μ

s

)

ds

⎞

⎠ +

Cd

(

ρ

n 0

,

μ

0

).

Hencewehave d

(

ρ

_tn

,

μ

t

)

≤

C

⎛

⎝

d

(

gn

,

g

)

+

t

0 d

(

ρ

_sn

,

μ

s

)

ds

⎞

⎠ +

Cd

(

ρ

n₀

,

μ0

),

i.e., d

(

ρ

n t

,

μ

t

)

≤

Cd

(

gn

,

g

)

+

Cd

(

ρ

0n

,

μ

0

),

whereC

>

0 isindependentofn.Wenowshowthestrongconvergenceofrn _to_r._For_this,_we_use_the_following_identities

rn

(

t

)

=

T×R eiϑ

ρ

n

_(ϑ,

_ω

_,

_t

₎

_gn

₍

_ω

₎

_d

_ϑ

_d

_ω

and r

(

t

)

=

T×R eiϑg

(

ω

)

μ

t

(

d

ϑ,

d

ω

)

,

toobtain

|

rn

(

t

)

−

r

(

t

)

| ≤

T×R cos

ϑ

ρ

_tngn

−

μ

tg

(

d

ϑ,

d

ω

)

+

T×R sin

ϑ

ρ

_tngn

−

μ

tg

(

d

ϑ,

d

ω

)

=:

I1

+

I2

.

(7)

Here I1 canbeestimatedas I1

≤

T×R cos

ϑ

ρ

n_t

−

μ

t

g

(

d

ϑ,

d

ω

)

+

T×R cos

ϑ

gn

−

g

ρ

_tn

(

d

ϑ,

d

ω

)

≤

d

(

ρ

n_t

,

μ

t

)

+

d

(

gn

,

g

),

duetocos

ϑ

g

(

ω

),

cos

ϑ

ρ

n

t

(ϑ,

ω

)

∈

S

.Similarly,wealsoﬁnd I2

≤

d

(

ρ

tn

,

μ

t

)

+

d

(

gn

,

g

)

,concludingthat

|

rn

(

t

)

−

r

(

t

)

| ≤

I1

+

I2

≤

2d

(

ρ

tn

,

μ

t

)

+

2d

(

gn

,

g

)

≤

Cd

(

gn

,

g

)

+

Cd

(

ρ

n

0

,

μ

0

)

→

0 as n

→ ∞.

Thiscompletestheproof.

2

Remark 2.3. AsimilarresulttoProposition2.2canbeobtainedifweapproximatetheinitialdata

μ

0

∈

M(T

× R)

andthe frequencydistribution gn

₍

_ω

₎

_∈

_L∞

₍

_R)

_as_follows:

_μ

n

0

∈

M(T

× R)

and gn

(

ω

)

∈

S

satisfying d

(

μ

n0

,

μ

0

)

→

0 and

gn

(

·)

−

g

(

·)

L∞(R)

→

0 asn

→ ∞

.

Remark 2.4. AsimilarresulttoProposition2.2canalsobeprovedfortheKuramotomodelwithnoise(1.5).Thestrategyof theproofisanalogousbutitusesstochasticprocessestechniquestowritethecorrespondingstochasticdifferentialequation systems.Moreover,one needstoresorttoWasserstein distancesinsteadoftheboundedLipschitzdistanceabovetomake thestabilityargumentofthesolutions.Wedonotincludethedetailsoftheproofsincethetechnicalitieslieoutsideofthe scopeofthepresentwork,see[15] forrelatedproblems.

NoticethatProposition2.2andRemark2.4allowustoworkwithcontinuousfrequencydistributionsforbothKuramoto models(1.2) and(1.5) byapproximation.Moreprecisely,wecanapproximateGaussiananduniformfrequencydistributions bysumsofDiracDeltasataﬁnitenumberoffrequencieswhileapproximatingtheinitialdatabysmoothfunctionsifthey arenotregularenough.ThisfactwillbeusedinthenumericalschemesinSection3andthesimulationsinSection4.

2.2. TheKuramotomodelwithdiffusion:stationarystatesandfreeenergies

Weﬁrstderiveanexplicitcompatibilityconditionforsmoothstationarystates

ρ

_∞ oftheequation(1.5).Weﬁrsteasily ﬁndfrom(1.5) and(2.1) that

D

∂

_ϑ2

ρ

_∞

= ∂

ϑ

((

ω

−

K r∞sin

(ϑ

−

ϕ

_∞

))

ρ

_∞

) ,

wherer_∞ and

ϕ

∞ aregivenby

r_∞eiϕ∞

₌

T×R

eiϑ

ρ

_∞

(ϑ,

ω

)

g

(

ω

)

d

ϑ

d

ω

.

Wenoticethatwecanset

ϕ

∞

=

0 inthestationarystatewithoutlossofgeneralitybychoosingtherightangularreference system.Thisyields

∂

ϑ

ρ

∞

(ϑ,

ω

)

−

ω

−

K r∞sin

ϑ

D

ρ

∞

=

c0

(

w

),

forsomefunctionc0

(

w

)

.Solvingtheabovedifferentialequation,weﬁnd

ρ

_∞

(ϑ,

ω

)

=

ρ

_∞

(

0

,

ω

)

exp

−

K r_∞

+

ω

ϑ

+

K r_∞cos

ϑ

D

×

⎛

⎝

1

+

c0

(

ω

)

e K r∞ D

ρ

_∞

(

0

,

ω

)

ϑ

0 exp

−

ω

ϑ

∗

+

K r∞cos

ϑ

∗ D

d

ϑ

_∗

⎞

⎠ ,

for

ϑ

∈ [

0

,

2

π

)

,where

ρ

_∞

(

0

,

ω

)

isﬁxedbythenormalization,i.e.,

T

(8)

Ontheotherhand,since

ρ

_∞

(

2

π

,

ω

)

=

ρ

_∞

(

0

,

ω

)

,weget 1

=

exp

2

π ω

D

⎛

⎝

1

+

c0

(

ω

)

e K r∞ D

ρ

_∞

(

0

,

ω

)

2π

0 exp

−

ω

ϑ

∗

+

K r∞cos

ϑ

∗ D

d

ϑ

_∗

⎞

⎠ ,

andsubsequently,thisimplies

c0

(

ω

)

e K r∞ D

ρ

_∞

(

0

,

ω

)

=

exp

(

−

2

π ω

/

D

)

−

1

2π 0 exp

(

−(

ω

ϑ

∗

+

K r∞cos

ϑ

∗

)/

D

)

d

ϑ

∗

.

Hencewehave

ρ

_∞

(ϑ,

ω

)

=

ρ

_∞

(

0

,

ω

)

exp

−

K r_∞

+

ω

ϑ

+

K r_∞cos

ϑ

D

×

1

+

(

exp

(

−

2

π ω

/

D

)

−

1

)

ϑ 0 exp

(

−(

ω

ϑ

∗

+

K r∞cos

ϑ

∗

)/

D

)

d

ϑ

∗

2π 0 exp

(

−(

ω

ϑ

∗

+

K r∞cos

ϑ

∗

)/

D

)

d

ϑ

∗

.

(2.3)

LetusdiscussfurtherpropertiesinthecaseofnoisyidenticalKuramotooscillators,whicharegovernedbytheequation (1.5) with g

= δ

0.Thenwecanset

ω

=

0 withoutlossofgeneralityin(2.3) toconcludethatitsstationarystateisgivenby

ρ

∞

(ϑ)

=

ρ

∞

(

0

)

exp

−

K r_∞

+

K r_∞cos

ϑ

D

,

where

ρ

∞

(

0

)

isagainﬁxedbythenormalization

T

ρ

∞

(ϑ)

d

ϑ

=

1

.

Notethatforbothidenticalandnon-identicaloscillatorsifr_∞

=

0,i.e.,incoherencestate,weobtain

ρ

_∞

(ϑ,

ω

)

=

ρ

(

0

,

ω

)

=

1

2

π

,

for all

ω

∈ R,

duetothenormalization.Inthiscasetheequation(1.5) hasthestructureofagradientﬂow.Moreprecisely,ifweset

ξ

[

ρ

](ϑ,

t

)

:=

K

(

W

ρ

)(ϑ,

t

)

−

D log

ρ

(ϑ,

t

),

withW

(ϑ)

=

cos

ϑ

,thenitiseasytocheckthat

∂

t

ρ

+ ∂

ϑ

(

ρ

∂

ϑ

ξ

[

ρ

]) =

0

.

Usingthisobservation,wenowestimatethefollowingfreeenergy

E

(

t

)

:= −

K 2

T

(

W

∗

ρ

)(ϑ,

t

)

ρ

(ϑ,

t

)

d

ϑ

+

D

T

ρ

(ϑ,

t

)

log

ρ

(ϑ,

t

)

d

ϑ.

ρ

beasmoothsolutiontotheequation(1.5) withg

= δ

0.Thenwehave

d

dt

E

(

t

)

= −

D

E

(

t

),

wherethedissipationrate

D

_E

(

t

)

isgivenby

D

E

(

t

)

:=

T

|∂

ϑ

ξ

[

ρ

]|

2

ρ

d

ϑ.

Proof. A straightforwardcomputationyields

d dt

E

(

t

)

= −

T

(

K

(

W

ρ

)

−

D log

ρ

) ∂

t

ρ

d

ϑ

= −

T

ξ

[

ρ

]∂

t

ρ

d

ϑ

= −

T

|∂

ϑ

ξ

[

ρ

]|

2

ρ

d

ϑ.

2

(9)

ρ

beasmoothsolutiontotheequation(1.5) withg

= δ

0.Thenwehave

˙

r

=

K r

T sin2

(

ϕ

− ϑ)

ρ

d

ϑ

−

Dr

.

Inparticular,ifthestrengthofnoiseD isstrongenoughsuchthatD

≥

K ,then

˙

r

≤

0 forallt

≥

0.

Proof. By deﬁnitionoftheorderparameterr,weget

˙

r

= ˙

ϕ

T sin

(ϑ

−

ϕ

)

ρ

(ϑ,

t

)

d

ϑ

+

T cos

(ϑ

−

ϕ

)∂

t

ρ

(ϑ,

t

)

d

ϑ

=

I1

+

I2

,

whereI1vanishessince

I1

=

˙

ϕ

r

T×T sin

(ϑ

− ϑ

∗

)

ρ

(ϑ,

t

)

ρ

(ϑ

∗

,

t

)

d

ϑ

d

ϑ

∗

=

0

.

FortheestimateofI2,weﬁnd

I2

=

T cos

(ϑ

−

ϕ

)

D

∂

_ϑ2

ρ

− ∂

ϑ

(

u

[

ρ

]

ρ

)

d

ϑ

= −

D

T cos

(ϑ

−

ϕ

)

ρ

d

ϑ

−

T sin

(ϑ

−

ϕ

)

u

[

ρ

]

ρ

d

ϑ

= −

Dr

+

K

T sin2

(ϑ

−

ϕ

)

ρ

d

ϑ.

Combiningtheabovetwoestimatesconcludesthedesiredresult.

2

Beforepassingtotheconstructionofnumericalmethodsthefollowingremarkshouldbemade.

Remark 2.5. Severalworksforthecontinuum Kuramoto modelarebased onthe g-weighted kinetic density f

(ϑ,

ω

,

t

)

:=

ρ

(ϑ,

ω

,

t

)

g

(

ω

)

(see[13] forexample).Inthisway,onecanrewritetheKuramotomodel(1.5) as

∂

tf

+ ∂

ϑ

(

v

[

f

]

f

)

=

D

∂

ϑ2f

,

with f

(ϑ,

ω

,

t

)

=

ρ

(ϑ,

ω

,

t

)

g

(

ω

),

(2.4) v

[

f

](ϑ,

ω

,

t

)

=

ω

−

K

T×R

sin

(ϑ

− ϑ

∗

)

f

(ϑ

∗

,

ω

,

t

)

d

ϑ

∗d

ω

.

Notethatthedistributionofthenaturalfrequenciesisnowgivenby

g

(

ω

)

=

T

f

(ϑ,

ω

,

t

)

d

ϑ.

Evenif,inthecontinuation,wewillusetheform(1.5) fortheconstructionofourstructurepreservingnumericalmethods, theycanbeeasilyreformulatedtotheform(2.4).InSection4wewillshowboth

ρ

(ϑ,

ω

,

t

)

and f

(ϑ,

ω

,

t

)

insomenumerical tests.

3. Structure preserving methods

The goalnow isto propose ﬁnitevolume numericalschemes preserving the structureof gradientﬂow to the caseof identicaloscillatorsandthataregeneralizableforoscillatorswithnaturalfrequenciesgivenbyadistributionfunction g

(

w

)

. Tostartwith,from(2.1) and(1.5),itfollowsthatthedensity

ρ

satisﬁesthefollowingcontinuityequation

∂

t

ρ

= ∂

ϑF

[

ρ

],

(3.1)

with

(10)

3.1. Semi-discretestructurepreservingschemes

Inspired by [17,18,22,44,51], we construct a discrete numerical scheme in the variable

ϑ

for the above equation as follows.Fori

=

1

,

· · · ,

N,weﬁrstintroduceauniform spatialgrid

ϑ

i

∈ T

suchthat

ϑ

i+1

− ϑ

i

= ϑ

and

ϑ

N+k

= ϑ

kfork

∈ R

. Withoutlossofgenerality,weset

ϑ

1/2

= ϑ

N+1/2

=

0

≡

2

π

,andwethendeﬁne

ρ

i

(

ω

,

t

)

:=

1

ϑ

i+1/2 ϑi−1/2

ρ

(ϑ,

ω

,

t

)

d

ϑ.

Weconsiderthefollowingapproximationsfor(3.1)

d

dt

ρ

i

(

ω

,

t

)

=

Fi+1/2

[

ρ

](

ω

,

t

)

−

Fi−1/2

[

ρ

](

ω

,

t

)

ϑ

for i

=

1

,

· · · ,

N

,

(3.3)

wherethenumericalﬂuxfunction Fi±1/2

[

ρ

](

ω

,

t

)

isgivenby

Fi+1/2

[

ρ

](

ω

,

t

)

:=

D

ρ

i+1

−

ρ

i

ϑ

−

ui+1/2

ρ

˜

i+1/2

,

(3.4) with ui+1/2

(

ω

,

t

)

:=

1

ϑ

ϑi+1

ϑi u

(ϑ,

ω

,

t

)

d

ϑ,

ρ

˜

i+1/2

:= (

1

− δ

i+1/2

)

ρ

i+1

+ δ

i+1/2ρi

,

and u

(ϑ,

ω

,

t

)

=

ω

+

K

ϑ

N

j=1 sin

(ϑ

j

− ϑ)

R

ρ

j

(

ω

∗

,

t

)

g

(

ω

_∗

)

d

ω

_∗

.

(3.5)

As in[22,44] asuitablechoiceoftheweightfunctions

δ

i+1/2 yieldsamethodthatmaintainsnonnegativityofthesolution (withoutrestrictionson

ϑ

)andpreservesthesteadystateofthesystemwitharbitraryorderofaccuracy.Wewillreferto theschemesobtainedinthiswayasChang–Coopertypeschemes[22].

First,observethatwhenthenumericalﬂux(3.4) vanishesweget

ρ

i+1

ρ

i

=

D

ϑ

+ δ

i+1/2ui+1/2 D

ϑ

− (

1

− δ

i+1/2

)

ui+1/2

.

(3.6)

Similarly,ifweconsidertheexactﬂux(3.2),byimposing F

[

ρ

]

≡

0,wehave

D

∂

ϑ

ρ

=

u

ρ

.

Integratingtheaboveequationonthecell

[ϑ

i

,

ϑ

i+1

]

weget

ϑi+1

ϑi 1

ρ

∂

ϑ

ρ

ϑ

=

ϑi+1

ϑi u Dd

ϑ,

whichgives

ρ

i+1

ρ

i

=

exp

ϑ

D ui+1/2

.

(3.7)

Therefore,byequating(3.6) and(3.7) werecover

δ

i+1/2

=

1

ξ

i+1/2

+

1 1

−

exp

(ξ

i+1/2

)

with

ξ

i+1/2

= −

ϑ

D ui+1/2

.

(3.8)

Wecanstatethefollowing.

Proposition 3.1. Thenumericalﬂuxfunction(3.4) with

δ

i+1/2deﬁnedby(3.8) vanisheswhenthecorrespondingﬂux(3.2) isequalto

(11)

Proof. The latterresultfollowsfromthesimpleinequalityexp

(

x

)

≥

1

+

x.

2

Remark 3.1. Since r cos

(

ϕ

− ϑ

i

)

= ϑ

N

j=1 cos

(ϑ

j

− ϑ

i

)

R

ρ

j

(

ω

,

t

)

g

(

ω

)

d

ω

,

weget ui+1/2

(

ω

,

t

)

=

ω

+

K N

j=1

cos

(ϑ

j

− ϑ

i+1

)

−

cos

(ϑ

j

− ϑ

i

)

R

ρ

j

(

ω

,

t

)

g

(

ω

)

d

ω

=

ω

+

K r

ϑ

(

cos

(

ϕ

− ϑ

i+1

)

−

cos

(

ϕ

− ϑ

i

)) .

Ontheother hand,the aboveexpressionforui+1/2 isnot thatusefulinpracticesinceitcontains r whichdependson

ρ

. Thus,itwouldbemoretechnicallyusefultowrite

ui+1/2

(

ω

,

t

)

=

ω

+

2K sin

( ϑ/

2

)

N

j=1 sin

(ϑ

j

− ϑ

i+1/2

)

R

ρ

j

(

ω

,

t

)

g

(

ω

)

d

ω

=

ω

+

2K sin

( ϑ/

2

)

⎛

⎝

cos

(ϑ

i+1/2

)

N

j=1

ρ

s_j

(

t

)

−

sin

(ϑ

i+1/2

)

N

j=1

ρ

c_j

(

t

)

⎞

⎠ ,

(3.9) whereweset

ρ

s_j

(

t

)

:=

sin

(ϑ

j

)

R

ρ

j

(

ω

,

t

)

g

(

ω

)

d

ω

and

ρ

cj

(

t

)

:=

cos

(ϑ

j

)

R

ρ

j

(

ω

,

t

)

g

(

ω

)

d

ω

.

Remark 3.2. Theresultingschemeissecondorderaccuratein

ϑ

forD

>

0,anddegeneratetosimpleﬁrstorderupwinding inthelimitcaseD

=

0.Infact,itisimmediatetoshowthatasD

→

0 weobtaintheweights

δ

i+1/2

=

0

,

ui+1/2

<

0

,

1

,

ui+1/2

>

0

.

Inthelemmabelow,weshowthatthenumericalschemeconservesthemass.

Lemma 3.1. Considerthenumericalscheme(3.3).Thenwehave

d dt N

i=1

ρ

i

(

ω

,

t

)

=

0 for

(

ω

,

t

)

∈ R × R

+

.

Proof. It followsfromtheperiodicityofdomainthat

ρ

N+1

=

ρ1

and

ρ

N

=

ρ0

.

(3.10)

Similarly, we getuN+1/2

=

u1/2 andsubsequentlythisimplies

ξ

N+1/2

= ξ

1/2,

δ

N+1/2

= δ

1/2,and

ρ

˜

N+1/2

= ˜

ρ

1/2.From the aboveproperties,wecaneasilyobtain

FN+1/2

[

ρ

](

ω

,

t

)

=

F1/2

[

ρ

](

ω

,

t

)

for

(

ω

,

t

)

∈ R × R

+

.

This,togetherwiththefollowingstraightforwardcomputation

d dt N

i=1

ρ

i

(

ω

,

t

)

=

1

ϑ

N

i=1

Fi+1/2

[

ρ

](

ω

,

t

)

−

Fi−1/2

[

ρ

](

ω

,

t

)

=

1

ϑ

FN+1/2

[

ρ

](

ω

,

t

)

−

F1/2

[

ρ

](

ω

,

t

)

=

0

,

(12)

We next providethe positivity preservation whose proof can be obtained by usingalmost sameargument asin [44, Proposition 1]. However, for the completeness of thiswork, we sketch the proof in the propositionbelow. For this, we introducethetimediscretizationtn

₌

_n

_{t with}

_t

_>

_{0 and}_n

_{∈ N}

0andconsiderthefollowingforwardEulermethod

ρ

n_i+1

(

ω

,

t

)

=

ρ

n_i

(

ω

,

t

)

+

tF n i+1/2

[

ρ

](

ω

,

t

)

−

Fni−1/2

[

ρ

](

ω

,

t

)

ϑ

for i

=

1

,

· · · ,

N

,

(3.11) where Fn_i₊₁_/₂

[

ρ

](

ω

,

t

)

:=

D

ρ

n i+1

−

ρ

n i

ϑ

−

u n i+1/2

ρ

˜

in+1/2

.

Proposition 3.2. Supposethatg iscompactlysupportedandthetimestep

t satisﬁes

t

≤

( ϑ)

2

(

C0

ϑ

+

D

)

where C0

= |

supp

(

g

)

| +

K

.

Thentheexplicitscheme(3.11) preservesnonnegativity,i.e.,

ρ

n+1

i

≥

0 if

ρ

in

≥

0 fori

=

1

,

· · · ,

N.

Proof. It followsfrom(3.11) that

Fn_i₊₁_/₂

[

ρ

](

ω

,

t

)

−

F_in₋₁_/₂

[

ρ

](

ω

,

t

)

=

ρ

_in₊₁

D

ϑ

−

u n i+1/2

(

1

− δ

ni+1/2

)

+

ρ

_in

un_i₋₁_/₂

(

1

− δ

_in₋₁_/₂

)

−

un_i₊₁_/₂

δ

n_i₊₁_/₂

−

2D

ϑ

+

ρ

n_i₋₁

D

ϑ

+

u n i−1/2

δ

in−1/2

.

Ontheotherhand,weeasilyﬁnd

D

ϑ

−

u n i+1/2

(

1

− δ

ni+1/2

)

=

D

ξ

i+1/2

ϑ

1

−

1 1

−

exp

ξ

i+1/2

≥

0

,

duetox

(

1

− (

1

−

ex

)

−1

)

≥

0 forx

∈ R

.Similarly,weget

D

ϑ

+

u n i−1/2

δ

ni−1/2

=

D

ϑ

ξ

i−1/2 exp

ξ

i−1/2

−

1

≥

0

.

Wealsonoticethat

max 0≤i≤N

un_i₋₁_/₂

(

1

− δ

_in₋₁_/₂

)

−

un_i₊₁_/₂

δ

n_i₊₁_/₂

−

2D

ϑ

≤

max 0≤i≤N

|

un_i₊₁_/₂

| + |

un_i₋₁_/₂

|

+

2D

ϑ

≤

2

|

supp

(

g

)

| +

K

+

D

ϑ

.

This, together with the property of convex combination, yields that the nonnegativity is preserved ifthe time step

t

satisﬁes

t

≤

ϑ

2

(

|

supp

(

g

)

| +

K

+

D

/ ϑ)

.

2

Remark 3.3. The parabolicstability restriction

t

=

O( ϑ

2

₎

_which_appears _in_Proposition_3.2_can_be_avoided _if_we _use_a semi-implicitschemeasin[44].Tobemoreprecise,letusconsider