An accurate front capturing scheme for tumor growth models with a free boundary limit

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Journal of Computational Physics

www.elsevier.com/locate/jcp

An accurate front capturing scheme for tumor growth models with a free boundary limit

Jian-Guo Liu

^a

, Min Tang

^b

, Li Wang

^c

, Zhennan Zhou

^d,∗

aDepartmentofMathematicsandDepartmentofPhysics,DukeUniversity,UnitedStates

bDepartmentofMathematics,InstituteofNaturalSciencesandMOE-LSC,ShanghaiJiaoTongUniversity,China

cDepartmentofMathematics,ComputationalandData-EnabledScienceandEngineeringProgram,StateUniversityofNewYorkatBuffalo, UnitedStates

dBeijingInternationalCenterforMathematicalResearch,PekingUniversity,China

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

Articlehistory:

Received29August2017

Receivedinrevisedform6March2018 Accepted7March2018

Availableonline9March2018

Keywords:

Hele–Shawequation Freeboundaryproblem Frontcapturingscheme Tumorgrowthmodel Prediction-correctionmethod

We consider a class of tumor growth models under the combined effects of density- dependent pressure and cell multiplication, witha freeboundary model as itssingular limitwhenthepressure-densityrelationshipbecomeshighlynonlinear.Inparticular,the constitutive law connecting pressure p and density ρ is p(ρ)= m^m−¹ρ^m−1, and when m^{1,thecelldensity}ρmayevolveitssupportaccordingtoapressure-drivengeometric motionwithsharpinterfacealong itsboundary.Thenonlinearityand degeneracyinthe diffusion bring great challenges in numerical simulations. Prior to the present paper, thereislackofstandard mechanismtonumericallycapture thefrontpropagationspeed as m^{1.In thispaper, wedevelop anumerical schemebased onanovel} prediction- correction reformulation that can accurately approximate the front propagation even when the nonlinearity is extremely strong. We show that the semi-discrete scheme naturallyconnects tothe free boundarylimit equationasm→ ∞^{.With proper spatial} discretization, thefully discrete schemehasimprovedstability, preserves positivity,and can beimplementedwithoutnonlinearsolvers.Finally,extensivenumericalexamples in both one and two dimensions are provided to verify the claimed properties invarious applications.

©^{2018ElsevierInc.Allrightsreserved.}

1. Introduction

In this paper, we are concerned witha tumor growth model for the cell density function

ρ

(x,t), whose governing equationisgivenby:

∂

∂

t

ρ _{− ∇ ·} ( ρ _∇

p

( ρ )) = ρ

G

(

c

,

p

),

x

∈ R

^d

,

t

≥

⁰

,

(1) wherec(x,t)representsthenutrientconcentrationandisauniformlyboundedfunction; p(

ρ

)isthepressurethatdepends onthecelldensitythroughastateequation.Inthepresentpaper,weconsider

p

( ρ ) =

^m

m

−

¹

ρ

^m−1

,

(2)

*

Correspondingauthor.

E-mailaddress:zhennan@bicmr.pku.edu.cn(Z. Zhou).

https://doi.org/10.1016/j.jcp.2018.03.013

0021-9991/©^{2018ElsevierInc.Allrightsreserved.}

as in[25] andother possible forms can be found in[28]. The tumor cells are transportedaccordingto the Darcy’s law, whosevelocityisdeterminedbythenegativegradientofthepressure,andtheyproliferatewithagrowthratethatdepends onboththenutrientconcentrationandthepressure.ThegrowthrateG(c,p)isassumedtobemonotonicallyincreasingin the c variable,monotonicallydecreasinginthe p variableandmaytakenegative values,i.e., _∂^∂_cG(c,p)≥^0, _∂^∂_p^G(c,p)≤^0.

Wecomplementthissystemwithaninitialconditionthatsatisfies

ρ (

0

,

x

) = ρ

ⁱⁿⁱ

(

x

)≥

⁰

, ρ

ⁱⁿⁱ

∈

^L1

∩

^L∞

.

(3) It has been proved in [25] that, under some assumptions for the initial density, when m→ ∞^{, the solution of (1)} convergestothesolutionofafreeboundaryproblemsupportedon(t).Thegeometricmotionof(t)isgovernedbythe limitingpressurep_∞.Tounderstandthis,wefirstderivetheequationforp.Multiplyingequation(1) byp(

ρ

),itbecomes

∂

∂

tp

= ρ

p

( ρ )

p

+ |∇

^p

|

²

+

^p

( ρ ) ρ

G

(

c

,

p

) ,

andsince p(

ρ

)=_m^m₋₁

ρ

^m−1,wefindtheequationforpthatwrites

∂

∂

tp

= (

m

−

¹

)

p

p

+ |∇

^p

|

²

+ (

m

−

¹

)

pG

(

c

,

p

) .

(4)

Sendingm→ ∞^{intheaboveequation,weformallyhavethefollowing}‘complementaryrelation’:

p_∞

p_∞

+

^G

(

c

,

p_∞

)

=

⁰

.

(5)

(t)isthesupportofp_∞ andthenormalvelocityofitsboundary∂isgivenby v= −∇^p∞· ˆ^n,wherenˆ(x,t)istheunit outer normaldirectionontheboundary.Inthelimit ofm→ ∞^,theconstitutiverelation(2) isnolongersatisfiedby

ρ

_∞

and p_∞.Instead,ifstarts withacharacteristicfunction,

ρ

_∞ remains acharacteristicfunctionof(t) alongdynamics,and p_∞satisfies

p_∞

∈

^P_∞

( ρ ) =

0

,

0

≤ ρ

_∞

<

1

,

[

⁰

, ∞ ), ρ

_∞

₌

1

.

⁽⁶⁾

The limit ofm→ ∞ connectstwo differentkinds ofdescriptions ofsolid tumor: one describesthedynamics ofthe cell population density,andtheother considersthe‘geometric’motionofthesolid tumorbyfree boundaryproblems.Similar limitshavealso beenconsideredin thecongested crowdtransport models,orthe congestedaggregation models[1,4,12].

Besides,thereareothertypesoftumor-growthmodelsthatconsistofCahn–Hilliardequationsthatcanyieldafreeboundary probleminthesharpinterfacelimit,e.g.[17,29].

In terms ofnumerics, there are severalchallenges insimulating the tumor growthmodels withm>1. The first one is due to the degeneracy of the diffusion in (1), whose solution profile has sharp interfaces near its support and the boundaries ofthe supportpropagate with afinite speed [30]. Owing to thelack ofsmoothness ofdegenerate problems, parabolic solvers may lose the convergence order, which results in incorrect propagation speed of the sharp interfaces.

Many numericalmethods havebeen proposed forthe simulations ofdegenerate parabolicequations, includingthefinite element method[2,6],finitevolumescheme[5,13],finitedifferencemethod[19,22],relaxationschemewhichexhibitsthe meritoftheJin–Xinrelaxationmodel[18,23],discontinuousGalerkin method[32],orsomeapproachbasedonperturbation andregularization [27]. However,to thebest ofourknowledge,no existing numericalmethodshaveever investigatethe possibilityofpreservingthefreeboundarylimitofthedegeneratereaction-diffusionequation.

The second challenge lies inthe nonlinearterm, which becomes moresevere when m^{1. Thereasonis that, when} m^{1,numericalerrorsinthedensity}

ρ

willbe greatlymagnifiedinthepressure pwhen

ρ

iscloseto1,seeFig.1for a schematicplot.Notehere,thepressure pstill takestheformof(2).Moreover,incorrectnumericalapproximationofthe pressure pimpliesincorrectsupportofthedensity

ρ

,whichconverselyresultsinnoticeableerrorinthedensity

ρ

.Onthe otherhand,ifonesimulatestheequationofpin(4) instead,smallerrorsinpwillinducelargeerrorsin

ρ

whenpisclose to0,asinFig.1.Thisresultsinnoticeableerrorsinthesupportof

ρ

.Therefore,strongnonlinearityduetolargemindeed raisesgreatcomputationalchallenge.

In general, thereare two traditional approachestohandle nonlinearterms.One is touse afully implicitschemeand solve theresultingdiscrete nonlinearsystemby iterativemethods. However,asm increases, thegrowingmultiplicity and stiffnessoftheJacobimatrixoftheresultingalgebraicsystemmaketheimplementationofiterativemethodsinfeasible.The other choice isto treat thenonlinear termsemi-implicitly,forexample,asinthe authors’previous work [20].Like most numericalschemes forporousmediatype equations,themethodin[20] leadsto satisfactoryresultsformoderatem,but whenmincreases,thereisafastdecreaseinthesizeofspatialgridsandtimestepsinordertogetaconsistentnumerical approximation.Thisisbecause,ifnotdoingso,eithertheschemeisunstableorthenumericalfrontpositiondeviatesfrom itstruelocation.Therefore,traditionalwaysoftreatingmoderatenonlinearityisnotenoughforthestrongnonlinearityhere, andnewmethodsneedtobedeveloped,whichisthegoalourpaper.

Fig. 1.Schematicplotoftheconstitutiverelationbetweenthedensityρandthepressurepforvariousm.Whenm1,theconstitutivelawgivesriseto greatnumericalchallenges.

Sincethenonlineardiffusiontermcausestheprimarychallengeinnumericalsimulations,toillustratethenoveltyofour scheme,weconsider

∂

t

ρ = ρ

^m

,

(7)

whichistheporousmediaequation[30],andagreeswiththetumorgrowthmodel(1) with G≡^{0.Theideaistointroduce} auxiliary quantities, like the velocity field u, then we can apply the semi-implicit time discretization to the augmented system,andthus,inthesemi-discretelevel,theaugmentedsystemnaturallyconnectsthefreeboundarylimit.

Ourapproachstarts withderivingtheequivalentvelocityequationforthecelldensitymodel.Specifically,werewritethe originalnonlineardiffusionequation (7) as

∂

t

ρ = ∇ ·

ρ (

m

ρ

^m−2

)∇ ρ := −∇ · ( ρ

u

) ,

whereusatisfies

u

= −

^m

ρ

^m−2

_∇ ρ _{= −}

^m

m

−

¹

∇ ρ

^m−1

,

canbeconsideredasvelocity.Recallthat _m^m₋₁

ρ

^m−1 canbeviewedastheeffectivepressure pandu= −∇^p.

Thenewvariableusuggestsawaytosingleoutthevelocityfromthedensityequation.Giventhefactthatvelocityfield playsthemajorroleinexpandingthefrontatthecorrectspeed,we comeupwiththefollowingnewsystemthat evolves thecelldensityandthevelocityfieldsimultaneously,iftheyareinitializedwithcompatibleprofiles:

∂

t

ρ + ∇ · ( ρ

u

) =

⁰

,

∂

tu

−∇

m

ρ

^m−2

∇ · ( ρ

u

)

=

⁰

.

Here theequation foruis obtainedbymultiplying both sidesofthedensityequation by p(

ρ

)=^m

ρ

^m−2 andtakingthe derivativewithrespecttox.

However, aswe shallshow in Section 2, thissystem isnot stable inthe sense that perturbations inthe constitutive relationshipu= −_m^m₋₁∇x

ρ

^m−1doesnotdecayintime.Forthisreason,weproposethefollowingrelaxationsystem

⎧ ⎨

⎩

∂

_t

ρ _{+ ∇ ·} ( ρ

u

) =

⁰

,

∂

tu

−∇

m

ρ

^m−2

∇ · ( ρ

u

)

= −

¹

²

⁽

^u

⁺

m

m

−

¹

∇ ρ

^m−1

),

⁽⁸⁾

where

1 is the relaxationparameter. Since we artificiallyexpand the size of the systemof equation, andthe extra relaxation term in the above system will reinforce the constitutive relation and thus helps to stabilize the discrepancy betweentheauxiliaryquantityuanditsconsistentrepresentation.

Weshallshowinthepaperthat,as →^{0,thetimesplittingmethodtotherelaxationsystem(8) leadstoasystemof} augmenteddifferentialalgebraicequations(DAEs),whichcanbeunderstoodasapredictioncorrectionmethod.Inaddition, forthetumorgrowthmodel,theproposednumericalmethodfortheaugmentedDAEsiscompatiblewiththefreeboundary limitofthecelldensitymodel,andisshowntocapturethecorrectfrontspeedofthemovingboundaries.

Therestofthepaperisorganizedasfollows.Therelaxationsystemofthecelldensitymodelareformulatedandanalyzed in Chapter2. We alsoshow augmented DAEs asthe limitingsystemof therelaxation system. In Chapter3, we propose

the prediction correction method of the augmented DAEs in the semi-discrete level, andproves how it connectsto the free boundarylimit—the Hele–Shawmodel.The fullydiscretescheme ispresentedinChapter4, alongwithsome further numerical analysisresults.At last, inChapter 5 we confirmthe propertiesof ournumerical methodwithnumerous test examples,andwealsoexploretheapplicationsoftheschemeforafewextendedmodels.

2. Relaxationsystemandthepredictioncorrectionformulation 2.1. Reformulatingtheuequation

In this section, we focus on the density equation (1), and formulate a relaxation systemout ofit, which provides a superior frameworkforaccurate numericalapproximations. Asbrieflymentioned intheintroduction,letusfirstintroduce thevelocityfieldthattransportsthecellpopulation:

u

= −∇

^p

( ρ ) = −

^m

m

−

¹

∇ ρ

^m−1

,

(9)

thentheequationfor

ρ

reads

∂

t

ρ + ∇ · ( ρ

u

) = ρ

G

(

c

) .

(10)

Obviously, together with(9), the reformulation (10) is as difficult to solve numerically asthe original one. However, as discussed inthe introduction,one canderive an equation forufromthe equation for

ρ

andevolve the celldensityand thevelocity fieldsimultaneously.When

ρ

anduareinitializedwithcompatibleprofiles,thefollowingtransportsystemis consideredinstead:

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

∂

_tu

=

^m

∇

ρ

^m−2

_{∇ ·} ( ρ

u

) − ρ

G

(

c

) ,

∂

t

ρ + ∇ · ( ρ

u

) = ρ

G

(

c

),

u

(

x

,

0

) = −

^m

m

−

¹

∇ ρ

^m−1

(

x

,

0

).

(11)

Note thatherethe firsttwo equationsareequivalent to theoriginal celldensityonlywhen thelast equationis satisfied.

Indeed,ifweintroducethediscrepancyterm W

=

^u

+

^m

m

−

¹

∇ ρ

^m−1

,

itisstraightforwardtoverifythat

∂

_tW

(

x

,

t

) =

⁰

,

W

(

x

,

0

) =

⁰

.

Therefore,weconcludeW(x,t)=^0,andtheconsistencycondition(9) isalwayssatisfied.

However,whensolvingthesystem(11) numerically,thediscretizationerrordestroystherelation(9).Infact,W deviates from0 at everytime step,andsincethere isnodampinginthe W equation, thelocaltruncationerroraccumulatesand leads to a largeglobalerror. Itis worth emphasizingthat, W isnot a directquantity ofinterest, however,itdetermines whetherthetransportsystemisanequivalentreformulationoftheoriginaldensityequation (1).

Toillustratethepropagationofthediscrepancywhennumericalerrorispresent,weaddasmallperturbationto(11):

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

∂

tu

=

^m

∇

ρ

^m−2

_{∇ ·} ( ρ

u

) − ρ

G

(

c

) + δ

1

,

∂

t

ρ + ∇ · ( ρ

u

) = ρ

G

(

c

) + δ

2

,

u

(

x

,

0

) = −

^m

m

−

1

∇ ρ

^m−1

(

x

,

0

).

(12)

Here, δ1,δ2 are smallperturbationfunctions(innumericalschemes, theyare justlocaltruncationerrors).Then by direct calculation,weget

∂

_tW

(

x

,

t

) = δ

₁

+

^m

∇ ( ρ

^m−2

δ

₂

).

Clearly,^W(·,^t)L^∞ mayincreaselinearlyintime,whichbreakstheconsistencycondition(9) inthetransportsystem(11).

Inparticular,^W(·,t)L^∞ mayincreasefasterwhenmislarge.

2.2. Therelaxationsystemanditsproperties

Toensure theconsistency condition, we insteadconsider thefollowing equationfor thediscrepancywithan artificial damping

∂

tW

(

x

,

t

) = −

¹

^2W

⁽

^x

^,

^t

^),

⁽¹³⁾

where0<

1 istherelaxationconstant.Thisdiscrepancyequationleadstothefollowingrelaxationsystem

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

∂

tu

=

^m

∇

ρ

^m−2

∇ · ( ρ

u

) − ρ

G

(

c

)

−

¹

²

u

+

^m

m

−

¹

∇ ρ

^m−1

,

∂

t

ρ _{+ ∇ ·} ( ρ

u

) = ρ

G

(

c

),

u

(

x

,

0

) = −

^m

m

−

¹

∇ ρ

^m−1

(

x

,

0

).

(14)

Correspondingly,therelaxationsystemwithperturbationsisgivenby

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

∂

_tu

=

^m

∇

ρ

^m−2

_{∇ ·} ( ρ

u

) − ρ

G

(

c

)

−

¹

²

u

+

^m

m

−

¹

∇ ρ

^m−1

₊ δ

₁

,

∂

t

ρ + ∇ · ( ρ

u

) = ρ

G

(

c

) + δ

2

,

u

(

x

,

0

) = −

^m

m

−

¹

∇ ρ

^m−1

(

x

,

0

),

(15)

whereagain,δ1,δ2 aresmallperturbationfunctionsthatmayrepresentlocaltruncationerrorsfromnumericalschemes.

Asweshallseebelow,theadvantageoftherelaxationsystem(14) canbeseenintwoways.Firstofall,when issmall, theconsistencycondition (9) iseffectivelypreservedalongthe dynamicseveninthepresenceofperturbations. Secondly, therelaxationtermin(15) canbe shownofthesameorderaslocalperturbations, sothatevenwithperturbationstheu equationin(15) stillwellapproximatestheuequationin(11).Thispropertyisimportantastheaccuracyinapproximating udeterminestheaccuracyofthefrontposition.Inparticular,intheabsenceofperturbation,therelaxationtermisexactly 0.

Morespecifically,weshowinthefollowingthat1)when →^0,W(x,t)→^{0 fort}≥^{0,whichindicatesthe}preservation oftheconsistencycondition(9);2)therelaxationtermsin(14) ^W(x,t)

² isofthesameorderastheperturbations.

For(15),wefindthat W(x,t)satisfies

∂

_tW

(

x

,

t

) = −

¹

^2W

⁽

^x

^,

^t

^{) + δ}

¹

⁺

^m

^{∇ (} ρ

^m−1

δ

₂

) ,

whosesolutioncanberepresentasfollows W

(

x

,

t

) =

^exp

−

^t

²

W

(

x

,

0

) +

t 0

exp

−

^t

− τ

²

^δ

¹

⁽ τ ) +

^m

∇ ( ρ

^m−1

δ

2

)( τ )

d

τ .

Then,withintegrationbyparts,weget W

(

x

,

t

)

²

⁼

1

^2exp

−

^t

^{2 W}

⁽

^x

^,

⁰

^{) −}

²

δ

1

(

0

) −

²m

∇( ρ

^m−1

δ

2

)(

0

)

+ δ

1

(

t

) +

^m

∇( ρ

^m−1

δ

2

)(

t

)

−

t 0

exp

−

^t

− τ

²

∂

_τ

δ

1

( τ ) +

m

∇( ρ

^m−1

δ

2

)( τ )

d

τ ,

whichindicatesthatwhen →^0, W

(

x

,

t

)

²

^{= δ}

¹

⁽

^t

^{) +}

^m

^∇( ρ

^m−1

δ

2

)(

t

) +

^O

(

²

).

Obviously, ^W(x,t)

² isatthesameorderofδ1 andδ2 forsmall ,andwhen →^0,W(x,t)→^{0 fort}≥^0.

We shallfurther explore therole of therelaxationterm by theChapman–Enskog expansion belowsimilar to what is donein[18].ConductingtheChapman–Enskogexpansiontotherelaxationsystem(14),weget

u

= −

^m

m

−

¹

∇ ρ

^m−1

−

²

∂

tu

+

²m

∇

ρ

^m−2

∇ · ( ρ

u

) − ρ

G

(

c

)

= −

^m

m

−

¹

∇ ρ

^m−1

−

²

∂

t

−

^m

m

−

¹

∇ ρ

^m−1

−

²

∂

tu

+

²m

∇

ρ

^m−2

∇ · ( ρ

u

) − ρ

G

(

c

) +

²m

∇

ρ

^m−2

∇ · ( ρ

u

) − ρ

G

(

c

)

= −

^m

m

−

¹

∇ ρ

^m−1

+

⁴

∂

t

∂

tu

−

^m

∇

ρ

^m−2

∇ · ( ρ

u

) − ρ

G

(

c

)

.

Ifwecontinuethecalculations,formally,allhigherordertermswillcanceleachother,whichimpliesthat(14) isequivalent to(1),notonlytotheleadingorder,butalsoateach higherorder.Thisisnotsurprisingsinceanalyticallywe alwayshave theconsistencycondition(9) andtheextrarelaxationtermissimplyzero,andthus(14) isequivalentto(11).Ontheother hand,whensolving(11) numerically,theconsistencycondition isnolongersatisfiedwithfullaccuracy,andtherelaxation term in(14) will add extra contributions to counteractthe perturbations causedby numericalapproximations sothat it reinforcestheconsistencyrelationbetweenuand

ρ

withoutaddinganydissipation.Therefore,therelaxationtermin(11), althoughremainingzeroanalytically,playsancrucialroleinnumericalapproximation.Weremarkthat,thesameresultcan alsobeobtainedbyapplyingChapman–Enskogexpansionto(13),whichisequivalentto

∂

t

u

+

^m

m

−

1

∇ ρ

^m−1

= −

¹

²

u

+

^m

m

−

1

∇ ρ

^m−1

.

2.3. Theprediction-correctionmethod

Basedontheanalysisintheprevioussections,wechoosetosolve(14) insteadandproposethefollowingtimesplitting method:

⎧ ⎨

⎩

∂

t

ρ + ∇ · ( ρ

u

) = ρ

G

(

c

),

∂

tu

=

^m

∇

ρ

^m−2

∇ · ( ρ

u

) − ρ

G

(

c

) ,

⎧ ⎨

⎩

∂

t

ρ =

⁰

,

∂

tu

= −

¹

²

⁽

^u

⁺

m

m

−

¹

∇ ρ

^m−1

) .

⁽¹⁶⁾

At every time step,given (

ρ

ⁿ,uⁿ),one solves the left systemin(16) foronetime step t andobtains the intermediate values(

ρ

^∗,u^∗),andthensolvethesecondsystemofequationsin(16) toget(

ρ

ⁿ⁺¹,uⁿ⁺¹).

When →^{0,thesecondstepin}(16) reducesto

∂

t

ρ =

⁰

,

u

(

x

,

t

) = −

^m

m

−

¹

∇ ρ

^m−1

(

x

,

t

),

(17)

whichcanbeunderstoodasaprojectionstep.Notethatintheprojectionstep,

ρ

isaconstantintime,namely

ρ

^∗=

ρ

ⁿ⁺¹. Therefore,thetimesplittingmethodforthefullyrelaxedsystem(

₌0)becomes:

⎧ ⎨

⎩

∂

_t

ρ _{+ ∇ ·} ( ρ

u

) = ρ

G

(

c

),

∂

tu

=

m

∇

ρ

^m−2

∇ · ( ρ

u

) − ρ

G

(

c

)

,

^u

(

x

,

t

) = −

^m

m

−

¹

∇ ρ

^m−1

(

x

,

t

).

(18)

We usethe first two equations to numericallyevolve thesystems while takingthe last equation asa constraint, which induces the followingprediction-correctionmethod: ateachtime step,we firstsolve theleft two equationsin(18) from

ρ

ⁿ,uⁿ

to obtain

ρ

ⁿ⁺¹,u^n∗

(herein

ρ

equation, the convection term istreated semi-implicitly, i.e., ∇ ·(

ρ

ⁿu^n∗)), and then usetherightequationtoenforceuⁿ⁺¹= −_m^m₋₁∇(

ρ

ⁿ⁺¹)^m−1.Hereu^n∗ isanintermediatevaluewhichisonlyusedto help compute

ρ

ⁿ⁺¹ better. Actually,u^n∗ can bea goodapproximation to u(tⁿ⁺¹), but, u^n∗ and

ρ

ⁿ⁺¹ maynot satisfy the consistencycondition (9).

Weremarkthat,theintroductionofu^n∗ givesusthefreedomtosolvefor

ρ

ⁿ⁺¹ stablyandaccuratelywithoutworrying aboutthe constraint(9). Hence, u^n∗ can be viewedasa prediction,whichis correctedby theconsistency condition. The correctionisessentiallyaprojectionontothesolutionmanifold(asshowninFig.2),whichisnotcarriedoutonu^n∗directly, butbyusing

ρ

ⁿ⁺¹ andtheexplicitconsistencycondition(9).Itisworthmentioningthatsimilarprojectionideashavebeen introduced tootherequationsinnumericalsimulations,liketheincompressibleflows ([10,11,15])andtheLandau–Lifshitz equation ([8,31]).

3. TimediscretizationandconnectionstotheHele–Shawmodel

Inthissection,weaimtoproposeasemi-discreteformof(18) thatyieldsagoodstabilitycondition.Notethatalthough in theprediction stepof(18),the two equationsare linearrespectively in

ρ

andu, they arestill nonlinear equations.If we naivelyupdate

ρ

andu explicitlythere,itisequivalenttoexplicitlyupdatingoriginalequation (1),whichissubjectto severe stability constraintsthathighly dependonthestrength ofthenonlinearityencodedby m.Instead,weconsideran implicit-explicitdiscretizationfor(18) asfollows:

u^n∗

−

^un

t

=

^m

∇

( ρ

ⁿ

)

^m−2

∇ · ( ρ

ⁿu^n∗

) − ρ

ⁿG

(

cⁿ

,

p

( ρ

ⁿ

))

,

(19)

ρ

ⁿ⁺¹

− ρ

ⁿ

t

= −∇ · ( ρ

ⁿu^n∗

) + ρ

ⁿ⁺¹G

(

cⁿ

,

p

( ρ

ⁿ

)) ,

(20) uⁿ⁺¹

= −

^m

m

−

¹

∇( ρ

ⁿ⁺¹

)

^m−1

.

(21)

Fig. 2.Schematicplotoftheprediction-correctionnumericalsimulationoftheaugmenteddifferentialalgebraicequations(18).Thecorrectionisonlydone ontheucomponentbyprojectingontothesolutionmanifold.

Inthisscheme,each equation can besolved consecutively,andeach ofthem isonlysemi-implicit, whichmeansthat no nonlinearsolverisneededinimplementingthescheme.Weremarkthatsemi-implicitmethodsforasimilarDarcy-coupled modelwithHele–Shawlimithasbeenconsideredin[14].Fortheeaseofanalysis,weassumeGisaconstantinthissection to conduct the numerical analysis. In the numerical examples, we shall instead consider more generaltime andspatial dependentG invariousexamples.

3.1. TheconnectiontotheHele–Shawmodel

Inthefreeboundarylimit,m→ ∞^{,withproperlypreparedinitial}conditions,thedensityfunction

ρ

behavesasatime dependentcharacteristicfunctionwithasharpfront,whosegeometricmotionisgovernedbytheHele–Shawmodelforthe pressure(seee.g. [25]).

In orderto examine theconnection of thesemi-discrete scheme to theHele–Shaw model,we aimto show that u^n∗, computedfrom(19) andusedin(20) topropagate

ρ

ⁿ⁺¹,satisfiestheHele–Shawmodelwhenm→ ∞^,witht fixed.We point outthat ourcomputation hereis formal,andforthisreason, we assumethat the limitsof

ρ

ⁿ and pⁿ existswhen m→ ∞^,i.e.,

ρ

ⁿ→

ρ

_∞ⁿ,pⁿ→^pn_∞^.

Form^{1,therighthandsideof(19) is}dominating,andtheleadingorderwithrespecttom reads

0

= ∇

( ρ

ⁿ

)

^m−2

∇ · ( ρ

ⁿu^n∗

) − ρ

ⁿG

= ∇

( ρ

ⁿ

)

^m−1

∇ ·

^un∗

+ ( ρ

ⁿ

)

^m−2

( ∇ ρ

ⁿ

) ·

^un∗

− ( ρ

ⁿ

)

^m−1G

.

Bythedefinitionofthepressurep(

ρ

),andsinceuⁿ= −_m^m₋₁∇(

ρ

ⁿ)^m−1 fromthepreviousstep,weobtain

0

= ∇

_m

−

¹

m p

( ρ

ⁿ

) ∇ ·

^un∗

−

¹

muⁿ

·

^un∗

−

^m

−

¹ m p

( ρ

ⁿ

)

G

.

Form^1,thiseffectivelyreducesto

0

= ∇

p

( ρ

ⁿ

)

∇ ·

^un∗

−

^G

,

whichindicatesthatthereexistsascalarconstantathatisindependentofspace,suchthat p

( ρ

ⁿ

)

∇ ·

^un∗

−

^G

=

^a

.

Notethat,duetothedegeneratediffusion,ifweinitiallystartwithadensitythat hascompactsupport,then

ρ

aswell as thepressurep(

ρ

ⁿ)remaincompactlysupported,whichimplies

p

( ρ

ⁿ

)

∇ ·

u^n∗

−

G

=

0

.

Therefore,ifwedenotethesupportof p(

ρ

ⁿ)byn,thenthevelocityfieldu^n∗ satisfies

∇ ·

^un∗

−

^G

=

⁰

,

x

∈

n

.

(22)

From (19), since p(

ρ

ⁿ)=^{0 outside of} n, we have u^n∗=^{un for x}∈R^d/¯n. From (21) of the previous step, uⁿ is supportedinn,soisu^n∗.Theabovelimitisforthevelocityfieldu^n∗,whichiscloselyrelatedtothelimitingpressure.We furthershowthatthevelocityfieldcanbewrittenasthegradientofascalarfunction,whichplaystheroleofthepressure.

Weobservethat,(19) canberewrittenas u^n∗

=

^un

+ ∇

m

t

( ρ

ⁿ

)

^m−2

∇ · ( ρ

ⁿu^n∗

) − ρ

ⁿG

,

and recall uⁿ= −_m^m₋₁∇(

ρ

ⁿ)^m−1,then u^n∗ can also be written asthe gradientof a scalarfunction, which we denoteby

−^{pn∗,namelyun∗}= −∇^pn∗.Thus,(22) becomes

−

p^n∗

=

^G

,

x

∈

_n

.

It is worth emphasizingthat we didnot assume p^n∗ is thesolution toa Hele–Shawflow model,theexistence of p^n∗ is onlyfromthefactthatu^n∗ isthegradientofascalarfunction.Besides,itisobviousthattheexistenceofp^n∗ isnotunique, whichleadstothefollowingdiscussiononthenonuniquenessoftheboundaryconditionson∂n.

Next,wetrytofindtheboundaryconditionfor p^n∗ on∂n byassumingthat p^n∗ iscontinuousacross n.Duetothe definitionof p^n∗ suchthat u^n∗= −∇^{pn∗ andthefactthatun∗ issupportedin}n,we concludethat p^n∗ isconstant along

∂n.Thenwehaveforsomeconstanta,

−

p^n∗

=

^G

,

x

∈

n

,

p^n∗

=

^a

,

x

∈ ∂

n

.

⁽²³⁾

The solutionsto (23) yieldthesameu^n∗= −∇^{pn∗ fordifferentconstanta.} Especially,we canview u^n∗ as−∇^pn₀^∗,where pⁿ₀^∗solvestheHele–Shawmodel

−

^pn₀^∗

=

^G

,

x

∈

n

,

pⁿ₀^∗

=

⁰

,

x

∈ ∂

n

.

⁽²⁴⁾

We remark that, forpositive G,due to theconcavity of pⁿ₀^∗ within n, although pⁿ₀^∗ is continuous in thewhole space,

∇^{pn∗ mayhavejumpsacross}n,whichreflectsthejumpsinu^n∗ acrossn.

To sum up, for m^{1 and fixed} t, due to (21) from the previous time step, and (19), the velocity field of the semi-discrete numerical schemeisthe sameasthevelocity field givenby thelimitingHele–Shawmodel.Then thefront propagatesaccordingtothevelocityfieldu^n∗ from(20),whichisthesameasthefreeboundarymodel.

3.2. Stabilityanalysis

Whenm>1,itisdifficulttoanalyzethestabilityofthesemi-discreteschemeduetononlinearity.However,form=2, thenonlinearityinthedegeneratediffusionisonlyquadratic,andtheintroductionofthevelocityumakesstabilityanalysis oftheaugmentedDAEs(19)–(21) feasible.Similarobservationhasbeenmadeinarecentwork[9].Forsimplicity,westill assumethat Gisaconstant,andtheextensionstomoregeneralGfunctionswillbediscussedlaterinthissection.

Heuristicargumentform>1:Inobtaining u^n∗ fromequation (19),therighthandside istreatedsemi-implicitly:only the velocityon therighthandside istakenimplicitlywhiletherestofthe termsare stilltakenexplicitly.Therefore, the stability may be compromised for efficiency. Besides, in solving for

ρ

ⁿ⁺¹ from equation (20) with u^n∗ and cⁿ⁺¹ given, the convectionterm onthe righthandside istreatedexplicitly. Therefore,the time stept needs to satisfy theregular hyperbolicCFLconstraint.

However, duetothehighnonlinearityofthesystem, itisimpossibleforusto derivethesufficientstability condition, butweshallnumericallyverifythat,withproperspatialdiscretization,theresultingschemeisuniformlyaccurateinmand the stability conditionis notvery sensitivetom (see examplesin Section5.1). Wewantto emphasizethat suchstability constraintsistotallyacceptable.Fromtheperspectiveofmodeling,wedonotneedm tobeverylarge,forexample,m=⁸⁰ is big enough tomake the cell densitymodelbehave almost the sameasthe free boundarydynamics (see examples in Section5.2and5.3).

Rigorousstabilityanalysisform=^2:Whenm=^2,thesemi-discretization(19)–(21) reducesto:

u^n∗

−

^un

t

=

²

∇

∇ · ( ρ

ⁿu^n∗

) − ρ

ⁿG

,

(25)

ρ

ⁿ⁺¹

₋ ρ

ⁿ

t

= −∇ · ( ρ

ⁿu^n∗

) + ρ

ⁿ⁺¹G

,

(26)

uⁿ⁺¹

= −

²

∇ ρ

ⁿ⁺¹

.

(27)

Clearly,(25) isequivalentto u^n∗

−

^un

t

=

²

∇∇ · ( ρ

ⁿu^n∗

) +

^unG

,

(28)

wherewehaveuseduⁿ= −²∇

ρ

ⁿ.Takethespatialgradientof(26) andmultiplyby−^2,weget uⁿ⁺¹

−

uⁿ

t

=

²

∇∇ · ( ρ

ⁿu^n∗

) +

^un+1G

.

(29)

Subtracting(28) from(29),wesee uⁿ⁺¹

−

^un∗

t

= (

uⁿ⁺¹

−

^un

)

G

,

(30)

whichimplies

u^n∗

=

^un+1

−

tG

(

uⁿ⁺¹

−

^un

).

(31)

Itisimportanttonotethatthisrelationtellsthatu^n∗ isindeedagoodpredictionofuⁿ⁺¹ sincetheirdiscrepancyisoforder O(t²)whenuⁿ⁺¹−^{un isatorder}t.

Nowmultiplyingbothsidesof(26) by

ρ

ⁿ⁺¹t,integratinginspace,onegets 1

2

ρ

ⁿ⁺¹

²

−

¹

2

ρ

ⁿ

²

+

¹

2

ρ

ⁿ⁺¹

− ρ

ⁿ

²

= −

^t

R^d

∇ · ( ρ

ⁿu^n∗

) ρ

ⁿ⁺¹dx

+

tG

ρ

ⁿ⁺¹

²

.

(32)

Here· ^{representstheL2 norm.Ifweassumethattheschemeispositivity}preserving,i.e.,

ρ

ⁿ≥^0,thenwithintegration bypartsand(27),welearnthat

−

R^d

∇ · ( ρ

ⁿu^n∗

) ρ

ⁿ⁺¹dx

= −

¹ 2

R^d

ρ

ⁿu^n∗

·

^un+1dx

.

Therefore,from(31),weobtain

−

¹ 2

R^d

ρ

ⁿu^n∗

·

^un+1dx

= −

¹ 2

R^d

ρ

ⁿ

|

^un+1

|

^2dx

+

^O

(

t

).

(33)

Hence,weconcludethat

1 2

−

^G

t

ρ

ⁿ⁺¹

²

_≤

¹

2

ρ

ⁿ

²

₊

O

(

t²

).

(34)

Thisimpliesthefollowingstabilityestimatefor ¹₂−^Gt>0,

ρ

ⁿ

²

≤ ¯

C1e^{2G T}

ρ

⁰

²

+ ¯

C2

≤ ¯

C e^{2G T}

ρ

⁰

²

,

n

t

<

T

,

whereC¯1,C¯2 andC¯ onlydependonthefinaltimeT.Here,wehaveassumedthat

ρ

⁰isfinite.

Inconclusion,wehaveprovedthefollowingstabilityestimate

Theorem3.1.ForsomeT>0andG>0,ifGt<¹₂,andifthesemi-discretescheme(25),(26),(27)ispositivitypreserving,thenthe solutionstothisschemesatisfy

ρ

ⁿ

L²

≤

^{C eG T}

ρ

⁰

L²

, ∀

ⁿ

t

<

T

,

(35) wheretheconstantC onlydependsonT andG.

Althoughthe theoremabovedoesnot necessarilyimplythe stability constraintofafullydiscrete scheme,it indicates thatourtimediscretizationrespectsthestabilityestimateoftheoriginaldensitymodel(1).

4. Thefullydiscretescheme

In thissection, weintroduce the spatial discretizationto obtain thefully discrete scheme.We firstconstruct thedis- cretization for (19)–(21) in one spatial dimension, and then extend it to 2D. Higher dimensional extensions should be similar.Wewillfocusourattentionexclusivelyonthediscretizationof

ρ

andubyassumingthatGisgiven,anddiscretiza- tionforcisstraightforward,see[21].Wealsolookintothescheme’spropertyonpositivityandestablishaformal energy estimate.