An asymptotic-preserving method for a relaxation of the Navier-Stokes-Korteweg equations

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An asymptotic-preserving method for a relaxation of the Navier-Stokes-Korteweg equations

Alina Chertock, Pierre Degond, Jochen Neusser

To cite this version:

Alina Chertock, Pierre Degond, Jochen Neusser. An asymptotic-preserving method for a relaxation

of the Navier-Stokes-Korteweg equations. Journal of Computational Physics, Elsevier, 2017, 335,

pp.387-403. �10.1016/j.jcp.2017.01.030�. �hal-02498895�

Contents lists available atScienceDirect

Journal of Computational Physics

www.elsevier.com/locate/jcp

An asymptotic-preserving method for a relaxation of the Navier–Stokes–Korteweg equations

Alina Chertock^a,Pierre Degond^b,∗, Jochen Neusser^c

aDepartmentofMathematics,NorthCarolinaStateUniversity,Raleigh,NC27695,USA bDepartmentofMathematics,ImperialCollegeLondon,LondonSW72AZ,UK

cUniversitätStuttgart,InstitutfürAngewandteAnalysisundNumerischeSimulation,Pfaffenwaldring57,D-70569Stuttgart,Germany

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

Articlehistory:

Received15December2015

Receivedinrevisedform20December2016 Accepted17January2017

Availableonline24January2017 Keywords:

Asymptotic-preservingscheme Diffuse-interfacemodel

Compressibleflowwithphasetransition

The Navier–Stokes–Korteweg (NSK)equations are a classicaldiffuse-interface model for compressibletwo-phaseflows.AsdirectnumericalsimulationsbasedontheNSKsystem are quiteexpensive andinsomecaseseven impossible,weconsiderarelaxationofthe NSKsystem,forwhichrobustnumericalmethodscanbedesigned.However,timestepsfor explicitnumericalschemes dependonthe relaxationparameterandthereforenumerical simulations inthe relaxationlimit are veryinefficient.To overcome thisrestriction, we proposean implicit–explicitasymptotic-preservingfinite volumemethod. Weprovethat thenewschemeprovidesaconsistentdiscretizationoftheNSKsystemintherelaxation limitanddemonstratethatitiscapableofaccuratelyandefficientlycomputingnumerical solutionsofproblemswithrealisticdensityratiosandsmallinterfacialwidths.

©^{2017TheAuthor(s).PublishedbyElsevierInc.Thisisanopenaccessarticleunderthe} CCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

There are in general two approaches to describe the behavior ofmulti-phase fluids, the sharpinterface (SI) andthe diffuse interface (DI) approach. The first approach represents multiple phases with different sets of equations that are coupledby some interface conditions.Thesecond approach,which usedtodescribe, e.g. themergingprocess ofdroplets andbubbles,needsonlyonesetofequationstomodelthephasesanddoesnotrequirethelocationoftheinterface tobe trackedexplicitly.

Inthispaper,weconsidertwoDImodelsforahomogeneoustwo-phasecompressiblefluid:TheNavier–Stokes–Korteweg system(NSK)andarelaxationsystemfortheNSKsystem.TheNSKsystemgoesbacktotheworkofKorteweg[31]andwas formulatedinitspresentformin[18,2].TheNSKsystemusesaVan-der-Waalslikepressurefunctiontoidentifytwodistinct phasesandathird-ordertermtomodelphasetransitions.Manyauthorsachievedanalyticalresultsonthewell-posedness oftheNSKsystemanditsvariants,e.g.[4,7,24,32,37].Whilesomenumericalmethodshavebeensuccessfullydevelopedand usedfortheseandrelatedfamiliesofproblems,see,e.g., [6,8,16,21,25,39],there arestill manyopenproblems,forwhich accurateandefficientnumericalmethodsareyettobedesigned.Up toourknowledge,therobustcomputationofrealistic densityvaluesforliquidandvaporphaseshasnotbeensuggestedfortheNSKmodel.Additionally,numericalmethodsalso failincaseswhenverysmallinterface widthsclosetoasharpinterface aretobeconsidered.Inbothcases,theoccurring problems arerelated tosteep densitygradients. Anothersource of difficulty one comesacross while numericallysolving

* Correspondingauthor.

E-mailaddresses:chertock@math.ncsu.edu(A. Chertock),p.degond@imperial.ac.ukm(P. Degond),jochen.neusser@mathematik.uni-stuttgart.de (J. Neusser).

http://dx.doi.org/10.1016/j.jcp.2017.01.030

0021-9991/©^{2017TheAuthor(s).PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).

NSKsystemsisrelatedtotheVan-der-Waalslikeformofthepressureequation.Thelatterpreventsonefromusingupwind hyperbolic solvers, whichhave beensuccessfullyapplied, e.g.,to stabilizecomputationsforNavier–Stokes equationswith highReynoldsnumbers.

The issueofvery smallinterfacesisespeciallyimportantbecausetheNSKmodelcanonly providethecorrectamount ofcapillaryforcesiftheinterfaceisextremelysmall[17,25,27].One ideatoloosenthestrict couplingbetweeninterfacial width andcapillaryforces istointroduce an additionalCahn–HilliardorAllen–Cahn type equation foranewphase field variable.Thiswasdoneforexamplein[1,5,42].AnotheransatztoavoidsomeofthedifficultiesfortheNSKsystemssuggests tointroducearelaxationoftheNSKsystem,inwhichthethird-ordertermisreplacedbyafirst-ordertermandaPoisson equation, that defines a newphase field parameter, see, e.g., [37]. This modelis parametrized by a so-calledKorteweg parameter α. IftheKorteweg parameter tends tozero, therelaxationsystem formallyconverges to theNSKsystem. The most importantfeature of the relaxationsystem isthe fact, that the first-order partis purely hyperbolic for sufficiently small Korteweg parameter. It should also be observed that the addition ofthe Poisson equation to the systemdoesnot increase thecomputationalcostofnumericalsimulationsastest casesdemonstratethatthetime savings,thatcomefrom the fact that one does not have to solve a third order system, are greater than the loss of time that comes from the numerical solutionofthe Poissonequation. Thesepropertiescan be exploitedtoconstructrobust numericalschemes for therelaxationsystem.In[34],ithasbeenshownthattheoverallapproachisrobustforproblemswithlargedensityratios andsmallinterfacialwidths.However,thenumericalschemeproposedin[34]isanexplicitschemeandthusthetimesteps decreaseastheKortewegparameter αtendstozero.

It is the main purpose of thiscontribution to constructan asymptotic-preserving (AP) scheme [28] in the Korteweg limit, that is,a schemefortherelaxationsystemthat provides aconsistent approximation oftheoriginal NSKsystemas theKortewegparameter α tendstozero.TheAPapproachwasdevelopedintheframeworkoflineartransportindiffusive regimes[22,29,33]andhasbeenappliedtomanydifferentareas,e.g.fluidanddiffusionlimitsofkineticmodels,relaxation methodsforhyperbolicsystemsandlow-Mach numberlimitsforcompressibleflowproblems;see,e.g.,[11,12,14,15,20,23, 30,35,36].In[23],thetimeandspatial discretizationoftheisentropicEulerandNavier–StokesEquationsinthelowMach numberlimitwas investigated.Inspiredbythisresearch,weconstructhereaschemethatcapturestheKorteweglimitfor therelaxationsystemandprovetheAPpropertyoftheproposedscheme.TheAPpropertyofthenewschemeisachieved bysplittingtherelaxationsystemintoanon-stiffnonlinear,compressiblehyperbolicNavier–Stokeslikesystemandasystem that canbetreatedbyaPoissonsolver,andallowstheuseoftimeandspatialstepsthatareindependentoftheKorteweg parameter. Asthe resulttheproposednumericalscheme isveryefficientforsmallvaluesoftheparameter α,whichisa significant improvementcomparedto anexplicitschemefrom[34].Beyondthat,we expectourschemeto beasymptotic preservinginthesharpinterfacelimit.Wecannotgiveanalyticalprooftothat,butwesupportthisstatementbyanumerical example.

The paperisorganized asfollows.InSection 2,weintroduce theNSKandrelaxationsystemsanddescribe their main propertiestogether withthe basicthermodynamicalframework. Wecommentontheadvantagesoftherelaxationsystem andpointout whyitisnecessarytointroduceanewschemeinordertosolvetherelaxationsystemefficiently.Section 3 contains the basic outcome of this contribution. We propose the AP scheme forthe relaxation systemand perform an asymptotic analysisto show that the schemetransforms into a schemefor the NSKequations in theKorteweg limit.In Section4,wedemonstratethatthealgorithmprovidesamassiveimprovementcomparedtoastandardexplicitschemefor anumberofproblemsinoneandtwospacedimensions.

2. Navier–Stokes–Kortewegequations 2.1. TheNavier–Stokes–Kortewegsystem

Letanopen set⊆R^d,d∈ {¹,2,3}^{uptothefinaltime T}>0 begiven.TheisentropicNavier–Stokes–Korteweg(NSK) equationsinarbitraryspatialdimensionaregivenby

∂tρ_{+ ∇ ·}(ρv)=⁰,

∂t(ρv)+ ∇ ·(ρv⊗^v)+ ∇(p(ρ))= ∇ ·^Tε[^v] +γ ε²ρ_∇ρ,(x,t)∈×(0,T), (2.1) where ρ₌ρ(x,t)isthe densityofthefluid,v=(v1(x,t),...,v_d(x,t))^T∈R^{d isitsmomentumand p}(x,t)isthepressure.

Notethat εistheReynoldsnumberand γ ε² isthecapillarynumber.Wereferto[26,41]foradetailedexplanationonthe physical meaningofthescaling ε→^{0 and} γ =O(¹).The matrixTε[^v]∈R^{d×d in(2.1)stands fortheviscous partofthe} stresstensorwhichisgivenfortheviscositycoefficients ν,μ ∈R^with μ ≥^{0 and3}ν+²μ >0 by

T^ε_{i j}:=εν∇ ·(v)δi j+2εμDi j, Di j:=¹ 2

vj,x_i+vi,x_j

, (i,j∈ {1,2}). (2.2)

Weaugment(2.1)withtheinitialdata

ρ(x,0)=ρ0, v(x,0)=^v0, x∈, (2.3)

Fig. 2.1.Pressurepas a function ff densityρfor the valuesb1=b=3,T∗=0.85, andR=8.

andboundaryconditionsthatcorrespondtoaboundedbox:

v=⁰, ∇ρ·ⁿ=⁰, x∈∂. (2.4)

Todescribeatwo-phasefluidwechoosetheVan-der-Waalstypepressure p(ρ)= ^{R T∗}ρ

b−ρ ^−b1ρ². (2.5)

Thereby,b,b1,R are positive constants and T_∗ isthe fixed temperature. If T_∗ is chosen smallenough, the pressure p is monotonedecreasinginsomenon-emptydensityinterval.Thisstructureallowsonetodefinephases.Ifthedensity ρ lies intheinterval(0,α1]^,((α1,α2)),{[α2,b)}^thecorrespondingfluidstateiscalledvapor(spinodal){liquid},seeFig. 2.1foran illustration.

Weobservethatthefirst-orderpartof(2.1)isnotpurelyhyperbolicforalldensityvalues.Indeed,consider,forinstance, theone-dimensional(1-D)case,d=^{1.Itiseasytocheckthatthe}eigenvaluesoftheJacobianofthefirstorderpartof(2.1) Df¹(ρ,ρv)∈R^2×2are

λ1(ρ,u)=^v−

p(ρ), λ2(ρ,u)=^v+

p(ρ) (2.6)

withcorrespondingeigenvectors K1(ρ,v)=

1 v−

p(ρ)

, K2(ρ,v)=

1 v+

p(ρ)

. (2.7)

Therefore,thefirstorderpartof(2.1)ishyperbolicifandonlyif ρ∈(0,b)\ [α1,α2]^.

Thelackofhyperbolicityinthefirst-orderpartof(2.1)andthepresenceofthethird-orderderivativeinthemomentum balancemake thenumericalsolutionof(2.1)tobe achallenging task:Explicit schemessufferfromextremelysmalltime stepswhile implicitdiscretizations leadtobadly conditionedalgebraicproblems.The non-monotonicityof p preventsthe useofmostmodernshock-capturingschemes.

2.2. ArelaxationfortheNavier–Stokes–Kortewegsystem

ToovercomesomeoftheshortcomingsoftheclassicalNSKsystemweproposearelaxationfortheNSKsystem[37]

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

∂_tρ^α_{+ ∇ ·}(ρ^αv^α)=⁰,

∂t(ρ^αv^α)+ ∇ ·(ρ^αv^α⊗^vα)+ ∇^p(ρ^α)= ∇ ·^Tε[^vα] + ¹

α²ρ^α∇(^cα−ρ^α), γ ε²c^α+ ¹

α²⁽ρ^α₋c^α)=⁰.

(2.8)

Here α>0 istheKortewegparameterandcα isanewunknown,thatisdefinedbytheadditionalPoissonequation,andthe stresstensorTε givenby(2.2).Therelaxationsystem(2.8)isaugmentedwiththeinitialconditions

ρ^α(x,0)=ρ0, v^α(x,0)=^v0, x∈, (2.9)

andtheboundaryconditions

v^α=⁰, ∇^cα·ⁿ=⁰, x∈∂. (2.10)

Thesystem(2.8)hasastructuraladvantagethatbecomesevidentwhenwerewritethetime-dependentequationsas

⎧⎪

⎨

⎪⎩

∂tρ^α_{+ ∇ ·}(ρ^αv^α)=⁰,

∂_t(ρ^αv^α)+ ∇ ·(ρ^αv^α⊗^vα)+ ∇

p(ρ^α)+ ¹ 2α²ρ²

= ∇ ·^Tε[^vα] + ¹

α²ρ^α∇^cα. ^(2.11)

Again,ifweconsiderthe1-Dcaseforthesakeofsimplicity,thentheeigenvaluesoftheJacobianofthefirst-orderpartof (2.11) Df¹(ρ^α,ρ^αvα)∈R^{2×2 are}

λ1(ρ^α,ρ^αv^α)=^vα−

p(ρ^α)+ ¹ α²ρ^α, λ2(ρ^α,ρ^αv^α)=^vα+

p(ρ^α)+ ¹ α²ρ^α,

(2.12)

withcorrespondingeigenvectors K₁(ρ^α,ρ^αv^α)=

1 v^α−

p(ρ^α)+_α¹2ρ^α

, K₂(ρ^α,ρ^αv^α)=

1 v^α+

p(ρ^α)+_α¹2ρ^α

.

(2.13)

Astraightforwardcomputation(see[38])showsthatweobtainapurelyhyperbolicsystemfor

1

α^{2>|min{p(s) : s∈(}α1,α2)|, ^(2.14)

where(α1,α2)istheintervalofthedecreasingpressures,i.e.,p(s)<0,seeFig. 2.1.Thesystem(2.8)–(2.10)canbeseenas anapproximationoftheclassicalNSKsystemwith(ρ^α,ρ^αvα,cα)→(ρ,ρv,ρ)fortheKorteweglimitα_→0,where(ρ,ρv) is the solution of thecorresponding initial boundary value problem(2.1), (2.3), (2.4). We refer to [8,9,13,19,21] for first rigorousresultsontheKorteweg limit.Itispossibletoshowthat(2.8)formallyconvergesto(2.1).Wetaketheasymptotic expansion

ρ^α=ρ⁽⁰⁾+αρ⁽¹⁾+α²ρ⁽²⁾+. . . ,

ρ^αv^α=ρ⁽⁰⁾v⁽⁰⁾+αρ⁽¹⁾v⁽¹⁾+α²ρ⁽²⁾v⁽²⁾+. . . , c^α=^c(0)+αc⁽¹⁾+α²c⁽²⁾+. . . ,

(2.15)

forsmall α andlookatthebalanceswithin theequationsofsystem(2.8).Thereforewecompute theTaylorexpansion at

ρ⁽⁰⁾forthepressure

p(ρ^α)=^p(ρ⁽⁰⁾)+^p(ρ⁽⁰⁾)(ρ^α−ρ⁽⁰⁾)+^p (ρ⁽⁰⁾)(ρ^α−ρ⁽⁰⁾)²+. . .

=^p(ρ⁽⁰⁾)+αp(ρ⁽⁰⁾)(ρ⁽¹⁾+αρ⁽²⁾+. . . )+α²p (ρ⁽⁰⁾)(ρ⁽¹⁾+αρ⁽²⁾+. . . )²+. . . ^(2.16)

Ashortcomputationleadstothefollowingtermsforthedifferentpowersof α: O(α⁻²):

ρ⁽⁰⁾₌c⁽⁰⁾ (2.17)

O(α⁻¹):

ρ⁽¹⁾₌c⁽¹⁾ (2.18)

O(1):

ρ_t⁽⁰⁾_{+ ∇ ·}ρ⁽⁰⁾v⁽⁰⁾ =⁰,

ρ⁽⁰⁾v⁽⁰⁾

t+ ∇ ·

ρ⁽⁰⁾v⁽⁰⁾⊗^v(0) + ∇

p⁽⁰⁾

= ∇ ·^Tε[^v(0)]

+ρ⁽⁰⁾c⁽²⁾−ρ⁽²⁾₊ρ⁽¹⁾_∇c⁽¹⁾−ρ⁽¹⁾₊ρ⁽²⁾_∇c⁽⁰⁾−ρ⁽⁰⁾, γ ε²c⁽⁰⁾=

c⁽²⁾−ρ⁽²⁾

.

(2.19)

Table 2.1

DiscreteL²()-distance.Thedistance decreasesasα does.The fifthlinecontainsthe experimentalorderof convergence(EOC)withrespecttoαandthethirdlinecontainstheCPUtime.

i 1 2 3 4 5

x=⁰.005,α⁻_i²= 1 10 100 1000 10000

CPU-time [s] 563 654 787 828 2178

D_hⁱ= uh−u^α_hL² 0.25 0.033 2.9e−3 2.4e−4 3.7e−5

EOCi=^lnln⁽(α^Dihi+^/1^D/α^ih+i¹)⁾ – 0.879 1.056 1.082 0.812

Wesubstitute(2.17),(2.18)into(2.19)andobtain

ρ_t⁽⁰⁾_{+ ∇ ·}(ρ⁽⁰⁾v⁽⁰⁾)=⁰,

(ρ⁽⁰⁾v⁽⁰⁾)_t+ ∇ ·(ρ⁽⁰⁾v⁽⁰⁾⊗^v(0))+ ∇^p(0)=εv⁽⁰⁾+γ ε²ρ⁽⁰⁾_{∇ ·}ρ⁽⁰⁾, ^(2.20)

whichistheclassicalNavier–Stokes–Kortewegsystem,see(2.1).

Weillustratethisresultbythefollowingnumericalexperimentthatistakenfrom[34].

Example2.1(NumericalverificationoftheKorteweglimit).

Weconsiderthe1-Dsystem(2.8) with ε=⁰.01 oninterval =(−¹,2)subjectto theboundaryconditions(2.10) and thefollowinginitialdata

ρ0(x)=

0.3, x∈(0.3,0.6)∪(0.85,1.05)

1.8, otherwise ,

v₀(x)=⁰.

(2.21)

Fromthephysicalpointofview,theseinitialconditionsdescribetwovaporbubblessurroundedbyliquidfluid.Anumerical solutionof thisinitial-boundaryvalue problem, denoted by uα

h =(ρ_h^α,ρ_h^αvα h,cα

h)^T andanumerical solutionofthe corre- spondingclassicalNSKsystem,denoted byu_h=(ρh,ρhv_h)^T.Both solutionswerecomputedin[34]usingan explicitlocal discontinuousGalerkin(LDG)method,see,e.g.,[3,10,16].

Thenumericalresults,presentedinTable 2.1,indicatethattherelaxedmodelisanO(α²)-approximationoftheoriginal system.As onecanalsoseefromthisTable,fordecreasingvaluesof α theCPU-timeisincreasingduetothedependence on α oftheeigenvalues(2.12).Themaximumwavespeedforsystem(2.8)is λmax= |^vαmax|+

p(ρmax^α )(where ρ_max^α and vα

maxarethemaximumvaluesofthedensityandvelocity,respectively).Foranexplicitschemeoneneeds

t=^cD Gmin x

λmax

,x²

ε

=^cD Gmin

x

|^vαmax| +

p(ρ_max^α ),x²

ε

, (2.22)

forsome 0≤^cD G<1 to satisfythe CFLcondition forstability.Forsmall α oneneeds t=O(αx). Thisrestrictionisa hugedrawbackfornumericalsimulationsandweareinterestedinfindingawaytocircumventthisrestriction.

3. Anasymptotic-preservingschemeintheKorteweglimit

Havinginmindtheshortcomingsoftherelaxationsystem(2.8),weproposeanewAPnumericalscheme.Notethatfrom hereonwesuppresstheindex α ofalloftheprimalvariablesinordertoshortenthenotation.

3.1. Ahyperbolicsplitting

Thenumericalsolutionoftherelaxationsystem(2.8) requiresa resolutionoftwo scales:the(fast)wavescale,coming fromthe Korteweg part andthe (slow)convection scale. Inorder to obtain an accurate andefficient numericalscheme, whichisabletohandlebothscales,weimplementasplittingapproach.Tothisendwefirstintroduce aparametera and rewritethesystem(2.8)inthefollowingform(byaddingandsubtractingaρ∇ρ):

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

∂tρ+ ∇ ·(ρv)=0,

∂t(ρv^α)+ ∇ ·(ρv^α⊗^v)+ ∇ ˜^p(ρ)= ∇ ·^Tε[^v] + ¹

α²ρ_∇(c−ρ)+^aρ_∇ρ, γ ε²c+ ¹

α²⁽ρ−c)=0.

(3.1)

Herethepressureisdefinedas

˜

p(ρ)=^p(ρ)+^a

2ρ² (3.2)