A generalization of the quantum Bohm identity: Hyperbolic CFL condition for Euler–Korteweg equations

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A generalization of the quantum Bohm identity:

Hyperbolic CFL condition for Euler–Korteweg equations

Didier Bresch, Frédéric Couderc, Pascal Noble, Jean-Paul Vila

To cite this version:

Didier Bresch, Frédéric Couderc, Pascal Noble, Jean-Paul Vila. A generalization of the quantum Bohm

identity: Hyperbolic CFL condition for Euler–Korteweg equations. Comptes Rendus. Mathématique,

Académie des sciences (Paris), 2015, �10.1016/j.crma.2015.09.020�. �hal-01870745�

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Contents lists available atScienceDirect

C. R. Acad. Sci. Paris, Ser. I

www.sciencedirect.com

Partial differential equations

A generalization of the quantum Bohm identity: Hyperbolic CFL condition for Euler–Korteweg equations

Généralisation de l’identité de Bohm quantique : condition CFL hyperbolique pour équations d’Euler–Korteweg

Didier Bresch^a^,¹, Frédéric Couderc^b,Pascal Noble^b^,²,Jean-Paul Vila^b

aLAMA–UMR5127CNRS,bâtimentLeChablais,campusscientiﬁque,73376LeBourget-du-Lac,France bIMT,INSAToulouse,135,avenuedeRangueil,31077Toulousecedex9,France

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

Articlehistory:

Received30March2015 Accepted24September2015 PresentedbyOlivierPironneau

Inthisnote,weproposeasurprisingandimportantgeneralizationofthequantumBohm potential identity. This formula allows us to design an original conservative extended formulationofEuler–Kortewegsystemsandtheconstructionofanumericalschemewith entropy stability property under a hyperbolic CFL condition in the multi-dimensional setting. To the authors’ knowledge, this generalization of the quantum Bohm identity stronglyimproveswhatisalreadyknownforsimulationofsuchadispersivesystemand is also important for theoretical studies on compressible Navier–Stokes equations with degenerateviscosities.

©²⁰¹⁵^Published^byÊlsevier^Masson^SASôn^behalfôfÂcadémie^des^sciences.^Thisîsân openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r é s um é

Danscettenote,onproposeuneimportantegénéralisationdel’identitéditedupotentiel de Bohm quantique. Cette dernière permet de déﬁnir une formulation augmentée des systèmes d’Euler–Korteweg, qui est sous forme conservative dans le cas multi- dimensionnel. Uneconséquencetrès importantede cetteformulationest laconstruction deschémasavecstabilitéentropiquesousconditionCFLhyperboliquedusystèmed’Euler–

Korteweg.Cettegénéralisationdel’identitédeBohmévitedoncledéveloppementd’ondes parasitespour cessystèmes detypedispersifetestaussiimportante,par exemple,dans l’étudedeséquationsdeNavier–Stokescompressiblesàviscositésdégénérées.

©²⁰¹⁵^Published^byÊlsevier^Masson^SASôn^behalfôfÂcadémie^des^sciences.^Thisîsân openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

E-mailaddresses:Didier.Bresch@univ-savoie.fr(D. Bresch),frederic.couderc@math.univ-toulouse.fr(F. Couderc),pascal.noble@math.univ-toulouse.fr (P. Noble),vila@insa-toulouse.fr(J.-P. Vila).

1 ResearchofD.B.waspartiallysupportedbytheANRprojectDYFICOLTIANR-13-BS01-0003-01.

2 ResearchofP.N.waspartiallysupportedbytheANRprojectBoNDANR-13-BS01-0009-01.

http://dx.doi.org/10.1016/j.crma.2015.09.020

1631-073X/©²⁰¹⁵^Published^byÊlsevier^Masson^SASôn^behalfôfÂcadémie^des^sciences.^Thisîsânôpenâccessârticleûnder^the^CC^BY-NC-ND^license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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40 D. Bresch et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 39–43

Versionfrançaiseabrégée

Danscettenote,nousprésentonsunegénéralisationimportantedel’identitédupotentielquantiquedeBohm.Nousmon- tronsensuitecommentcetterelationpermetd’introduirede nouvellesformulationsaugmentées(sousformeconservative) dusystèmed’Euler–Korteweg(2)–(4)enplusieursdimensionsd’espace.Selonlechoixdescoefficientsdecapillarité,cetype desystèmeintervientdanslamodélisationdesmélangesdetypeliquide–vapeur,dessuper-fluidesoudel’hydrodynamique quantique par exemple.Ces nouvellesformulations permettent de construire un schéma numérique d’ordre1 à stabilité entropique sousconditionCFLhyperboliquealorsquelesystèmeprimalestdispersif.Àtitred’illustration,nousprésentons desrésultatsnumériquespourdesfilmsmincesavectensiondesurfacemodélisésparleséquationsdeSaint-Venant.Nous expliquons égalementrapidementcommentcette généralisationde l’identitéde Bohmpeut êtreutiliséedans lecadrede résultatsthéoriquessurNavier–Stokescompressiblesàviscositésdégénérées.

1. IntroductionandgeneralizationofthequantumpotentialBohmidentity

In thisnote, wepresentan importantgeneralization ofthequantumpotential Bohmidentity. Then weshow howthis relation allows usto introduce a newextended formulationof (2)–(3) by considering the conservativevariables u,w insteadofu,w.ItallowsonetotransformtheEuler–Kortewegsystemintoahyperbolicsystemperturbedbyasecond-order skew symmetricterminamulti-dimensionalsetting.Themainmotivationistheconstructionofanumericalschemethat iseasilyproved“entropy”stableunderahyperbolicCFLcondition.Wepresentpreliminarynumericalresultsforthinﬁlms with surface tension modeled by the shallow water equations illustrating this important property. We then ﬁnish by a remarkontheimportanceofthegeneralizationofthequantumpotentialBohmidentityforthecompressibleNavier–Stokes systemwithdegenerateviscosities.

1.1. GeneralizationofthequantumpotentialBohmidentity

LetusﬁrstpresentanextensionofthequantumpotentialBohmidentity 2div

√ /√

=div

∇∇log

stronglyusedinquantumﬂuidmechanics.Moreprecisely,wecanprovethefollowingresult.

Lemma1.1.Let ϕ,K andF bethreesmoothfunctionsfromR⁺→^R⁺^such^that

√ϕ()=

K(), F()= K().

Then

∇ K() (

0

K(s)ds)

=^div

F()∇∇ϕ() − ∇

F()−^F()ϕ()

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Thisrelationisanon-trivialextensionofthequantumBohmidentity,whichcorrespondstothecaseK()=^c/.Itsproof reliesonsomealgebraiccalculations.

2. Euler–Kortewegsystem

UsingthenewgeneralizationofthequantumpotentialBohmidentity,letusnowintroduce(new)extendedformulations oftheso-calledEuler–Kortewegsystem,which,inseveralspacedimensions,reads

∂t+div(u)=0, (2)

∂t(u)+^div(u⊗^u)+ ∇^p()=^div(K), (3)

where denotesthefluiddensity,uthefluidvelocity, p()thefluidpressureandKtheKortewegstresstensordefinedas K=

div(K()∇)+¹

2(K()−K())|∇|² Î_Rⁿ−^K()∇⊗ ∇. (4) with K() thecapillarycoefficient. Then we show the importance ofsuch formulations froma numericalpoint ofview, allowing tocontrol numericalinstabilities.Thesemodels comprisea liquid–vapormixture(forinstancehighly pressurized andhotwaterinnuclearreactorscoolingsystem)[8],superfluids(heliumneartheabsolutezero)[7]orevenregularfluids at sufficiently small scales (think ofripples on shallow waters) [9].In quantum hydrodynamics, the capillarycoefficient

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is chosen so that K()=^constant: ⁱⁿ ^this^case, ^the Euler–Kortewegequations correspond to thenonlinear Schrödinger equationafterMadelungtransform.Inclassicalﬂuidmechanics,thecapillarycoeﬃcientK()ischosenconstant.

Thesystem(2)–(3)admitstwoadditionalconservationslaws.Oneconservationlawissatisﬁedbytheﬂuidvelocity

∂tu+^u· ∇^u+ ∇(δE)=⁰, (5)

withE ^the^potentialênergyândδE îtsvariationalgradient E(,∇)=^F0()+¹

2K()|∇|², δE=^F₀()−¹

2K()|∇|²−^K(). (6)

The local existence of strongsolution to (2),(5) is proved in [1]. Forthat purpose,the authors introduced an extended formulationbyconsideringanadditionalvelocityw= ∇ϕ()with√ϕ()=

K():

∂tu+^u· ∇^u+ ∇

F₀()−|^w|²

2 = ∇

a()div(w)

, ∂tw+ ∇ u^Tw

= −∇

a()div(u)

, (7)

wherea()=

K().Thisformulationis particularlyadapted to thederivation ofa prioriestimates,the veryﬁrst one beingaconservationlawonthetotal(kinematic+ ^potential)^energy:

∂t

2|^u|²+E(,∇)

+^div u

2|^u|²+E(,∇)+^p()

=^div

F()(∇^{w u}− ∇^{u w})

−^div

(F()−F())(div(w)u−^div(u)w)

. (8)

withF()=ϕ().Remarkthattheleft-handsideof(1)correspondstotheEuler–Lagrangequantityfromthecapillarity term

K()|∇_|²/2.Itsuﬃcestoobservethat

K()|∇_|²_{= |∇}( 0

K(s))ds|²

andthusasobservedin[3]thevariationalgradientofthepotentialenergymaybewritten

δE=^F₀()− K() (

0

K(s)ds) .

Thisexplainstheimportance ofthegeneralizationofthequantum Bohmpotential.Following nowthestrategyof[1],we introduce a “good” additionalunknown, homogeneous to a velocity. We denotethisadditional velocity w= ∇ϕ() with

√ϕ()=

K().InordertowriteasuitableextendedformulationoftheEuler–Kortewegmodel,wealsodeﬁne F()so that F()=

K().UsingthegeneralizationofthequantumBohmpotential(1),theEuler–Kortewegsystemadmitsthe extendedformulation:

⎧⎪

⎨

⎪⎩

∂t₊div(u)=⁰,

∂t(u)+div(u⊗u)+ ∇p()=div(F()∇w^T)− ∇

(F()−F())div(w) ,

∂t(w)+^div(w⊗^u)= −^div(F()∇^u^T)+ ∇

(F()−F())div(u) .

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Thetotalenergyisthentransformedintoaclassicalentropyoftheﬁrst-orderpartof(9)

2^u²+E(,∇)=

2

^u²+ ^w²

+^F0():= ¯E(,u,w),

whereasthesecond-orderpartisskewsymmetric.Asaconsequence,theenergyconservationlaw(8)isobtainedthrough acomputationsimilartothatusedintheﬁrst-ordercase.

2.1. Application:stableschemesunderhyperbolicCFLcondition

In thissection, we introducea numerical schemefor theimportant extendedformulation (9). Thisstrongly improves whathasbeendone in[10].Thenumericaldomainisarectangledeﬁnedby 0≤^x≤^Lx and0≤^y≤^Ly,whichisdivided into N=nx×ny rectangularcells. Forthesake ofsimplicity,we consideruniformgridwithconstantspatial stepsδx and δy.Wefocusonthespatial discretizationofthesecond-orderterms:they arewrittenas∂α(f()∂βu)with(α,β)∈ {^x,y} and f()=^F(),F(),F()−F().Forthatpurpose,weintroducethefollowingﬁnitedifferenceoperators:

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42 D. Bresch et al. / C. R. Acad. Sci. Paris, Ser. I 354 (2016) 39–43

(d1u)i,j=^uⁱ⁺¹^/²^,^j−ui−¹/2,j

δx , (d⁺₁u)i+¹/2,j=^uⁱ⁺¹^,^j−ui,j

δx , (_d¯₁_u)i,j=^uⁱ⁺¹^,^j−ui−¹,j

2δx , (d₂u)i,j=^uⁱ^,^j⁺¹^/²−^ui,j−¹/2

δy , (d⁺₂u)i,j+¹/2=^uⁱ^,^j⁺¹−^ui,j

δy , (d¯₂u)i,j=^uⁱ^,^j⁺¹−^ui,j−¹

2δy . (10)

Asaresult, thedifferentialoperatorT()u=^div(F()∇^u^T)+ ∇

(F()−^F())div(u)

isapproximatedby Th()deﬁned as

Th()u_i_,_j=

⎧⎪

⎨

⎪⎩ d1

F()d⁺₁u1

i,j+ ¯d2

F()_d¯₁_u₂

i,j+ ¯d1

(F()−F())_d¯₂_u₂

i,j, d¯1

F()_d¯₂_u₁

i,j+ ¯^d2

(F()−^F())_d¯₁_u₁

i,j+^d2

F()d⁺₂u2

i,j

Wediscretize(9)asfollows:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ⁿ_i_,⁺_j¹−ⁿ_i_,_j

δt +^d1(Fⁿ_,₁)i,j+^d2(Fⁿ_,₂)i,j=⁰, (u)ⁿ_i_,⁺_j¹−(u)ⁿ_i_,_j

δt +^d1(Fuⁿ,1)i,j+^d2(Fuⁿ,2)i,j=Th(ⁿ⁺¹)wⁿ_i_,⁺_j¹, (w)ⁿ_i_,⁺_j¹−(w)ⁿ_i_,_j

δt +^d1(F_wⁿ_,₁)i,j+^d2(F_wⁿ_,₂)i,j= −Th(ⁿ⁺¹)uⁿ_i_,⁺_j¹,

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whereFⁿ_,_k^,F_uⁿ_,_k^,F_wⁿ_,_k^(k=¹,2)areclassicalRusanovﬂuxesevaluatedat ⁿ,uⁿ,wⁿ.Moreprecisely,theconvectionpart istreatedexplicitly,whereasthecapillarytermsaretreatedimplicitly.Remarkthattherearenocapillarytermsinthemass conservationlawsothat theimplicitstepamountstosolvea linear sparsesystemandiseasily provedentropystable. As a consequence,one can prove, by using discrete duality propertiesof the discrete second-order operators, the following entropystabilityresult.

Theorem2.1.Suppose(11)iscompletedwithperiodicboundaryconditions.Assumethehyperbolicscheme(system(11)withF=⁰⁾ isentropystablethenthefullyhyperbolic/capillaryscheme(11)isentropystable:

nx

i=⁰ ny

j=⁰

E¯(ⁿ_i_,⁺_j¹,uⁿ_i_,⁺_j¹,wⁿ_i_,⁺_j¹)≤

nx

i=⁰ ny

j=⁰

E¯(ⁿ_i_,_j,uⁿ_i_,_j,wⁿ_i_,_j).

Thismeansthatthenumericalscheme(11)isentropystableunderaclassicalhyperbolicCourant–Friedrichs–Lewycon- dition.Asanapplication,wecarriedoutanumericalsimulationofathinﬁlmfallingdownaninclined plane.Aconsistent shallowwatermodel[2]isgivenby

∂th+^div(hu)=⁰, (12)

∂t(hu)+^div(hu⊗^u)+ ∇(^p(h))+

gsin(θ ) ν

2

∂x

2h⁵

225 e₁=^S(h,u)+σh

ρ ^∇(^h^). ⁽¹³⁾

with p(h)=^g^cos(θ )h²/2 and S(h,u)=^gh^sin(θ )e1−³νu/h ande1 thefirst vectorof thecanonical basedirected down- stream. Here g=⁹.8 is thegravity constant, ρ,ν, σ are respectively thefluid density,the kinematic viscosity, andthe surfacetension,whereasθ istheinclinationoftheplane.Wepickedthevaluesfoundin[9]forasolutionwith31%glycerin byweight: ρ=¹.07×¹⁰³ ^{kg m}⁻³^, ν=².3×¹⁰⁻⁶^m²^s⁻¹ând σ=⁶⁷×¹⁰⁻³^{kg s}⁻²^.^The^source^termîs^treatedimplicitly:

since the sourceterm isonly intheequation for uandis linearwithrespect to u, theimplicit stepremains linear. We ﬁrst carryout anumericalsimulationoftheoriginalexperiencein[9],butimposed periodicboundaryconditionsinboth directions(seeFig. 1).

In order totest the robustness of thescheme, we alsocarried various numericalexperiments ofa drop falling down a plane inorderto dealwithwet/dryfronts. Forthatpurpose,we introduced aprecursor ﬁlmwitha thicknessof1.0× 10⁻⁵ mm (seeFig. 2).

Wewilldealtheproblemsofconsidering physicalboundaryconditions,deriving higherorderschemesandconsidering wet/dryfrontsinaforthcomingpaper[5].Thiswillbeusefultocomputeinstabilitiesinmovingcontactlines.

3. CompressibleNavier–Stokesequationswithdegenerateviscosities

Notethat ourformulation(mainly thegeneralizationofthe quantumBohmpotential)maybe coupledwiththeresult recently obtained in [6], allowing one to write an augmented formulation to the following compressible Navier–Stokes systemwithdragandcapillaryterms

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Fig. 1.(Coloronline.)Numericalsimulationofaroll-waveinpresenceofsurfacetension.Ontheleft:one-dimensionalroll-wavewithouttransversepertur- bations.Ontheright:atwo-dimensionalroll-wave.

Fig. 2.(Coloronline.)Dropfallingdownaninclineplane(θ=⁶⁰^◦⁾ât^time^t=^{0 and}^t=^{1 s.}^The^fluid^density,^kinematic^viscosityând^surface^tensionâre respectivelyρ=¹.0×¹⁰³^{kg m}⁻³^,ν=¹.0×¹⁰⁻⁶^m²^s⁻¹ândσ=⁶⁷×¹⁰⁻³^{kg s}⁻²^.

∂t+^div(u)=⁰, (14)

∂_t(u)+^div(u⊗^u)−^2div(μ()D(u)− ∇(λ()divu)+ ∇^p()+^r1_|u|^α^u=^div(K), (15) withλ()=²(μ()−μ())andK()=^c(μ())²/.Suchacompatiblesystemmaybeusedtoprovetheglobalexistence ofweaksolutionstothecompressibleNavier–Stokesequationswithdegenerateviscositieswithoutcapillaryanddragterms, see[4].Thisgivesageneralizationoftheresultin[11]wheretheyconsiderthecase μ()=μ andλ()=^{0 and} ^where theystronglyusetheusualquantumBohmidentityintheapproximatesystem.

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