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Contents lists available atScienceDirect

Journal

of

Computational

Physics

www.elsevier.com/locate/jcp

A

multigrid

multilevel

Monte

Carlo

method

for

transport

in

the

Darcy–Stokes

system

Prashant Kumar

a

,

b

,∗

,

Peiyao Luo

c

,

Francisco

J. Gaspar

a

,

Cornelis

W. Oosterlee

a

,

c aCWICentrumWiskunde&Informatica,Amsterdam,theNetherlands

bFacultyofAerospaceEngineering,DelftUniversityofTechnology,Delft,theNetherlands cDIAM,DelftUniversityofTechnology,Delft,theNetherlands

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received4September2017 Receivedinrevisedform16May2018 Accepted28May2018

Availableonline31May2018

Keywords: UQ Darcy–Stokesflow Contaminanttransport MLMC Multigrid Uzawasmoother

AmultilevelMonteCarlo(MLMC)methodforUncertaintyQuantification(UQ)of advection-dominated contaminant transport in a coupledDarcy–Stokes flow system is described. In particular, we focus on high-dimensionalepistemic uncertainty due to an unknown permeabilityfield in the Darcy domain that is modelled as a lognormalrandom field. This paper explores different numerical strategies for the subproblems and suggests an optimal combination for the MLMC estimator. We propose a specific monolithic multigridalgorithmto efficientlysolve thesteady-stateDarcy–Stokesflowwithahighly heterogeneous diffusion coefficient. Furthermore, we describe an Alternating Direction Implicit(ADI)basedtime-steppingfortheflux-limitedquadraticupwindingdiscretization forthetransport problem. Numericalexperiments illustratingthe multigridconvergence andcostoftheMLMCestimatorwithrespecttothesmoothnessofpermeabilityfieldare presented.

©2018ElsevierInc.Allrightsreserved.

1. Introduction

TransportinaDarcy–Stokesflowsystemcanbeusedtoanalyzealargenumberofdynamicalprocesses.Thismodelis,for example,ofimportanceinthestudyofaccidentaldischargeofradioactivecontaminantsorchemicalspillageinsurfacewater bodiesandthesubsequenttransporttotheconnectedaquifers.AcoupledDarcy–Stokessystemisthenusedtosimulatethe interactionbetweenthesurfacewaterandthegroundwaterflow.Thecouplingisachievedbyimposinginterfaceconditions basedonmassconservation,balancingthenormalstressandaspecialcondition,calledtheBeavers–Joseph–Saffman(BJS) interface condition [1,2] thatrelatesthe shearstress andtangentialvelocity along theinterface. Thesteady-statevelocity fieldderivedfromthismodelisthenutilizedfortheadvectionofchemicalcomponentsinatransportmodel.Manyrelevant physicalphenomena concerningthemasstransportsuchasmolecular diffusion,mechanicaldispersion,adsorption,canbe convenientlyincorporatedinthismodel.

The mathematicaltheoryandanalysisoftransportina coupledDarcy–Stokesflowiswell developedanda numberof stable,convergentnumericalmethodshavebeenproposed.Theseincludefinitevolume(FV),mixedfiniteelement(MFE)or moreadvanced,locallyconservativediscontinuousGalerkin(DG)schemes,see[3–10] andreferencestherein.Theseschemes requiretheknowledge ofphysicalquantitieslikefluidviscosity, permeabilitiesandexperimentallymeasured BJSinterface parametersforapproximatingtheDarcy–Stokesflow.Furthermore,thechemicaltransportequationalsorequiresinputsuch

*

Correspondingauthor.

E-mailaddresses:pkumar@cwi.nl(P. Kumar),p.luo@tudelft.nl(P. Luo),gaspar@cwi.nl(F.J. Gaspar),c.w.oosterlee@cwi.nl(C.W. Oosterlee). https://doi.org/10.1016/j.jcp.2018.05.046

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asinitialandinflowboundaryconditions.Inmanycases,completeinformationofthesephysicalquantitiesisnot available andtheymaybe modelled inaprobabilistic framework.Forinstance,itis wellknownthat thepermeabilityfield canbe modelled asalognormalrandomfield[11–13].Oncetheseuncertaintiesareincorporatedinthemathematicalmodels,the goalistoobtainthestatisticsofcertainquantitiesofinterest,forexample,themeanspatialconcentrationofcontaminants inanaquiferafteracertainintervalfromthetimeofdischarge.

Many Uncertainty Quantification(UQ) techniques exist in theliterature butthey are not universally applicable to all engineeringproblems.Thesetechniques arebroadlycategorizedinto intrusiveandnon-intrusiveapproaches. Notablyone ofthemostpopularintrusivetechniquesisthestochasticGalerkinmethodbasedonageneralizedPolynomialChaos(gPC) expansion[14,15]. ThestochasticGalerkinformulationforacoupled setofPDEsishowevernot straightforward,as exist-ing iterativesolvers cannot be directly utilizedandmayneed severe modifications.Another classof well-establishedUQ techniquesincludesstochastic collocationmethods[15,16] thatarebasedonadeterministic samplingapproachwherethe nodesin the random spaceare computedusing cubaturerules. Themain advantage of thestochastic collocation method is its non-intrusiveness, hence,existing deterministic solvers can be utilized.The computational cost is governed by the numberofnodesthathoweverincreasesrapidlywithanincreaseinthestochasticdimensions.

Forourproblemtohaveanypracticalrelevance,thestochasticmodelshouldconsiderrandomfieldswithlowspatial reg-ularityandasmallcorrelationlength.Thisusuallytranslatestoahigh-dimensionalUQproblem[17] andtheaforementioned deterministic approaches may not be computationally tractable. Historically, Monte Carlo (MC) type methods havebeen proventobeeffectiveUQtoolsforsuchproblemsastheydonotsufferfromthecurseofdimensionality,areeasyto imple-mentandhaveahighparallelizationpotential.ForthestandardMCmethod,thestatisticalerrorconvergesas

(

V

ar

[

Q

]

/

N

)

12, whereN isthenumberofMCsamplesand

V

ar

[

Q

]

isthevarianceoftheQuantityofInterest(QoI).Thisslowconvergence withrespect to N is themain drawbackof themethod.Problems involving a spatio-temporal grid,ensuring a low root-mean-squareerror(RMSE)willrequirealargenumberofsamplesonaveryfinemeshmakingtheestimatorexpensive.

Overthepastfewyears,alargenumberofsamplingandvariancereductiontechniqueshavebeenproposedtoovercome thislimitation. Forinstance,theauthorsin [18] have applieda Quasi-MonteCarlo(QMC) samplingtechnique to improve theconvergencerateforthestochasticDarcyflowsubproblem.Forthesameproblem,toreducethegridsizerequirements, high-ordernumericalschemeswereappliedby[19] thatresultedinlowerasymptoticcosts.Morerecently,theMonteCarlo methodhasbeengeneralizedtomultiplegridlevels,exhibitingan exceptionalimprovementoverthestandardMC[20,21]. The improvedefficiencyofthesemultilevelMonte Carlo(MLMC)methods comes frombuildingtheestimate fortheQoI, ona hierarchyofgrids, by exploitingthe linearity oftheexpectationoperator, i.e.

E

[

QL

] =

E

[

Q0

] +

L

=1

E

[

Q

Q−1

]

.

On thecoarsest grid expectationsare inexpensive tocompute accurately andforlarge values of

,where the numerical solutioniscomparativelyexpensive, fewersamplesarerequiredasthevarianceofthecorrectionterm

V

ar

[

Q

Q−1

]

is significantlysmallercomparedtothepure samplingvariance,

V

ar

[

Q

]

.Whileofferinglargesavings overthestandardMC

method,MLMCretainsalltheimportantpropertiesofMCmethodslikeparallelizationpotentialandcombinationwithother complementaryvariancereductiontechniques[18,22].

ThepurposeofthispaperistodescribeinasystematicmanneranumericalstrategytodesignanefficientMLMC esti-matorforUQofstochastictransportintheDarcy–Stokessystem.AnefficientMLMCestimatorrequirescarefulconsideration ofthenumericaltechniquesfortheapproximationoftheQoI.Thepaperisorganizedinthefollowingway:

InSection2,we describethestochastic transportintheDarcy–Stokesflow.ThemixedformulationoftheDarcy equa-tion is used that is coupled with the Stokes flow using three interface conditions. The stochastic extension of the problemisobtainedbymodelling thepermeabilityasalognormalrandomfield.Thespatialcovarianceoftherandom fieldisderivedfromtheparameterizedMatérnfunction[23] thatallowssimulationoffieldswithdifferentdegreesof smoothness.

InSection3,theFiniteVolume(FV)discretizationoftheDarcy–Stokesflow onastaggeredgridisdescribed. Aspecial discretizationalongtheinterfaceisproposedtakingintoaccounttheBeaver–Joseph–Saffmanninterfacecondition that dependsontherandompermeabilityalong theinterface anda specificparameter. Theoptimalchoice forthe spatio-temporaldiscretizationofthetransportequationisbasedontheregularityoftheDarcy–Stokessolution.Typically,the errorintheFVapproximationofthevelocity fielddependsonthespatialregularityofthepermeabilityfieldandalso onthe smoothnessofboundary conditionsalong thetwo domains.Thus, the discretizationschemeforthe transport equationshouldberelatedtotheaccuracyofthevelocityapproximation.Usinghigher-orderschemesforlowregularity problemsmayleadtoanexpensiveMLMCestimatorwithoutanyimprovementinthenumericalaccuracy.Wealsoshow that,foradvectiondominatedtransportwithsharpgradientsanddiscontinuities,low-orderschemesareverydiffusive andarelesssuitedforMLMCapplications.

AmonolithicmultigridsolverfortheDarcy–StokesproblembasedonanUzawasmoother[24] isproposedinSection4. Thissmootheremploysanequation-wisedecoupledrelaxationforthepressureandvelocityunknowns.Forthevelocity asymmetricGauss–SeideliterationisemployedwhereasforthepressureaRichardsoniterationisapplied.The Richard-soniterationtakesintoaccountlocalfluctuationsinthepermeabilityfield toderivetheoptimalrelaxationparameter. The multigridalgorithm is basedon cell-centred permeabilitycoefficientsanddirect coarse-grid discretizationbased oncell-centredaveraging.The proposedsolverisrobust andalsoworkswell onverycoarsegrids. Wegeneralizethis multigridmethod tomulti-block problemsby thegridpartitioningtechnique [25]. Toincorporatethecoarse time-step inthe MLMChierarchy,an implicittime stepping is desiredforthecontaminant transport equation.We consider an

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Fig. 1.GeometryoftheDarcy–Stokesproblemcoupledwiththetransportequation.SubdivisionofthedomainDintoafree-flowsubregionDsanda

porousmediumsubdomainDd,byaninternalinterface

ds.

AlternatingDirectionImplicit(ADI)basedsolverforthediscretetransportequationwhichbreakstheproblemintotwo 1Dproblems,greatlyreducingthecostoftime-stepping.

Single and multilevel Monte Carloestimators are defined in Sections 5 and 6, respectively. Section 7 is devoted to numerical experiments performed on two test problems: a 2-block problemwith no-slip interface condition anda morerealistic 4-blockcasewiththeBJSinterface condition.Wethoroughlytestthemultigridmethodwithrespectto differentMatérn parametersto identifytheoptimalcyclingstrategy. Finally,wecombinethesecomponentstoobtain the solution of the stochastic transport problem using MLMC estimator and compare the asymptotic cost with the standardMonteCarlomethod.

2. StochastictransportinDarcy–Stokessystem

We consider thetransport equation defined inthe boundeddomain

D

R

2,with boundary

D

and ina finitetime interval

T

=

(

0

,

T

]

, for T

<

.The transport equation coupled withthe steady-state Darcy–Stokesflow on

D

, is subdi-vided into a porous medium

D

d

R

2, where the flow is described by Darcy’s law and the free-flow region

D

s

R

2, governed bytheStokes equationswithboundaries

D

d and

D

s,respectively(Fig. 1).Theinternal interfaceisdefinedas

ds

=

D

d

D

s.Furthermore,we denoteby

ω

an eventintheprobability space

(,

F

,

P)

,where

isthesamplespace with

σ

-field

F

andprobability measure

P

. Ourdescriptionofthestochastic flow modelfollowsfrom[26–28] wherethe deterministiccounterpartofthisproblemisconsidered.

Porousmediumdescription.The steady-statesingle-phaseflow ina porousmedium can be modelled byDarcy’slaw andtheincompressibilitycondition

ηK

−1ud

+ ∇

pd

=

0 in

D

d

×

, (

a

)

∇ ·

ud

=

fd in

D

d

×

.

(

b

)

(2.1)

Thefluidpressureisrepresentedby pd andthevelocityvectorud

=

(

ud

,

vd

)

denotesthehorizontalandvertical

compo-nents.Thefluidviscosityisdenotedbythepositiveconstant

η

andK isthespatiallyvariablepermeabilityfield.Theknown source(sink)termisindicatedby fd

L2

(

D

d

)

,whereL2

(

D

d

)

isthespaceofsquare-integrablefunctionsin

D

d.

We divide the boundary

D

d

\

ds

into two disjointsets,

D

dD and

D

dN, where deterministic Dirichletand Neumann boundaryconditionsareprescribed,

pd

=

gdD on

D

dD

,

(

a

)

ud

·

n

=

gdN on

D

dN

, (

b

)

(2.2)

respectively,withntheoutwardnormaltotheboundary

D

dN andgd

D

L2

(∂

D

dD

)

,gdN

L2

(∂

D

dN

)

.

We willoperateundertheassumption thatthe solidframework isrigidandthereisno interactionbetweenthefluid andthesolidmatrixoftheporousmedium.

We model the permeability field by means of a lognormalfield, i.e. logK

(

x

,

ω)

:=

Z

(

x

,

ω)

is a zero-mean Gaussian randomfieldforx

D

d

=

D

d

D

dand

ω

.Therefore

E

[

Z

(x,

·

)

] =

0

,

cov(Z

(x

1

,

·

),

Z

(x

2

,

·

))

=

E

[

Z

(x

1

,

·

)

Z

(x

2

,

·

)

]

,

x1

,x

2

D

d

.

(2.3)

The lognormalpropertyensures apositivepermeabilitythroughoutthedomain.Forfurthersimplification,weconsideran

isotropicandstationaryGaussianprocess,whichcanbeobtainedfromahomogeneouscovariancefunctionC

:

R

R

such that

cov(Z

(x

1

,

·

),

Z

(x

2

,

·

))

=

C

(

r

)

with r

=

x1

x2

2

.

(2.4) Fortheproblemtobewell-posed,weassume

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TogeneratesamplesoftheGaussianrandomfield Z,thecovariancematrixisderivedfromthefamilyofMatérnfunctions

[29] characterizedbytheparameterset

=

(νc

,

λc,

σ

2

c

)

,andhasthestandardform

C

(

r

)

=

σ

c22 1−νc

(

ν

c

)

2

ν

c r

λc

νc Kνc

2

ν

c r

λc

.

(2.6)

Here,

isthegammafunctionandKνc isthemodified Besselfunctionofthesecondkind.Theparameter

νc

1

/

2 defines

the smoothness,

σ

2

c

>

0 isthe variance and

λc

>

0 isthe correlation length ofthe Gaussian process. TheMatérn model

hasgreatflexibilityinmodelling spatialprocessesbecauseofparameter

νc

,whichgovernsthedifferentiabilityofthe ran-domfield. For

νc

=

1

/

2,theMatérnfunction correspondsto anexponential modelandfor

νc

→ ∞

to aGaussian model. Furthermore,the other two parameters

λc

and

σ

2

c dictate the numberof peaksand theamplitude ofthe random field,

respectively.TherealizationsofGaussian randomfieldsareHöldercontinuousi.e., Z

(

x

,

·

)

(

D

d

)

almost surelywiththe exponent0

<

κ

<

νc

[30].For

νc

>

1,therealizationsarecontinuouslydifferentiable.

Free-flowdescription.We consider acreeping, steady-stateflow connected tothe porous flow regime. We assume a viscous,incompressibleNewtonianfluidflowwhichcanbemodelled bytheStokesequations,consistingofthemomentum andcontinuityequations

− ∇ ·

σ

s

=

fs in

D

s

×

,

(

a

)

∇ ·

us

=

0 in

D

s

×

,

(

b

)

(2.7)

respectively, withus

=

(

us

,

vs

)

asthefluid velocity components andfs

=

(

fs

1

,

f2s

)

L2

(

D

s

)

2 is theforce. The fluid stress tensor

σ

s isdefinedby

σ

s

= −

psI

+

2

η

R(us

),

(

a

)

R

:=

1 2

(

u s

+

(

us

)

T

), (

b

)

(2.8)

wherepsdenotesthefluidpressureandRisthestraintensor.In2D,thecomponentsofthestresstensor(2.8)(a)aregiven

by

σ

s

:=

σ

xxs

σ

xys

σ

s yx

σ

y ys

,

(2.9)

andisrelatedtotheprimitivevariablesvia

σ

s xx

= −

ps

+

2

η

us

x

,

σ

s xy

=

σ

yxs

=

η

us

y

+

vs

x

,

σ

s y y

= −

ps

+

2

η

vs

y

.

(2.10)

Theexteriorboundary

D

s

\

dsispartitionedintotwodisjointsets,

D

sD and

D

s

N,andthefree-flowmodeliscompleted

byimposingthefollowingboundaryconditions

us

=

gsD on

D

sD

,

(

a

)

σ

s

·

n

=

gs

N on

D

sN

, (

b

)

(2.11)

wherenistheoutwardnormaltotheboundary

D

sN andgs

D

L2

(∂

D

sD

)

2,gsN

L2

(∂

D

sN

)

2.

Interfaceconditions. The coupling betweenthe two problems is attained by imposing three interface conditions for massconservation,normalstressbalanceandthethirdconditionrelatingtheslipvelocitytotheshearstressalong

ds

us

·

ns

=

ud

·

ns on

ds

×

,

(

a

)

ns

·

σ

s

·

ns

=

pd on

ds

×

,

(

b

)

us

·

τ

+

K αB J

τ

·

σ

s

·

ns

=

0 on

ds

×

, (

c

)

(2.12)

respectively.Wedenoteby

τ

andns theunittangential andnormalvectorstotheinterface

ds

,respectively.Thethird in-terfacecondition(2.12)(c)wasoriginallyderivedbyBeaverandJoseph[1] onthebasisofexperimentationanddimensional analysis,andfurtherhasbeenmathematicallyprovenbySaffmann[2].ItisoftenreferredtoastheBeaver–Joseph–Saffmann (BJS) condition. Theliterature on BJS isgrowing rapidly asitis well able torepresentthe physics along theinterface of viscousfluid flowandaporousmedia, formoredetailssee[8,31,32].Notethatfrictionconstant

K

αB J dependsonthe

per-meabilityalongtheinterfaceandisthusherearandomvariable.Parameter

αB J

>

0 isadimensionlessquantitymeasured experimentallyandcanalsobepronetouncertainty.

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Fig. 2.Staggered grid location of unknown and corresponding control volumes.

Alternatively,oneobtainsano-slipinterfaceconditionbyneglectingthesecondtermfrom(2.12)(c)

us

·

τ

=

0 on

ds

×

.

(2.13)

Transport model. The generic single-component transport equation with random initial data gives us the following equation:

φ

c

t

+ ∇ ·

(

cu

D

c

)

=

φ

f t in

D

×

T

×

, (

a

)

c

=

c0 in

D

×

,

t

=

0

,

(

b

)

(2.14)

where c denotes the concentration of the chemical component, typically expressed in terms ofmoles per unit volume;

φ

(

0

,

1

]

istheknownporosityofthemediumwhile ft

L2

(

D

)

isanetvolumetricsourceforc.Theinitialconcentration

c0

:=

c0

(

x

,

ω)

L2

(

D

)

correspondstoarandomfield.Thesteady-statevelocityfieldu

=

us

udisderivedfromthesolution of the coupledDarcy–Stokes problem. In the Darcydomain, the hydrodynamical dispersion isrepresented by the tensor field

D

accountingforthemolecular diffusionandmechanicaldispersion.Molecular diffusionwhichtakesplaceduetoa concentrationgradientthatismoresignificantcomparedtothemechanicaldispersionresultingfrommicro-scalevariations in thevelocityfield compared totheaverageflow. Inthispaper,we considera diffusiontensordefinedin[33,34] foran isotropicporousmediawithtensorcomponents:

Dxx

=

DLu 2 |u|

+

DTv 2 |u|

+

D

,

Dy y

=

DLv 2 |u|

+

DT u2 |u|

+

D

,

Dxy

=

Dyx

=

(

DL

DT

)

uv|u|

,

(2.15)

where DL,DT

>

0 arelongitudinalandtransversedispersivity,respectively,and D

>

0 istheeffectivemoleculardiffusion.

IntheStokesdomain,thedispersiontensorisusuallytakentobeisotropic,i.e.

D

=

DI.

ThetransportmodeliscompletedbyapplyingaCauchyboundaryconditionattheinflowboundaryandanon-dispersive massfluxconditionatoutflowboundaries,formallyexpressedas

(

cu

D

c

)

·

n

=

(

cinu)

·

n

D

in

×

T

, (

a

)

D

c

·

n

=

0

D

out

×

T

,

(

b

)

(2.16)

respectively.Theinflowboundaryisdefinedas

D

in

:= {

x

D

:

u

·

n

<

0}andtheoutflowboundaryas

D

out

:=

D

\

D

in.

Foreaseofpresentation,weconsiderasimplifiedchemicaltransportmodelignoringtheadsorptionphenomenaandalso assumethattheinjectionofthechemicalcomponentdoesnothaveanyeffectonthesteady-stateDarcy–Stokesflowfield. 3. Finitevolumediscretization

Forthespatialdiscretizations,weemploythefinitevolumeschemeonastaggeredgridfortheDarcy–Stokessystem.The domainissubdividedintosquareblocksofsizeh

×

hconformingwith

D

and

ds

.Thelocationsandindexingofunknowns

u

,

vandp

,

calongwiththeirrespectivecontrolvolumes

D

1

h,

D

2h and

D

h3 areshowninFigs.2–3.

As the same variables describe different physics in the two subdomains, discretization along the interface becomes involved. A large numericalerror or evena reduction inthe order ofgrid convergence may be encountered ifinterface conditions arenot handledproperly.Astaggered arrangementoftheunknownsgreatlysimplifies thediscretizationalong theinterfaceandhasbeenproventobeeffectiveinreducingnumericalerroralongtheinterface[28].Moreover,astaggered gridisalsoaconvenientwayofavoidingspuriousoscillationsinthenumericalsolution[35] andobtainingconservationof massthroughoutthesystem,alsoonarelativelycoarsegrid.

Tomakethispaperself-contained,webrieflydiscussthespatio-temporaldiscretizationofthecoupledproblem.Forthe Darcy–Stokesapproximation,wewillcloselyfollowthedescriptionfrom[28].

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Fig. 3.ControlvolumesD1 h(i+ 1 2,j)(left),D 2 h(i,j+ 1 2)(middle)andD 3

h(i,j)(right)fortheprimaryunknownsu,vandp,crespectively,togetherwith

thecorrespondingindexingforeachvariable.

3.1. DiscretizationofDarcy–Stokesflow 3.1.1. DiscretizationofDarcyequation

Thediscreteequationsforvelocitiesandpressureinthemixedformulationareeasytodetermine.Thediscretizationfor thehorizontalvelocityisobtainedbyintegratingtheDarcyequation(2.1)(a)inthecontrolvolume

D

h1

(

i

+

12

,

j

)

ηK

−1 i+1 2,j ud i+12,j

+

pdi+1,j

pdi,j h

=

0

.

(3.1)

Similarly, the discrete equations forthe vertical velocities are obtained by integrating theDarcy equation in the control volume

D

h2

(

i

,

j

+

12

)

.Forthepressure,weintegratethecontinuityequation(2.1)(b)inthevolume

D

h3

(

i

,

j

)

ud i+12,j

u d i−12,j h

+

vd i,j+12

v d i,j−12 h

=

f d i,j

.

(3.2)

Forboundarieswherethepressureisprescribed(2.2)(a),weintegratetheDarcyequationoverhalfvolumes.Atboundaries forwhichthevelocitiesareknown,(2.2)(b)isdirectlyapplied.

3.1.2. DiscretizationofStokesequation

Integratingthefirstcomponentofthemomentumequation(2.7)(a)inthecontrolvolumes

D

2h

(

i

,

j

+

12

)

,

(

σ

xx)i+1,j

(

σ

xx)i,j h

+

(

σ

xy)i+1 2,j+12

(

σ

xy)i+12,j−12 h

=

(

f1s

)

i+1 2,j

.

(3.3)

Thecomponentsintheaboveequationareapproximatedusing(2.10) as

(

σ

xx)i+1,j

= −

psi+1,j

+

2

η

us i+32,j

u s i+12,j h

,

(

σ

xx)i,j

= −

psi,j

+

2

η

us i+1 2,j

us i−1 2,j h

,

(

σ

xy)i+1 2,j+12

=

η

u s i+1 2,j+1

us i+1 2,j h

+

vs i+1,j+1 2

vs i,j+1 2 h

,

(

σ

xy)i+1 2,j−12

=

η

u s i+1 2,j

us i+1 2,j−1 h

+

vs i+1,j−1 2

vs i,j−1 2 h

.

(3.4)

Usingtheseapproximationsin(3.3),wefinallyget

2

η

h2

(

u s i+32,j

2u s i+12,j

+

u s i−12,j

)

η

h2

(

u s i+12,j+1

2u s i+12,j

+

u s i+12,j−1

)

η

h2

(

v s i+1,j+1 2

vs i,j+1 2

vs i+1,j−1 2

+

vs i,j−1 2

)

+

1 h

(

p s i+1,j

psi,j

)

=

(

f1s

)

i+1 2,j

.

(3.5) Similarly,thesecondcomponentofthemomentumequationisintegratedinthevolume

D

h2

(

i

,

j

+

12

)

.Thecontinuity equa-tion(2.7)(b)isasgivenin(3.2) overthevolume

D

3h

(

i

,

j

)

.

The Dirichlet boundary condition (2.11)(a)can be directly utilized forthe approximation of the stress componentin (3.4). Ifthestress componentsare prescribed alongthe boundary(2.11)(b) thenthe Stokesequations areintegratedover halfvolumesalongthatboundary.

(7)

Fig. 4.Locationsofunknownsrequiredfordiscretizationalongtheinterface.(Forinterpretationofthecoloursinthefigure(s),thereaderisreferredtothe webversionofthisarticle.)

3.1.3. Discretizationofinterfaceequations

An appropriate interfacediscretizationis crucialforachievinga strongnumericalcouplingbetweentwosub-solutions. As theverticalvelocitiesliealongtheinterface, weintegratetheStokesequation forvsover thehalfvolume indicatedin redinFig.4,

(

σ

xy

)

i+1 2,j+12

(

σ

xy

)

i−12,j+12 h

+

(

σ

y y)i,j+1

(

σ

y y)i,j+1 2 h

/

2

=

(

f2s

)

i,j+1 2

.

(3.6)

Now,wedescribethecomputationofthesestress componentsintheaboveequation.Toapproximatetheseterms,wewill utilizetheinterfaceconditions(2.12)(a)–(c).

Forthenormalstresscomponent

(σy y

)i

,j+1,weuse

(

σ

y y)i,j+1

= −

psi,j+1

+

2

η

vs i,j+32

v s i,j+12 h

.

(3.7)

Next, thenormalstress

(σy y

)

i,j+1

2 attheinterface isderived by equilibratingthe normalstress withthepressure atthe interface

(

σ

y y)i,j+1 2

=

p d i,j+12

.

(3.8) Pressure pd

i,j+12 atthe interface is not known andis obtained by integrating the Darcy equation over the half volume

indicatedbythegrayboxinFig.4,as

η

K−1 i,j+12v d i,j+1 2

+

pd i,j+12

p d i,j h

/

2

=

0

.

(3.9)

Equation(3.8) canthenberewrittenas

(

σ

y y)i,j+1 2

= −

p d i,j

+

ηh

2Ki,j+1 2 vd i,j+12

.

(3.10)

Thestresscomponent

(σxy)

i+1 2,j+ 1 2 isapproximatedas

(

σ

xy

)

i+1 2,j+12

=

η

u s i+12,j+1

u s i+12,j+12 h

/

2

+

vs i+1,j+12

v s i,j−12 h

.

(3.11)

Here,thehorizontalcomponentofthevelocityattheinterfaceus i+1

2,j+12

isderivedusingtheBJScondition(2.12)(c). There-fore,

Ki,j+1 2

α

B J

−1 us i+1 2,j+12

η

u s i+1 2,j+1

us i+1 2,j+ 1 2 h

/

2

+

vs i+1,j+1 2

vs i,j+1 2 h

=

0

.

(3.12)

Using(3.11) and(3.12),oneobtains:

(

σ

xy

)

i+1 2,j+12

=

2

η

mi,j+1 2 h u s i+1 2,j+1

+

η

mi,j+1 2 vs i+1,j+12

v s i,j+12 h

,

(3.13)

(8)

wherem

i,j+1 2

=

1

2ηKi,j+1 2 hαB J+2ηKi ,j+1 2

.Thefinalcomponent

(σxy)

i1 2,j+

1

2 iscomputedinasimilarmanner.Usingallstress componentsderivedabove,werewrite(3.6) as

2

η

mi,j+12 h2

(

u s i+12,j+1

u s i−12,j+1

)

η

mi,j+1 2 h2

(

v s i+1,j+12

2v s i,j+12

+

v s i−1,j+12

)

+

2 h

(

p s i,j+1

pdi,j

)

4

η

h2

(

v s i,j+32

v s i,j+12

)

+

vs i,j+1 2 Ki,j+1 2

=

(

f2s

)

i,j+1 2

.

(3.14)

3.2. Discretizationoftransportequation

The numericalscheme toapproximate theadvection–diffusion part ofthe transport equation dependson thetype of problem.In thecaseofcontaminanttransport intheDarcy–Stokessystem,convectionis thedominantcausefortheflow movement [33,34]. Animportantrequirementfordesigningan efficientMLMCestimatorisusingnumericalschemes that arestableoncoarsemeshes.Forexample,fortime-steppinganimplicitmethodisfavourableduetounconditionalstability withrespecttotime-stepsize.Similarly,anupwindbaseddiscretizationcangivestablesolutionsonrelativelycoarsegrids comparedtothecentraldifferencingbasedmethods.Webriefly discusstheimplementationdetailsofthespatio-temporal discretizationofthetransportequation.

Ineachofthecontrolvolumes

D

h3

(

i

,

j

)

(seeFig.3),theintegralformulationof(2.14)(a)takestheform

t

D3 h(i,j)

φ

cdx

+

D3 h(i,j)

∇ ·

(

cu

D

c

)

dx

=

D3 h(i,j)

φ

ftdx. (3.15)

The velocity field u is assumed to be exact but for the current problem this is derived from the solution of coupled Darcy–Stokessystem.Also,forsimplicity,wewillassume Dxy

=

Dyx

=

0 resultinginadiagonaldispersiontensor

D

. 3.3. Spatialdiscretization

UsingtheGaussdivergencetheorem,thesecondintegralin(3.15) isreformulatedasaboundaryintegralfortheboundary

D

3h

(

i

,

j

)

=

4k=1

D

3h

(

i

,

j

,

k

)

,suchthat

D3 h(i,j)

∇ ·

(

cu

D

c

)

dx

=

4

k=1

∂Dh3(i,j,k)

(

cu

D

c

)

·

nkdSk

,

(3.16)

where nk denotes theunit normal vectorto thecorresponding face. Flow intheStokes domain iswell-behavedandthe

centraldifferencing schemetoevaluate thefluxesin(3.16) willresultinasecond-order accurate solutionlocallyonvery fine grids. However, in the Darcy domain, a permeabilityfield withvery small correlation lengths mayexhibit a highly fluctuatingflowandmaygiverisetonon-physicaloscillationsinthesolutionevenonfinermeshes.Thequalityofsolution willdependonthecellPécletnumberisdefinedas

P eh

(

i

,

j

)

:=

min

Duxx

i−12,j

,

u Dxx

i+12,j

,

v Dy y

i,j−12

,

v Dy y

i,j+12

h

.

(3.17)

ToavoidoscillationsitisdesiredtohaveP eh(i

,

j

)

2 ineachcell.Forthecurrentproblem,thevelocitiesuh,vharerandom

variablesanditisdifficultto boundthecellPécletnumber,especially oncoarsergrids. Afirst-orderUpwindDifferencing Scheme(UDS)fortheapproximationoftheconvectivefluxesseemsaconvenientchoice.Themaindisadvantageofthe first-orderupwindschemeishoweverexcessivenumericaldiffusionthatsmearsoutsharpfeaturesinthesolution.Therefore,to preventthisweuseaQuadraticUpwindInterpolationforConvectiveKinematics(QUICK)[36] schemefortheapproximation ofthe convective fluxes. Forthediffusion partwe willuse thecentral differencing scheme.Thisschemeis lessdiffusive comparedtothefirst-ordersolvesandisalsohighlystable.Furthermore,theQUICKschemeisformallythird-orderaccurate inspacethus can capturetheadditionalsmoothnessinthe Stokes region.The QUICKschemeishowever, notmonotone, which means that flux limiters will be employed for problems with sharp gradients. Many versions of the flux-limited QUICKmethodexistintheliterature [37,38] mostlybasedonthe deferredcorrection(orsometimes referredto asdefect correction)frameworkwherethefirst-orderschemeisimprovedusingahigher-ordercorrection.Inthisarticle,weconsider theimplicitversionforthesame.Thefluxfortheface“BC”(seeFig.5)isdefinedasfollows

F i+12,j

:=

Fi+12,j

U D S

+

(

r i+12,j

)

F i+12,j

Q U I C K

F i+12,j

U D S

,

(3.18)

(9)

Fig. 5.Stencil for the QUICK scheme for purely convection problem.

where

F i+12,j

U D S and

F i+12,j

Q U I C K are face-averaged convective fluxes computedusing the UDS andQUICK scheme,

respectively,andarecomputedas

F i+12,j

U D S

=

u i+12,j

ci,j

,

if ui+1 2,j

0

,

u i+12,j

(

ci+1,j), if ui+12,j

0

,

(3.19) and

F i+12,j

Q U I C K

=

u i+12,j

3 8ci+1,j

+

3 4ci,j

1 8ci−1,j

,

if u i+12,j

0

,

u i+12,j

3 8ci,j

+

3 4ci+1,j

1 8ci+2,j

,

if u i+12,j

0

.

(3.20)

Thelimiterfunctionis

(

r

i+12,j

)

,whereri+12,jisthemeasureofthelocalsmoothness.Thisiscomputedbytakingtheratio

ofsuccessivegradientsalongthestream-wisedirectiontodecidetheweightsforthefirst-orderandQUICKschemes,where

˜

r i+12,j

=

ci+1,j

ci,j ci,j

ci−1,j

,

if u i+12,j

0

,

ci+2,j

ci+1,j ci+1,j

ci,j

,

if u i+12,j

0

,

(3.21) and r i+12,j

:=

min

(

r

˜

i+12,j

,

r

˜

−1

i+12,j

)

.The inverse of

˜

r is usedto findlarge denominatorsin (3.21). The choice ofthe limiter

function

is somewhat arbitrary. We havenumerically testeda number ofclassical TVD-limiterfunctions (for e.g., see [39])andtheyperformedreasonablywell.Inthecurrentapplication,wherewewillencounterdiscontinuousboundaryand initial conditions,theKorenlimiter[40] with

(

r

)

:=

max

[

0

,

min

(

2r

,

(

2

+

r

)/

3

,

2

)

]

wasabletoreconstructthepropagating sharp fronts.The gradients canbe computedusingthe solutionfromtheprevious time-step andthen,modify thefluxes using(3.18) toobtaintheimprovedsolution.NotethatthestaggeredarrangementofvariablesfortheDarcy–Stokesflowis idealhereasthevelocitiesatthecellfacesaredirectlyobtainedfromthesolutionofdiscreteDarcy–Stokesflow.Theinflow andoutflowboundaryconditions(2.16)(a)–(b)areutilizedtocomputefluxesforcontrolvolumesthatliealongthedomain boundary.

3.4. Temporaldiscretization

Forthetemporaldiscretizations,manywellestablished methodsareavailableintheliterature.Inourapplication, flex-ibility inthetime-stepsizeisneededandthereforewe preferanimplicittime-steppingscheme.Wecanformallyrewrite thetransportequation(2.14)(a)inasemi-discreteformas

ch

t

+

L

hch

=

fh in

D

h

×

T

×

,

(3.22)

with

L

h denoting the spatial discretizationmatrix. Forease ofpresentation, we have omittedthe porosity

φ (

x

)

termin

(10)

grid points as tm

:=

m

t and will denote by cmh the approximation of c

(

·

,

tm

)

. To solve (3.22), we propose to use an AlternatingDirectionImplicitbasedsolutionmethodthatdecomposesthesemi-discreteform(3.22) intotwolinearsystems eachrequiringinversionofa tridiagonalmatrixforUDSorapentadiagonal matrixforQUICKscheme.TheADIsolveruses twohalftime-steps.First,anintermediatequantitycm−1/2isgeneratedbyperforminganimplicitEulertime-steppingalong thex-coordinateandanexplicitEuleralong y-direction.Usingthisintermediatequantity,thereverseisperformedforthe nexthalfstep.MotivatedbytheclassicalPeaceman–Rachfordapproach[41],onecanexpressADImethodas

Ih

+

2t

L

xh chm−1/2

=

2t fhm−1/2

+

Ih

2t

L

hy cmh−1

, (

a

)

Ih

+

2t

L

y h cmh

=

t 2 fhm

+

Ih

2t

L

hx c m−1/2 h

,

(

b

)

(3.23)

where Ih is an identity matrix,

L

y h and

L

x

h are tridiagonal (pentadiagonal) matrices derived from discretizations along x-coordinateand y-coordinate, respectively.Therearemanystandard algorithmsavailableforinvertingtridiagonalor pen-tadiagonalmatrices,forexample,theThomasalgorithmcanefficientlyinverta tridiagonalmatrixwithonly8floatingpoint operationsperunknown.Furthermore,thisADIschemeissecond-orderaccurateintimewhichcanbeeasilyverifiedusing Taylor’sexpansion.

4. MultigridsolverforDarcy–Stokesproblemwithlognormaldiffusion

Multigridmethodsare generallyrecognizedasfastefficientsolutionmethodsforalargeclass oflinearandnon-linear problems. We propose a monolithicmultigridalgorithm to solve the coupled Darcy–Stokes system with a heterogeneous stochasticpermeabilityfield.Thismultigridalgorithmstemsfromtheworkdonein[28],whereafixedpermeabilityvalue wasusedthroughouttheDarcydomain.Weextendthemethodtothecaseswithhighlyheterogeneouspermeabilityfields. Solving theDarcyflow withhighly fluctuating anddiscontinuous diffusionparameteris quitechallenging. In thecontext of geometric multigrid methods, basically two approaches exist, based on either the cell-centred or the vertex-centred location ofunknowns, see[19,25,42] and referencestherein.The vertex-centredapproach requirestransferoperators that are dependenton thediffusion parameter.Such multigrid implementationsmaybe expensiveasthesetransfer operators needtobemodifiedonallgridlevels.Duetothis,thecell-centredversionissomewhatmorebeneficialasitispossibleto achieveadecentmultigridconvergenceusingconstanttransferoperators[19,42].Wewilldemonstratethatthecell-centred approachcanbeextendedtothecoupledDarcy–Stokessystem.

Wewouldalsoliketopointoutthatanothercommonalternativeforsolvingthelinearsystemsderivedfromacoupled modelistheDomain DecompositionMethod(DDM).Ina DDM,onesplits themainboundaryvalue problemintosmaller boundaryvaluesubproblems.Typically,theboundaryinformationbetweensubproblemsisexchangedduringeveryiteration until the converged solution is reached. Contrary to this, monolithic solution approachestreat the coupled systemas a single problem.An importantaspectofthisapproachis thatthecouplingvariables betweenthesubproblemsare treated simultaneously, thus afterevery iteration, thethree fields

(

u

,

v

,

p

)

T are updated throughoutthe domain.The monolithic approachhasproventobeveryefficientwhenthesubproblemsarestronglycoupled[43,44].

4.1. Uzawasmootherforsaddle-pointsystem

Discretizationofeachsubproblem,themixedformofDarcyflowandtheStokesequations,yieldsasaddle-pointsystem. This saddle-point structure can be maintained forthe coupled Darcy–Stokes systemby ordering the velocity unknowns togetherforboththesubproblemsfollowedbythepressureunknowns.Thisresultsinthefollowinglinearsystem

A

h

B

hT

B

h 0

uh ph

=

ghf h

,

(4.1)

where uh

=

(

udh

,

ush

)

T, ph

=

(

pdh

,

psh

)

T, gh

=

(

0

,

fhs

)

T and fh

=

(

fd

,

0

)

T. For both subproblems,

B

hT and

B

h representthe

discretegradientandminusdiscretedivergenceoperators,respectively,and

A

h isthediscreterepresentationofeitherthe

Laplacianoperator

η

fortheStokesequations,or

η

Kh−1Ih fortheDarcyequation.

The multigridmethodproposed hereusesa specialclassofrelaxationmethods calledtheUzawa smoothers[24]. The Uzawasmootherisbasicallyanequation-wise,decoupledsmootherwherethevelocitycomponentsintheDarcyandStokes domainsarefirstupdated,afterwhichthepressurefieldisupdated.Inthefollowing,weprovidethedetailsofthissmoother.

Considerasplittingofthematrixcoefficientsofthesaddlepointsystem(4.1) as

A

h

B

hT

B

h 0

=

M

h 0

B

h

ζ

h−1Ih

M

h

A

h

B

hT 0

ζ

h−1Ih

,

(4.2)

where

M

h isa smootherfor theoperator

A

h and

ζh

isa positive parameter.For agiven approximationof thesolution

(

ukh−1

,

phk−1

)

T,therelaxedapproximation

(

uk

h

,

pkh

)

T,canbedefinedinthefollowingway:

M

h 0

B

h

ζ

h−1Ih

!

uk h pk h

"

=

M

h

A

h

B

hT 0

ζ

h−1Ih

!

uk−1 h pkh−1

"

+

gh fh

.

(4.3)

(11)

Intermsofvelocityandpressurevariables,therelaxationstepcanbewrittendownas:

ukh

=

ukh−1

+

M

−1h

gh

A

hukh−1

B

hTpkh−1

, (

a

)

pkh

=

pkh−1

+

ζ

h

(

B

hukh

fh

),

(

b

)

(4.4)

respectively.In[27,28],thechoiceof

M

h wasbasedonasymmetricGauss–Seideliterationfor(4.4)(a)thatgives

M

h

=

(h

+

Lh

)

−1h

(h

+

Uh), (4.5)

where

h

,LhandUharethediagonal,strictlylowerandstrictlyupperpartsof

A

h,respectively.AsymmetricGauss–Seidel

iterationcomprisesaforwardandabackwardsweepforvelocityintheentiredomain.

Next, forsmoothingofthe pressure variable,a Richardsoniteration (4.4)(b) withappropriate relaxationparameters is applied.Foranycontrolvolume

D

h3

(

i

,

j

)

,weusethefollowingrelaxationparameters

ζ

h

(

i

,

j

)

=

η

in

D

s

,

ηh

2 5Kh

(

i

,

j

)

in

D

d

,

(4.6)

where

η

isthefluidviscosityandKh(i

,

j

)

isderivedbyapplyingahalf-weighting(HW)operator Kh(i

,

j

)

=

1

8

[

4Ki,j

+

Ki−1,j

+

Ki+1,j

+

Ki,j−1

+

Ki,j+1

]

(4.7)

Thisisageneralizationoftheoptimalrelaxationparametersthat werederivedusingaLocalFourierAnalysis(LFA)in[27,

28]. Usually,an LFAisperformedonthemultigrid componentsobtainedbyfreezingthecoefficient field.Inthe caseofa variablecoefficientfield,

ζh

needs tobemodifiedlocally.Inourexperience,usingtheweighted average(4.7) resultedina robustconvergenceratethatwewilllaterdemonstratenumerically.

Remark4.1. Wewouldliketo mentionthatobtaining an analyticbound onthesmoothingfactoroftheUzawa smoother forthestochasticDarcy–Stokesflowisratherinvolved.Weleavethisforourfuturework.

4.2. Multigridalgorithmwithgridpartitioning

Tocoverrealisticproblems,weproposeamulti-blockmultigridmethodwhichisbasedonthegridpartitioningtechnique [25].Amulti-blockalgorithmrequirescommunicationbetweentheStokesandDarcydomainsduringthemultigriditeration. In ourcase,thestencils donotusevariableslocatedmorethan onecellaway fromthe currentcontrol volume,therefore padding theStokesblock withoneextra rowtostorethe variablesfromtheDarcyblock issufficientto achievethe data exchange.

The multigridhierarchyisbased onuniformcoarsening,i.e.thecell-widthisdoubledineach coarseningstep.A two-blocktwo-gridmethodwithvariablepermeabilityfieldcanbedescribedbythefollowingsteps:

1.Finegridpre-smoothing.Relax the velocities in two blocks using thesymmetric Gauss–Seidel smoother.The vertical velocity fromtheStokesblockalongtheinterfaceistransferredtotheDarcyblockafterwhichthepressureisupdatedby performingRichardsoniterationswithoptimalrelaxationparametersgivenby(4.6).TheupdatedDarcypressureunknowns arethentransferredtotheStokesoverlapcontrolvolumes.

2.Defectcomputationandrestriction.The defect (residual) iscomputed foreach variable. Next,the residuals from the Darcy block are transferred to the Stokes overlap region. We use fixed restriction operators to transfer residuals to the coarse grid. Forthe velocities, we usea six-point stencilwhereas for pressures a four-point restriction isapplied. These stencilsaredenotedby

[

Ih2h

]

u

=

1 8

1 2 1

1 2 1

2h h

,

[

Ih2h

]

v

=

1 8

12

12 1 1

2h h

,

[

I2hh

]

p

=

1 4

1

1 1 1

2h h

,

(4.8) respectively.

3.Coarsegridrelaxation.The coarsegriddefectequation issolved exactlybya directmethod,e.g.,Gaussianelimination. Alternately, one can performmultiple smoothingiterations asdescribed instep 1to getan approximate solutionon the coarsegrid.Weusedirectdiscretizationtoobtainthediscretesystemonthecoarsegridsandthecoarsegridrepresentation ofpermeability K2h isalsoobtainedbyusingtherestrictionoperator

[

Ih2h

]

p givenin(4.8).Finally,verticalStokesvelocity

unknownsattheinterfacearetransferredtotheDarcyblock.

4.Coarsegridcorrection. The error is interpolatedto the fine grid, where it isused correct the fine grid solution.The prolongationoperatorsarechosenastheadjointoftherestrictionoperators.

(12)

5.Finegridpost-smoothing.The interpolationprocess introduces newerrorsin thesolutionthat need tobe eliminated. Forthis,severalpost-smoothingstepsareperformedwiththesameprocedureasusedinpre-smoothinggiveninstep1.

Althoughtheabovemulti-blockalgorithmresemblesthedomaindecompositionmethod,themaindifferenceisthatthe communicationbetweenthetwodomainsisperformedonallmultigridlevelsandnotjustonthefinestandcoarsestgrids. Thismakesmulti-blockmethodtypicallymoreefficientthanthetraditionalDDM.

The two-block method can be easily extended to more than two blocks that may have more interfaceswithout any deteriorationofthemultigridconvergencerate.Furthermore,thistwo-grid routineiseasilyextendedtoV- and W-cycling strategies.Numericaltestsshowedthatforthisproblem,theW(2

,

2)-cycleexhibitsafastconvergencerateregardlessofthe roughnessofthecoefficientfields.

5. Single-levelMonteCarlomethod

Inthissection,wedescribestepstoformulateasingle-levelMCestimatoralongwiththeboundsfortheerrorsincurred fromthefinitevolumeandstatisticalapproximationsofthecoupledproblem.Theseerrorestimateswillbeneededforthe developmentandanalysisoftheMLMCestimator.

5.1. MCestimator

Weareinterestedinestimatingthestatisticalmomentsoftherandomconcentrationfieldatsome finaltimeT.We de-noteby c

(

·

,

tm

)

thesolutionatadiscretetime tm

=

T wherem

=

T

/

t.Forsimplicity,weconsideraspatio-temporalgrid

withh

=

t anddenotebycmh

(ωi)

theapproximationoftheconcentrationfieldwiththerandominput

ωi

.The single-levelMCestimator

E

hMC employs N independent,identicallydistributed(i.i.d.)realizations ofthesolutionfield

{

c

m h

(ωi)

}

N i=1 andisdefinedas

E

[

c

(

·

,

tm

)

] ≈

E

NMC

[

cmh

] :=

1 N N

i=1 cmh

(

ω

i). (5.1) To compute a sample, cm

h

(ωi)

, we proceed as follows:First, a random permeability field and other associated stochastic

parameters are generated.Then, usingthe multiblock multigridmethod,we solve the Darcy–Stokesflow forthe velocity field. With thesevelocities, the linear system for the transport model is derived. Next, the ADI solver is employed for time-steppingandaftermtimesteps,weobtainasample ofcm

h.Thesesamplesarethenaveraged.Thehighermomentsof

theQoIcanbecomputedinasimilarfashionfromthesampleensemble.

Theproposed MCestimatorcontainstwoadditive errorsdueto thefinitesamplingandtheFVapproximation andone mayseparatethesetwoerrorsas:

E

[

c

(

·

,

tm

)

] −

E

NMC

[

cmh

]

L2(;D)

E

[

c

(

·

,

tm

)

] −

E

NMC

[

c

(

·

,

tm

)

]

L2(;D)

+

E

NMC

[

c

(

·

,

tm

)

] −

E

NMC

[

cmh

]

L2(;D)(5.2)

,

respectively.Here,thefunctionspaceL2

(

;

D

)

correspondstothespaceofsquareintegrablemeasurablefunctions f

:

L2

(

D

)

forthegivencompleteprobabilityspace

(,

F

,

P

)

definedearlier.Thesespacesareequippedwiththenorm

||

f

(x,

t

,

ω

)

||

L2(;D)

:=

E

'

||

f

(x,

t

,

ω

)

||

2L2(D)

(

1 2

=

||

f

(x,

t

,

ω

)

||

2L2(D)d

P

(

ω

)

1 2

.

(5.3)

Theaboveerrornormisregardedtobetherootmeansquarederror(RMSE)andiswidelyusedintheliteraturetoquantify theerrorassociatedwithMonteCarloestimators[45,46].

5.1.1. Boundfordiscretizationerror

Thespatio-temporaldiscretizationintroducesabiasintheMCestimator.Therefore,itisimperativetoquantifythisbias asa functionofthemeshparameters.Theapproximationsofthesubproblems (2.1)(a)–(2.16)(b) mayexhibit differentgrid convergenceratesdependingontheregularityofthesolutionvariablesandthediscretizationorderofthenumericalscheme employed.Theerrorinthetransportequationsdependsontheerrorintheapproximationofthevelocityin

D.

Thus,the secondtermintheright-handsideof(5.2) canbeformallyexpressedas

E

NMC

[

c

(

·

,

tm

)

cmh

]

L2(;D)

C0

hαd

+

hαs

+

hαt1

+

hαt2

.

(5.4)

Here,theconstantC0

>

0 isindependentofhandconvergencerates

αd,

αs,

αt

1

,

αt

2 arealsopositive.Therates

αs

and

αd

are usedtoexpresstheerrors fromtheapproximationof theStokesandDarcyequations,respectively. Theerrors dueto thespatialandtemporaldiscretizationsofthetransportequationsare quantifiedusingtherates

αt

1 and

αt

2,respectively. Clearly,theasymptoticconvergencewillbedictatedbythetermwhichdecaysattheslowestrate,i.e.,

Figure

Fig. 1. Geometry of the Darcy–Stokes problem coupled with the transport equation. Subdivision of the domain D into a free-flow subregion D s and a porous medium subdomain D d , by an internal interface  ds .
Fig. 2. Staggered grid location of unknown and corresponding control volumes.
Fig. 3. Control volumes D 1 h ( i + 1 2 , j ) (left), D 2 h ( i , j + 1 2 ) (middle) and D h 3 ( i , j ) (right) for the primary unknowns u, v and p, c respectively, together with the corresponding indexing for each variable.
Fig. 4. Locations of unknowns required for discretization along the interface. (For interpretation of the colours in the figure(s), the reader is referred to the web version of this article.)
+7

References

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