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 aCWI–CentrumWiskunde&Informatica,Amsterdam,theNetherlandsbFacultyofAerospaceEngineering,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
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 isthenumberofMCsamplesandV
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 overthestandardMCmethod,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 anFig. 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 intervalT
=
(
0,
T]
, for T<
∞
.The transport equation coupled withthe steady-state Darcy–Stokesflow onD
, is subdi-vided into a porous mediumD
d⊂
R
2, where the flow is described by Darcy’s law and the free-flow regionD
s⊂
R
2, governed bytheStokes equationswithboundaries∂
D
d and∂
D
s,respectively(Fig. 1).Theinternal interfaceisdefinedasds
=
∂
D
d∂
D
s.Furthermore,we denotebyω
an eventintheprobability space(,
F
,
P)
,whereisthesamplespace with
σ
-fieldF
andprobability measureP
. Ourdescriptionofthestochastic flow modelfollowsfrom[26–28] wherethe deterministiccounterpartofthisproblemisconsidered.Porousmediumdescription.The steady-statesingle-phaseflow ina porousmedium can be modelled byDarcy’slaw andtheincompressibilitycondition
ηK
−1ud+ ∇
pd=
0 inD
d×
, (
a)
∇ ·
ud=
fd inD
d×
.
(
b)
(2.1)Thefluidpressureisrepresentedby pd andthevelocityvectorud
=
(
ud,
vd)
denotesthehorizontalandverticalcompo-nents.Thefluidviscosityisdenotedbythepositiveconstant
η
andK isthespatiallyvariablepermeabilityfield.Theknown source(sink)termisindicatedby fd∈
L2(
D
d)
,whereL2(
D
d)
isthespaceofsquare-integrablefunctionsinD
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 andgdD
∈
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
dandω
∈
.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 thatcov(Z
(x
1,
·
),
Z(x
2,
·
))
=
C(
r)
with r=
x1−
x22.
(2.4) Fortheproblemtobewell-posed,weassumeTogeneratesamplesoftheGaussianrandomfield Z,thecovariancematrixisderivedfromthefamilyofMatérnfunctions
[29] characterizedbytheparameterset
=
(νc
,
λc,
σ
2c
)
,andhasthestandardformC
(
r)
=
σ
c22 1−νc(
ν
c)
2√
ν
c rλc
νc Kνc 2√
ν
c rλc
.
(2.6)Here,
isthegammafunctionandKνc isthemodified Besselfunctionofthesecondkind.Theparameter
νc
≥
1/
2 definesthe smoothness,
σ
2c
>
0 isthe variance andλc
>
0 isthe correlation length ofthe Gaussian process. TheMatérn modelhasgreatflexibilityinmodelling spatialprocessesbecauseofparameter
νc
,whichgovernsthedifferentiabilityofthe ran-domfield. Forνc
=
1/
2,theMatérnfunction correspondsto anexponential modelandforνc
→ ∞
to aGaussian model. Furthermore,the other two parametersλc
andσ
2c dictate the numberof peaksand theamplitude ofthe random field,
respectively.TherealizationsofGaussian randomfieldsareHöldercontinuousi.e., Z
(
x,
·
)
∈
Cκ(
D
d)
almost surelywiththe exponent0<
κ
<
νc
[30].Forνc
>
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 inD
s×
,
(
a)
∇ ·
us=
0 inD
s×
,
(
b)
(2.7)respectively, withus
=
(
us,
vs)
asthefluid velocity components andfs=
(
fs1
,
f2s)
∈
L2(
D
s)
2 is theforce. The fluid stress tensorσ
s isdefinedbyσ
s= −
psI+
2η
R(us),
(
a)
R:=
1 2(
∇
u s+
(
∇
us)
T), (
b)
(2.8)wherepsdenotesthefluidpressureandRisthestraintensor.In2D,thecomponentsofthestresstensor(2.8)(a)aregiven
by
σ
s:=
σ
xxsσ
xysσ
s yxσ
y ys,
(2.9)andisrelatedtotheprimitivevariablesvia
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
σ
s xx= −
ps+
2η
∂
us∂
x,
σ
s xy=
σ
yxs=
η
∂
us∂
y+
∂
vs∂
x,
σ
s y y= −
ps+
2η
∂
vs∂
y.
(2.10)Theexteriorboundary
∂
D
s\
dsispartitionedintotwodisjointsets,
∂
D
sD and∂
D
sN,andthefree-flowmodeliscompleted
byimposingthefollowingboundaryconditions
us
=
gsD on∂
D
sD,
(
a)
σ
s·
n=
gsN on
∂
D
sN, (
b)
(2.11)
wherenistheoutwardnormaltotheboundary
∂
D
sN andgsD
∈
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 onds
×
,
(
a)
−
ns·
σ
s·
ns=
pd onds
×
,
(
b)
us·
τ
+
√K αB Jτ
·
σ
s·
ns=
0 onds
×
, (
c)
(2.12)respectively.Wedenoteby
τ
andns theunittangential andnormalvectorstotheinterfaceds
,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.Fig. 2.Staggered grid location of unknown and corresponding control volumes.
Alternatively,oneobtainsano-slipinterfaceconditionbyneglectingthesecondtermfrom(2.12)(c)
us
·
τ
=
0 onds
×
.
(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 inD
×
T
×
, (
a)
c=
c0 inD
×
,
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.Theinitialconcentrationc0
:=
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 fieldD
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
andds
.Thelocationsandindexingofunknownsu
,
vandp,
calongwiththeirrespectivecontrolvolumesD
1h,
D
2h andD
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].
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)inthevolumeD
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+
2η
us i+32,j−
u s i+12,j h,
(
σ
xx)i,j= −
psi,j+
2η
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
−
2η
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,thesecondcomponentofthemomentumequationisintegratedinthevolumeD
h2(
i,
j+
12)
.Thecontinuity equa-tion(2.7)(b)isasgivenin(3.2) overthevolumeD
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.
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+
2η
vs i,j+32−
v s i,j+12 h.
(3.7)Next, thenormalstress
(σy y
)
i,j+12 attheinterface isderived by equilibratingthe normalstress withthepressure atthe interface
(
σ
y y)i,j+1 2=
p d i,j+12.
(3.8) Pressure pdi,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 2α
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=
2η
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)wherem
i,j+1 2=
1−
2ηKi,j+1 2 hαB J+2ηKi ,j+1 2.Thefinalcomponent
(σxy)
i−1 2,j+1
2 iscomputedinasimilarmanner.Usingallstress componentsderivedabove,werewrite(3.6) as
−
2η
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)
−
4η
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 resultinginadiagonaldispersiontensorD
. 3.3. SpatialdiscretizationUsingtheGaussdivergencetheorem,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,vharerandomvariablesanditisdifficultto 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)Fig. 5.Stencil for the QUICK scheme for purely convection problem.
where
F i+12,jU 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
(
ri+12,j
)
,whereri+12,jisthemeasureofthelocalsmoothness.Thisiscomputedbytakingtheratioofsuccessivegradientsalongthestream-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˜
−1i+12,j
)
.The inverse of˜
r is usedto findlarge denominatorsin (3.21). The choice ofthe limiterfunction
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 inD
h×
T
×
,
(3.22)with
L
h denoting the spatial discretizationmatrix. Forease ofpresentation, we have omittedthe porosityφ (
x)
termingrid points as tm
:=
mt 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],onecanexpressADImethodasIh
+
2tL
xh chm−1/2=
2t fhm−1/2+
Ih−
2tL
hy cmh−1, (
a)
Ih+
2tL
y h cmh=
t 2 fhm+
Ih−
2tL
hx c m−1/2 h,
(
b)
(3.23)where Ih is an identity matrix,
L
y h andL
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
hB
hTB
h 0uh 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 andB
h representthediscretegradientandminusdiscretedivergenceoperators,respectively,and
A
h isthediscreterepresentationofeithertheLaplacianoperator
−
η
fortheStokesequations,orη
Kh−1Ih fortheDarcyequation.The multigridmethodproposed hereusesa specialclassofrelaxationmethods calledtheUzawa smoothers[24]. The Uzawasmootherisbasicallyanequation-wise,decoupledsmootherwherethevelocitycomponentsintheDarcyandStokes domainsarefirstupdated,afterwhichthepressurefieldisupdated.Inthefollowing,weprovidethedetailsofthissmoother.
Considerasplittingofthematrixcoefficientsofthesaddlepointsystem(4.1) as
A
hB
hTB
h 0=
M
h 0B
h−
ζ
h−1Ih−
M
h−
A
h−
B
hT 0−
ζ
h−1Ih,
(4.2)where
M
h isa smootherfor theoperatorA
h andζh
isa positive parameter.For agiven approximationof thesolution(
ukh−1,
phk−1)
T,therelaxedapproximation(
ukh
,
pkh)
T,canbedefinedinthefollowingway:M
h 0B
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)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)thatgivesM
h=
(h
+
Lh)
−1h(h
+
Uh), (4.5)where
h
,LhandUharethediagonal,strictlylowerandstrictlyupperpartsofA
h,respectively.AsymmetricGauss–Seideliterationcomprisesaforwardandabackwardsweepforvelocityintheentiredomain.
Next, forsmoothingofthe pressure variable,a Richardsoniteration (4.4)(b) withappropriate relaxationparameters is applied.Foranycontrolvolume
D
h3(
i,
j)
,weusethefollowingrelaxationparametersζ
h(
i,
j)
=
⎧
⎨
⎩
η
inD
s,
ηh
2 5Kh(
i,
j)
inD
d,
(4.6)where
η
isthefluidviscosityandKh(i,
j)
isderivedbyapplyingahalf-weighting(HW)operator Kh(i,
j)
=
18
[
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,verticalStokesvelocityunknownsattheinterfacearetransferredtotheDarcyblock.
4.Coarsegridcorrection. The error is interpolatedto the fine grid, where it isused correct the fine grid solution.The prolongationoperatorsarechosenastheadjointoftherestrictionoperators.
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-temporalgridwithh
=
t anddenotebycmh
(ωi)
theapproximationoftheconcentrationfieldwiththerandominputωi
∈
.The single-levelMCestimator
E
hMC employs N independent,identicallydistributed(i.i.d.)realizations ofthesolutionfield{
cm h
(ωi)
}
N i=1 andisdefinedasE
[
c(
·
,
tm)
] ≈
E
NMC[
cmh] :=
1 N N i=1 cmh(
ω
i). (5.1) To compute a sample, cmh
(ωi)
, we proceed as follows:First, a random permeability field and other associated stochasticparameters 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)dP
(
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) canbeformallyexpressedasE
NMC[
c(
·
,
tm)
−
cmh]
L2(;D)≤
C0hαd
+
hαs+
hαt1+
hαt2.
(5.4)Here,theconstantC0