stability of spatial interpolation functions in finite element one dimensional kinematic wave...

VNUJournalofScience,EarthSciences24(2008)5765

57

Stabilityofspatialinterpolationfunctionsinfiniteelementonedimensionalkinematicwaverainfallrunoffmodels

LuongTuanAnh1,*,RolfLarsson2

1ResearchCenterforHydrologyandWaterResources,InstituteofHydrometeorologicalandEnvironmentalSciences

2WaterResourcesEngineeringDepartment,LundUniversity,Box118,S22100Lund,Sweden

Received27May2008;receivedinrevisedform5July2008

Abstract. This paper analyzes the stability of linear, lumped, quadratic, and cubic spatialinterpolationfunctionsinfiniteelementonedimensionalkinematicwaveschemesforsimulationofrainfallrunoff processes. Galerkins residual method transforms the kinematic wave partialdifferentialequationsintoasystemofordinarydifferentialequations.Thestabilityofthissystemisanalyzedusingthedefinitionofthenormofvectorsandmatrices.Thestabilityindex,orsingularityofthesystem,iscomputedbytheSingularValueDecompositionalgorithm.Theoscillationofthesolution of the finite element onedimensional kinematic wave schemes results both from thesources,andfromthemultiplicationoperatorofoscillation.Theresultsofcomputationexperimentand analysis show the advantage and disadvantage of different types of spatial interpolationfunctionswhenFEMisappliedforrainfallrunoffmodelingbykinematicwaveequations.

Keywords:Rainfallrunoff;Kinematicwave;Spatialinterpolationfunctions;Singularvaluedecomposition;Stabilityindex.

1.Introduction1

Theneed for toolswhichhave capabilityofsimulatinginfluenceofspatialdistributionof rainfall and land use change on runoffprocesses initiated the development ofhydrodynamic rainfallrunoff models [1, 8].Oneofthebasicassumptionsforsuchmodelsregardstheexistenceofacontinuouslayerofwatermovingover thewhole surfaceof thecatchments.Althoughobservationsshowthatsuchconditionsarerare, theassumptioncan

_______ *Correspondingauthor.Tel.:844917357025.Email:[email protected]

berelaxedbyconsideringthetotalflowtobethe resultof the flow frommanysmallplotsdrainingintoafinenetworkofsmallchannels.

Theactualphysicalflowprocessesmaybequitecomplicated,but forpracticalpurposesthere is nothing to be gained fromintroducingcomplexityintothemodels.Asacommonway of getting optimal results, theonedimensional kinematicwavemodels [2,5, 8, 11] are often selected. These can besolvedbydifferentmethods,oneofwhich isthe finite element method (FEM) which isanalyzedinthispaper.

TheFEMmodelsarenormallyderivedbythe weighted residuals method, which is

LuongTuanAnh,RolfLarsson/VNUJournalofScience,EarthSciences24(2008)576558

based on the principle that the solutionresiduals should be orthogonal to a set ofweightingfunctions[7]:

0 = i)W

f)h(( ,

where: f(h) = :partialdifferentialequationofh;

i iNiah :estimatedsolution;

Wi:setofweightingfunctions;Ni:functionsofspatialordinate; ia :functionsoftime.According to Peyret and Taylor [9], the

weighted residualmethod is a general andeffective technique for transforming partialdifferential equations (PDE) into systems ofordinarydifferentialequations (ODE).When

ii ah , and iN are functions defined on aspatialinterval(element)themethodiscalledFEM.Thespecialcaseofweightingfunctions

ii NW = iscalledGalerkinsresidualFEMandit is often used for solving onedimensionalkinematicwaverainfallrunoffmodels.

The numerical solutions of the finiteelement schemes for overland flow andgroundwater flow in one dimensionalkinematic wave rainfallrunoff models mayoften run into problems with stability andaccuracy due to oscillation of the solution.The schememay be considered stablewhensmalldisturbancearenotallowedtogrowinthe numerical procedure. The reasons foroscillationof theGalerkinsFEMmethod forkinematic wave equations have beendiscussedbyJaberandMohtar[5].

Oneimportantfactorwhichinfluencesthestability characteristics of themethod is thechoiceofspatial interpolation function. Jaberand Mohtar [5] used linear, lumped andupwind schemes for spatial approximationand the enhanced explicit scheme fortemporal discretization. They analyzed thestabilityofdifferentschemesthroughFourier

analysis and concluded that the lumpedscheme is themost suitable for solution ofkinematicwaveequations.

Blandford et al [2] investigated linear,quadratic, and cubic interpolation functionsfor simulation of onedimensional kinematicwave by FEM and found that quadraticelementsproducedthemostaccuratesolutionwhen the implicit interactionprocedurewasusedfortemporaldiscretization.

The results of these researches and themathematical implication ofGalerkins FEMshow that the stability and accuracy of thefiniteelementschemesdoesnotonlydependonthetypeofspatialinterpolationfunctions,but also on the temporal integration of thesystem of ODE occurring when FEM isapplied for overland flow kinematic waveandgroundwaterBoussinesqequations.

In theworks cited above, the numericalschemes have been based on equidistantspatialelements.Inpracticalapplications,itisoften necessary to use elements of differentsize, where the discretization reflects thevariationofphysicalpropertiesofthechannelor the catchments beingmodeled.Themainpurposeofthispaperistoanalyzetheeffectsof varying size of spatial elements on thestability of the solution. Furthermore, theoriginofinstabilitywillbediscussed.

In theanalysis, thenumerical stabilityofthe various schemes will be evaluated byinvestigating associated matrices using theSingular Value Decomposition (SVD)algorithm. The following types of spatialinterpolation functions are investigated:linear,lumped,quadratic,andcubic.

2. Finite element schemes for onedimensionalkinematicwaveequations

The onedimensional kinematic wave

LuongTuanAnh,RolfLarsson/VNUJournalofScience,EarthSciences24(2008)5765 59

equations have been used for simulation ofthe rainfallrunoff process in small andaverage size river basinswith steep slopes.Theyhavebeenappliedinnumerousstudiesforhydrologicaldesign,floodforecastingetc.[2, 3, 6, 8, 11, 12]. The onedimensionalkinematic wave equations are normallywrittenintheformofthecontinuityequation:

t)r(x,xq

th

=

+ , (1)

and the equation of motion for (quasi)uniformflow:

hq = , (2)

where:h:flowdepth(m); q :unitwidthflow(m2/s); ),( txr :effectiverainfallorlateralflow(m/s); nSo /

2/1= ; 5/3; = n : Manning

roughnesscoefficient( /sm1/3 ); oS :thesurfaceorbottomslopethatequalstofrictionslopeinthecaseofkinematicwaveapproximation; x :spatialcoordinate(m);and t :time(s).

Equations(1)and(2)arepartialdifferentialequations which have no general analyticalsolution.However,withgiveninitialconditionh(t=0)andboundaryconditionh(x=0),numericalsolutions canbe found.Thekinematicwaveresultsfromthechangesinflowandsinceitisunidirectional(fromupstreamtodownstream),onlyoneboundaryconditionisrequired.

Principlesofspatialdiscretization for theonedimensional kinematic wave modelusing theFEMmethodhavebeenpresentedbyRossetal[11].Thesurfaceareaoftheriverbasin is divided in the crossflow directioninto strips. Each strip is then divided intocomputational elements based on thecharacteristics(e.g.slope)ofthebasinsothateachelementisapproximatelyhomogeneous.

For each computational element, thevariablesh(x,t)andq(x,t)areapproximatedintheform:

=

=

=

=

n

1i(t)i(x)qiNqt)q(x,

n

1i(t)i(x)hiNht)h(x,

;

(3)

where: )(xNi : space interpolation function(shapefunctionorweightingfunction).

Itisnotedthattheexpressions(3)shouldsatisfy not only Equation (1) but also theinitialconditionandtheboundarycondition.

The Galerkins residual methodnormalizes the approximated error withshapefunctionoverthesolutiondomain:

=

=

+

M

1i0dxiNirx

iNiqiNdt

idh .(4)

Theapproximation(3)combinedwiththeintegral (4) transforms thepartialdifferentialEquation (1) into a system of ordinarydifferentialequations,whichforeachelement(4)takestheform:

0(e)fq(e)Bdtdh(e)A =+ . (5)

For the linear scheme, the spatialinterpolationfunctionscanbedefinedas:

y(x)N =11 ,and y(x)N2 = ,where lxy /= ; l isthelengthoftheelement.

In this case, thematrices ofEquation (5)arewrittenas:

( ) ; 1111

21

=eB

;

36

63)(

= ll

lleA ),(

2

2)( txrl

le

=f

The lumped scheme [5] is based on thespatial interpolation functions expressed intheforms:

= 2

lsH1* 1jN ;

=

2lsH*jN

TheheavysidefunctionH(x)isdefinedas:H(x) = 0ifx < 0;H(x) = 1ifx 0;s:distancefromnodej1.


Thematrices for the lumped scheme ofEquation(5)canbeestimatedintheform:

0

021)(

=

lleA

Thematrix )(eB and vector )(ef remainthesameaslinearscheme.

In the case of quadratic scheme [2], thespatialinterpolationfunctionsare:

.2

;44

;231

23

22

21

yyN

yyN

yyN

+=

=

+=

Thematrices foroneelementaredefinedasfollowing:

;

152

1530

15158

15

3015152

)(

=

lll

lll

lll

eA

;

21

32

61

320

32

61

32

21

)(

=eB ),(

6

326

)( txr

l

l

l

e

=f

For cubic scheme (one element, fournodes),spatial interpolation functionscanbeexpressedintheforms: 321 5.495.51 yyyN += 322 5.135.229 yyyN += 323 5.13185.4 yyyN += 324 5.45.4 yyyN +=

The matrices for one element areintegratedandarepresentedas:

;

1058

56033

1403

168019

56033

7027

56027

1403

1403

56027

7027

56033

168019

1403

56033

1058

)(

=

llll

llll

llll

llll

eA

;

21

8057

103

807

80570

8081

103

103

80810

8057

807

103

8057

21

)(

=eB

),(

8

83838

)( txr

l

l

l

l

e

=f

For the whole domain containing theelements,Equation(5)hastheform:

0=+ fBqhAdtd (6)

In the case of using lumped scheme,matrices A; B and vector f for the domain(strip)containingnelementscanbepresentedintheforms:

.

2000..0000

022

00..0000

0022

00..000..........

0..022

00000

0..0022

0000

0..00022

000

00..00022

00

00..000022

0

00...00002

1

12

65

54

43

32

2

1

+

+

+

+

+

+

+

=

n

nn

nn

l

l

ll

ll

ll

ll

ll

ll

ll

l

A

=

21

21000...00

210

2100...00

0210

2100..00

..........

..........

0..0210

21000

0..00210

2100

00..00210

210

00...00210

21

00...00021

21

B

Crf =

+

+

+

=

2

22

22

22

2

11

3322

2211

11

nn

nnnn

rl

rlrl

rlrl

rlrl

rl


Foroverlandflow,thesystemofordinarydifferential equations (6), can be written intheform:

0=+ CrBqhAdtd , (7)

where:C:sparsematrixcontainingthesizeofelements;r:vectorofeffectiverainfall.

The solution of Equation (7) can beobtainedbyvariousnumericalmethods,oneofwhichisthestandardRungeKuttamethodand Successive Linear Interpolation forsolutionofODEwithboundaries[4,10].

Inorder toanalyzehow thestabilityandaccuracyofthesolutionschemesdependsonthe choice of spatial interpolation functions,equation (7) has been transformed into asystemoflinearalgebraicequations:

yAx = , (8)

where: xh =t

:unknownvector;

BqCry = :givenvectorforexplicittemporal differential scheme and estimatedvectorforimplicitinteractiveschemeforeachtimestep.

3.Stabilityanderroranalysis

In order to evaluate the stability ofvarious finite element schemes, the SingularValueDecomposition(SVD)algorithmwillbeapplied. Itwill be introduced anddescribedbelow together with the definition of someessentialvectorandmatrixconcepts:

(i)AccordingtotheSVDalgorithm[4.10],thematrixA (mm) can be expressed in theform:

TVUA = , (9)where U, V: square orthogonal matrices(mm), :diagonalmatrixwith 0ii calledsingularvaluesofmatrixA.

(ii)Thenormofthevectorxisdefinedas:2/1)( xxx = T (10)

(iii)Thenormof thematrixA isdefinedas themaximumcoefficientofextensionandcanbeexpressedas:

max===TT VUVUA (11)

ThephysicalimplicationofEquation(8)isthatonevector,x,inlinearspaceistransformedby A into another vector, y. Thistransformation takes three different forms:extension,compression,andturning.

The stability index, or singularity of thematrix A, can be defined as the ratio ofmaximumextensioncapacityovertheminimumcompressioncapacity,expressedas[4]:

min

max

min

max

min

max)(

=

==

x

xTVU

x

xTVU

xAxx

Ax

A

x

x

x

xCond ,(12)

where max , min : maximum and minimumsingularvaluesofArespectively.

Now,inordertostudythestabilityofthesolution scheme, a disturbance (oscillation)y is introduced. This results in acorrespondingdisturbance (oscillation)x inthe solution. The system of linear algebraicequations (8) with and without oscillationbecomes:

yAx = xxAy max= (13)

yyxxA +=+ )( xy min ,where: yx, : oscillation vector of solutionandoscillationvectoroferrorsrespectively.

Thismeansthat:

yy

Axx

)(Cond (14)

The relationship (14) shows that thestabilityofthesolutionofsystem(8)dependson thestability indexof thematrixAwithahigh value of the index indicating lowerstability.Therelationship(14)alsomeansthatthe stability index (or singularity ofA)maybe considered as the multiplication ofoscillationy:


qBrCy = . (15)Theupper limitofoscillation (15)canbe

estimated by applying the definition of thenormofvectorsandmatrices:

qrqBrC

qBrCy

++

=BCmaxmax

(16)

where: Bmax : maximum singular value of

matrix B; Cmax :maximum singular value ofmatrixC.

Expression (16) shows that the source ofoscillation include oscillation in the sourcetermr(effectiverainfall)aswellasoscillationintheadvectiontermaccumulatedduringthecomputation process. The upper limits ofthese oscillations depend on the chosenspatial interpolation function, and they arerelatedwith the structure of thematrices Band C respectively. These values will becomputed and the resultswill be discussedbelow for the selected typesof interpolationfunctions.

The solution of the system (8) normallyrequires to inversematrixA [5, 12].We canshow that the singularity of the (square)matrixAhasthesamevalueasthesingularityoftheinversematrixA1byusingEquation(9):

TUVA 11 = . (17)ApplicationofSingularValueDecomposition

ofA-1gives:T'''1 VUA = . (18)

The decompositions (9) and (18) arealmostunique [10]. Itmeans that '1 = ,and:

)1/()1(

)()(

maxmin

1

min

max

=

=== AA CondCond (19)

Therelationships (14)and (19)show thatthe stability and accuracy of solution ofsystem (8) are directly related with thesingularityofthehardmatrixA.

4.Numericalexperiments

Inordertoverifythemethodology,somebasic investigations are made for differenttypesof interpolationschemes insection4.1.Insection4.2, theeffectofusingelementsofvarious lengths is investigated. Finally, insection4.3,theinfluenceofdifferentdisturbancesourcesisanalyzed.

4.1.Stabilityindexofmatrix Afordifferenttypesofspatialinterpolationfunctions

Nowweassume that thestudiedstripofsurface area is divided into elements of(equal)unit length.The index of stability ofmatrix A has been computed for variousnumbers of elements for each type ofinterpolation function. The results of thecomputationsarepresentedinFig.1.

0.0

2.0

4.0

6.0

8.0

10.0

12.0

1 2 3 4 5 6 7Elements

Cond

(A)

LinearQuadraticCubicLumped

Fig.1.ThechangeofstabilityindexofmatrixA.

Thenumericalexperimentsshowthattheindexofstabilityisvirtuallyconstantforeachtype of interpolation scheme when thenumberofelementsistwoorhigher.Itisalsoclearthatthelumpedschemegivesthelowestvalue of stability index, while linear,quadraticandcubic schemesgive2,3and4times higher values respectively. Inconclusion, the lumped scheme has the


highest order of stability among the fourstudiednumericalschemes.

The results of numerical experimentspresentedaboveagreewellwiththeresultsofanalytical Fourier stability analysis forconsistent (linear) and lumped schemes thathavebeenpresentedintheworkbyJaberandMohtar[5].

4.2.Theimpactoffiniteelementapproximations

Numerical experiments have beenconducted in order to assess the effect ofelement size on stability of the four finiteelement schemes: linear, lumped, quadraticand cubic.The calculationshavebeenmadeforastripof1000m length,whichhasbeenapproximated by two elements. The lengthsof the two elements have been chosenaccording to three different options, withmoreor lessasymmetricproportions:option1withproportions1:1,option2withproportions1:9,andoption3withproportions1:99.

The stability index ofmatrix A and themaximum extension capacity of errors ofmatrices B andC have been computed andare shown inTable 1.The results show thatthe stability of the finite element onedimensional kinematic wave schemes doesnot only depend on the type of spatialinterpolationfunction,butalsoonthespatialdiscretizationof thesurfacestripconsidered.Forall four interpolation schemes, the lowerthestabilityis,themoredisproportionatetheelements are.At the same time for all threeoptions, each with different geometricproportions,thestabilityishigherforlumpedand linear schemes than that for quadraticandcubicschemes.

Another conclusion is that there are twomain causes for oscillation of the solution.One is the oscillation sources, and the otherone is the multiplication operator.

Furthermore,itshouldbepointedoutthattheefficiency of the algorithm is an importantaspect with regards to the choice ofinterpolationschemeforpracticalapplications.The linear and lumped schemes require n+1equations,whilequadraticandcubicschemesrequire2n+1and3n+1equationsrespectivelyforsolvingaproblemwithnelements.

Table1.StabilityindexofmatrixAandmaximumcoefficientofoscillation

Casesofstudy

LinearLumped

Quadratic

Cubic

Bmax 0.866 0.866 1.29 1.67Cmax 404.5 404.5 334.2 198.7

Option1

Cond(A) 3.73 2.00 5.83 8.13Bmax 0.866 0.866 1.29 1.67Cmax 452.8 452.8 618.5 355.8

Option2

Cond(A) 14.6 10.0 41.2 63.1Bmax 0.866 0.866 1.29 1.67Cmax 495.0 495.0 680.3 391.3

Option3

Cond(A) 149.6 100.0 448.8 688.6

4.3. The upper limit of oscillation sources fordifferenttypesofspatialinterpolationfunctions

Iftheoscillationoccurringatagiventimestep are supposed to be equal for differenttypes of spatial functions, then the upperlimit of source of oscillationwill be relatedwith the maximum singular values ofmatrices B and C. The structure of thesematrices is depended on the type ofinterpolation functions. The maximumsingularvaluesofBandC forunitelementsofequal lengthhavebeencomputedandarepresentedinTable2.

The results show that for advectionoscillation, both the linear and the lumpedschemes give values that are nearlyindependent of the number of elements,whilethequadraticandcubicschemesexhibit


increasing values for increasing number ofelements (see Fig. 2). The experiment alsoshows that linearand lumped schemeshavethesamesourceofoscillation.Theycanalsocontrol the advection oscillation better thanquadratic and cubic ones. However, theoscillation of effective rainfall component isbetter controlled by quadratic and cubicschemesthanbylumpedandlinearones.

Table2.Maximumcoefficientsofsourceofoscillation

Numberofelements

Parameters

LinearLumped

Quadratic

Cubic

Bmax 1.0 1.0 1.16 1.551Cmax 0.500 0.500 0.667 0.375B

max 0.866 0.866 1.29 1.672Cmax 0.809 0.809 0.689 0.398B

max 1.0 1.0 1.33 1.713Cmax 0.901 0.901 0.689 0.398B

max 0.951 0.951 1.34 1.734 C

max 0.940 0.940 0.689 0.398B

max 1.0 1.0 1.35 1.745Cmax 0.960 0.960 0.689 0.398B

max 0.975 0.975 1.35 1.756Cmax 0.971 0.971 0.689 0.398B

max 1.0 1.0 1.35 1.757Cmax 0.978 0.978 0.689 0.398

0.0

0.5

1.0

1.5

2.0

1 2 3 4 5 6 7Elements

Max

. ext

ensi

on c

apac

ity

Lumped/Linear

Quadratic

Cubic

Fig.2.Thechangeofmaximumextensioncapacity

ofmatrixB.

5.Conclusions

This paper analyses the sources andcauses of oscillation of solutions for finiteelement one dimensional rainfallrunoffmodels when different types of spatialinterpolation functions is applied foroverland flowkinematicwave simulation. Itdoessobyapplyingthedefinitionofnormofvectorsandmatricesand theSingularValueDecomposition(SVD)algorithm.

ThestructureofmatrixA,whichcontainssizes of the finite elements, is related to thetypeofspatialinterpolationfunctionwhichisapplied. From the above presented resultsanddiscussions, itcanbeconcluded that thestability indexorsingularityofmatrixAcanbeconsideredasaneffectofmultiplicationofoscillation occurring during computationprocess. It will affect the stability andaccuracyofthesolutionoffiniteelementonedimensionalkinematicwaveschemes,and itis actually one of the main causes ofoscillationofsolutions.

The results of computation experimentshow the advantage and disadvantage ofdifferent types of spatial interpolationfunctionswhen FEM is applied for rainfallrunoffkinematicwavemodels. If the reasonfor growing oscillation is seen as the mostimportant criterion for assessing stability ofnumerical schemes, the lumped and linearschemes have higher order of stability thanthe quadratic and cubic schemes.However,whenthelumpedschemeisused,thematrixA becomes a diagonalmatrix and then thealgorithmismoreefficientthanallotherthreetypesofschemes.

The results also show that the finiteelement onedimensional kinematic waveschemes can be improved by choosing themost suitable spatial interpolation functionfordecreasingthesingularityofmatrixAand


minimizethesourceofoscillation.Thespatialinterpolationfunctionsofhigherorderdonotalways give improved results when finiteelementmethod is used for kinematicwaverainfallrunoffmodels.

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