discrete artiﬁcial boundary conditions vorgelegt von...

DiscreteArtificial Boundary Conditions

vorgelegt vonDiplom–Technomathematiker MatthiasEhrhardt

ausBerlin

Vom FachbereichMathematikderTechnischenUniversitat Berlin

zurErlangungdesakademischenGradesDoktorderNaturwissenschaften(Dr. rer. nat.)

genehmigteDissertation

Promotionsausschuß:Vorsitzender:Prof.Dr. UdoSimon,TU BerlinBerichter: Prof.Dr. AntonArnold, UniversitatdesSaarlandesBerichter: Prof.Dr. Rolf DieterGrigorieff, TU Berlin

TagderwissenschaftlichenAussprache:25.5.2001

Berlin 2001D 83

Contents

Acknowledgement iii

Abstract v

Zusammenfassung vii

Introduction 1

Chapter1. TheSchrodingerEquation 51. TransparentBoundaryConditions 52. DiscreteTransparentBoundaryConditions 93. DTBC for non–compactlysupportedInitial Data 264. NumericalInverse

–Transformations 33

Chapter2. TheConvection–DiffusionEquation 411. TransparentBoundaryConditions 412. DiscreteTransparentBoundaryConditions 443. NumericalResults 57

Chapter3. TheWide–AngleEquationof UnderwaterAcoustics 671. Introductionto UnderwaterAcoustics 672. TransparentBoundaryConditions 723. CoupledModelsfor UnderwaterAcoustics 764. TheTransparentBoundaryConditionfor anElasticBottom 785. DiscreteTransparentBoundaryConditions 826. NumericalExamples 91

ConclusionsandPerspectives 101

Appendix 103TheLaplaceTransformation 103TheInverseLaplaceTransformation 104The

–Transformation 105

TheInverse

–Transformation 106

Bibliography 109

CurriculumVitae 115

i

ii CONTENTS

Acknowledgement

I want to expressmy deepgratitudeto my Ph.D.supervisorProf.Dr. Anton Arnold for hisperseveringsupportduring the lastyears.I alsothankProf. Dr. PeterMarkowich for offeringmeapositionat theTechnicalUniversityin Berlin.

Specialthanksto Prof. Dr. JuanSolerfor inviting measa TMR–Predocto Granada.I amgratefulto Prof.Lynessfor answeringall my questionsaboutENTCAFandProf.RonMickensfor many helpful commentson differenceequations. I would like to thank Prof. Makrakisfor inviting meto a workshopon underwateracousticson CreteandProf. Finn Jensenfor hisadviceon testcasesfor underwateracoustics.I thankProf. Mireille Levy for a fruitful emailcommunication,Prof.JohnPapadakisandDr. FrankSchmidtfor helpfuldiscussions.

This work wasfinanciallysupportedby theGermanResearchCouncil (DFG) underGrantNo. MA 1662/1–3andMA 1662/2–2.

The lastsentenceis dedicatedto my parentswho madeit possiblefor me to studymathe-matics.

iii

iv ACKNOWLEDGEMENT

Abstract

Whencomputingnumericallythesolutionof apartialdifferentialequationin anunboundeddomainusuallyartificial boundariesareintroducedto limit thecomputationaldomain.Specialboundaryconditionsarederived at this artificial boundariesto approximatethe exact whole–spacesolution. If the solutionof the problemon the boundeddomainis equalto the whole–spacesolution(restrictedto the computationaldomain)theseboundaryconditionsarecalledtransparentboundaryconditions(TBCs).

Thisdissertationis concernedwith transparentboundaryconditionsfor convection–diffusionequationsandgeneralSchrodinger–typepseudo–differentialequationsarisingfrom “par abolic”equation(PE)modelswhich have beenwidely usedfor one–way wave propagationproblemsin variousapplicationareas,e.g.seismology, opticsandplasmaphysics.As a specialcasetheSchrodingerequationof quantummechanicsis included.

Existingdiscretizationsof theseTBCsinducenumericalreflectionsat this artificial bound-ary andalsomaydestroy thestability of theusedfinite differencemethod.To overcomebothproblemswe proposea new discreteTBC which is derived from the fully discretizedwhole–spaceproblem. This discreteTBC is reflection–freeandconservesthe stability propertiesofthewhole–spacescheme.While we shallassumea uniform discretizationin time, the interiorspatialdiscretizationmaybenonuniform.Thesuperiorityof thenew discreteTBC overexistingdiscretizationsis illustratedonseveralbenchmarkproblems.

v

vi ABSTRACT

Zusammenfassung

Bei der numerischenBerechnungder Losungeiner partiellen DifferentialgleichungaufeinemunbeschranktenGebietwerdengewohnlichkunstlicheRandereingefuhrt,umdasRechen-gebietzu beschranken. SpezielleRandbedingungenwerdenan diesenkunstlichenRandernhergeleitet,um die exakteGanzraumlosungzu approximieren.Falls die LosungdesProblemsauf dembeschranktenGebietmit der Ganzraumlosung(eingeschrankt auf dasRechengebiet)identischist, werdendieseRandbedingungenals transparenteRandbedingungen(TRB)beze-ichnet.

Die vorliegendeDoktorarbeitbefaßtsichmit transparentenRandbedingungenfur Konvek-tions–Diffusionsgleichungen und allgemeine Pseudodifferentialgleichungen vomSchrodingertyp.Diesesogenannten“Parabolischen” Gleichungenfindenweit verbreiteteAn-wendungbei1–Weg–Wellenausbreitungsproblemenin vielenBereichen,z.B.Seismologie,Op-tik undPlasmaphysik.Als Spezialfall ist die SchrodingergleichungderQuantenmechaniken-thalten.

ExistierendeDiskretisierungendieserTRB fuhren zu numerischenReflektionenan denkunstlichenRandernundzerstorenhaufigdie Stabilitat derzugrundeliegendenfinite Dif feren-zenMethode.Um beideProblemezu losen,fuhrenwir eineneuediskreteTRBein, die direktvom diskretisiertenGanzraumproblemhergeleitetwird. DiesediskreteTRB ist reflektions-frei underhalt dieStabilitatseigenschaftendesGanzraumschemas.Wahrendwir eineuniformeDiskretisierungin derZeit voraussetzenmussen,kanndie innereDiskretisierungnichtuniformim Ort sein. Die Uberlegenheitder neuendiskretenTRB gegenuber anderenexistierendenDiskretisierungenvonTRBenwird anhandvonmehrerenBeispielenillustriert.

vii

viii ZUSAMMENFASSUNG

Intr oduction

Many physicalproblemsare describedmathematicallyby a partial differential equation(PDE)which is definedonanunboundeddomain.If onewantsto solvesuchwhole–spaceevo-lution problemsnumerically, onefirst hasto make it finite dimensional.Thestandardapproachis to restrictthecomputationaldomainby introducingartificial boundaryconditionsor absorb-ing layers.Furtherpossiblemethodsthatcanbeappliedareboundaryelementmethods(BEM)(cf. [U17]) or infinite elementmethods(IEM) (seefor example[U14]). In this dissertationwefocuson theapproachof artificial boundaryconditions.

If theinitial datais supportedon a finite domainΩ, onecanconstructboundaryconditions(BCs)on ∂Ω with theobjective to approximatetheexactsolutionof thewhole–spaceproblem,restrictedto Ω. SuchBCs arecalledabsorbingboundaryconditions(ABCs) if they yield awell–posed(initial) boundaryvalueproblem(IBVP), wheresome“energy” is absorbedat theboundary. If the approximatesolutioncoincideson Ω with the exact solution,onereferstotheseBCsastransparentboundaryconditions(TBCs). Of course,theseboundaryconditionsshouldleadto a well–posed(initial) boundaryvalueproblem.Additionally, it is desirablethattheBCsarelocal in spaceand/ortime to allow for anefficientnumericalimplementation.

Hereweareconcernedwith theconstructionanddiscretizationof TBCsfor generalSchro-dinger–typepseudo–differentialevolutionequationsin 1D of theform

iψt p0 p1

∆ V

x t

1 q1

∆ V

x t 1 ψ x IR t 0

ψx 0 ψI x

(GS)

wherethe real coefficients p0, p1, q1 areconstant,∆ denotesthe Laplacianand the complexvalued“potential”V is assumedto begiven.As aspecialcase(q1 0,V IR) theSchrodingerequationof quantummechanicsis included.

Equationsof theform (GS)arisefrom “par abolic” equation(PE)models,whichhavebeenwidelyusedfor 1–waywavepropagationproblemsin variousapplicationareas,e.g.seismology[U4], [U5], opticsandplasmaphysics(cf. thereferencesin [U3]). In underwateracousticstheyappearaswide angleapproximationto the Helmholtzequationin cylindrical coordinatesandarecalledwideangleparabolicequations(WAPE) [U18].

The usual strategy of employing TBCs for solving a whole–spaceproblemnumericallyconsistsof first deriving ananalyticTBC at theartificial boundary. TheseTBCs aretypicallynonlocalin time(of “memory–type”)andcanbeapproximatedby a local–in–timeBC. Finally,thecontinuousBC mustbediscretizedto useit with aninterior discretizationof thePDE.TheanalyticTBC for theSchrodingerequationwasindependentlyderivedby severalauthorsfromvariousapplicationfields [T1], [T6], [T12], [T16], [T18], [T20].

While continuousTBCs fully solve the problemof cutting off the spatialdomainfor theanalyticalequation,their numericaldiscretizationis far from trivial. Up to now the standardstrategy is to derive first the TBC for theanalyticequation,thento discretizeit, andto useit

1

2 INTRODUCTION

in connectionwith someappropriatenumericalschemefor the PDE.The defectof this usualapproachof discretizingcontinuousTBCs(discretizedTBCs) is thattheinnerdiscretizationofthePDEoftendoesnot “match” thediscretizationof theTBCs. Therearetwo major problemsof theseexistingconsistentdiscretizationsof thecontinuousTBC:

P1: The discretizedTBCs for the Schrodinger–typeequation(GS) often destroy the stabil-ity of the whole–spacefinite differencescheme.Especiallyfor the Schrodingerequa-tion of quantummechanicstheunconditionalstabilityof theunderlyingCrank–Nicolsonschemeis destroyed[T16] andtheoverallnumericalschemeis renderedonly condition-ally stable[T6], [T16], [T19], [T26].

P2: The availablediscretizationsoften suffer from reducedaccuracy (in comparisonto thediscretizedwhole–spaceproblem)andinducenumericalreflectionsat theboundary, par-ticularly whenusingcoarsegrids.

In thisdissertationwediscussin detailarecentlydevelopedapproach(first outlinedin [T1])whichovercomesboththestabilityproblem(P1)andtheproblemof reducedaccuracy(P2). Inour discreteapproach weproposeto changetheorderof thetwo stepsof theusualstrategy, i.e.we first considerthediscretizationof thePDEon thewholespaceandthenderive theTBC forthedifferenceschemedirectlyon apurelydiscretelevel.

Thereareseveraladvantagesof thediscreteapproach. It completelyavoidsany numericalreflectionsat theboundary:no additionaldiscretizationerrorsdueto theboundaryconditionsoccur. The discreteTBC is alreadyadaptedto the inner schemeandthereforethe numericalstability is oftenbetter–behavedthanfor a discretizeddifferentialTBC. An additionalmotiva-tion for this discreteapproacharisesfrom thefact thatthenumericalschemeoftenneedsmoreboundaryconditionsthantheanalyticalproblemcanprovide (especiallyhyperbolicequations,systemsof equationsandhigh–orderschemes).

In the literaturethe discreteapproachdid not gain muchattentionyet. The first discretederivationof artificial boundaryconditionswaspresentedin [D2, Section5]. Thisdiscreteap-proachwasalsousedin [D5], [D6], [D7] for linearhyperbolicsystemsandin [D3] for thewaveequationin onedimension,alsowith error estimatesfor the reflectedpart. In [D6] a discrete(nonlocal)solutionoperatorfor generaldifferenceschemes(strictly hyperbolicsystems,withconstantcoefficientsin 1D) is constructed.Lill generalizedin [D4] theapproachof EngquistandMajda[D2] to boundaryconditionsfor aconvection–diffusionequationanddropsthestan-dardassumptionthat the initial datais compactlysupportedinsidethecomputationaldomain.However, the derived

–transformedboundaryconditionswereapproximatedin orderto get

local–in–timeartificial boundaryconditionsaftertheinverse

–transformation.Hereweconstructdiscretetransparentboundaryconditions(DTBC) for aCrank–Nicolson

finite differencediscretizationof (GS)suchthattheoverallschemeis unconditionallystableandasaccurateasthediscretizedwhole–spaceproblem.TheresultingDTBC is ageneralizationoftheDTBC for theSchrodingerequationin [T1]. Thesamestrategy appliesto theθ–schemeforconvection–diffusion equations[P2] andwasalsousedin [T10] for the wave equationin thefrequency domain.

Although this work concentrateson the discretederivation of BCs, we will alsoconsiderthe continuousproblem,sincethe basicideasof the constructionandderivationcarry over tothe discretecase,e.g.we canusediscreteversionsof the L2–estimates.Moreover the well–posednessof the continuousproblemis necessaryfor the stability of the numericalscheme.Justlike theanalyticTBC, thediscreteTBC will benonlocalin thetimevariable.

INTRODUCTION 3

The dissertationis organizedasfollows: In Chapter1 we introduceour new approachofderiving DTBCs:wefirst review thecontinuousTBC for theSchrodingerequationin onespacedimensionandderive andanalyzethe discreteTBCs which arein the form of a discretecon-volution. In orderto obtainanefficient implementationonecaneasilylocalizethesenonlocalin time DTBCs just by cutting off the rapidly decayingsequenceof the convolution coeffi-cients. Variousnumericalexamplesin Section2 illustratethe superiorityof our DTBC overexisting discretizations.At theendof the first Chapterwe discussthe DTBC in thecasethatthe initial datais not supportedin the computationaldomainand show how to apply a nu-mericalinverse

–transformationif theexact inverse

–transformationcannotbedetermined

analytically. Chapter2 showshow thepresentedmethodcanbeappliedto a linearconvection–diffusionequationsandto a moregeneralfinite differencescheme.Finally, the third Chaptergivesa morepracticalapplicationof theDTBCsto underwatersoundpropagation.We discusstheDTBCsfor theCrank–Nicolsonschemefor theWAPE (which is of theform (GS)). In theconclusionwesummarizetheobtainedresultsanddiscusstopicsfor furtherresearch.

4 INTRODUCTION

CHAPTER 1

The Schrodinger Equation

In this Chapterwe want to clarify the approachof deriving discretetransparentboundaryconditions.For simplicity we considerthetime–dependentSchrodingerequationin onespacedimension.Thisapproachwasintroducedby Arnold in [T1].

1. Transparent Boundary Conditions

In this Sectionweshallsketchthederivationof theTBC anddiscussthewell–posednessoftheresultingIBVP. Herewewill treatthecaseof theSchrodinger equation

i ψt 2

2∆ψ V

x t ψ x IR t 0

ψx 0 ψI x(1.1.1)

whereψI L2 IR , V t L∞

IR andVx is piecewisecontinuous.

1.1. Derivation of theTBC. Ourgoalis to designtransparentboundaryconditions(TBCs)at x 0 andx L, suchthat the resultingIBVP is well–posedandits solutioncoincideswiththesolutionof thewhole–spaceproblemrestrictedto

0 L .

Wemake thefollowing two basicassumptions:

A1: Theinitial dataψI is supportedin thecomputationaldomain0 x L.A2: The given electrostaticpotentialis constantoutsidethis finite domain: V

x t 0 for

x 0,Vx t VL for x L.

REMARK 1.1. Withoutthefirst assumptioninformationwouldbelost,andthewhole–spaceevolution couldnot be reproducedon the finite interval 0 L . The secondassumptionallowsto explicitly solve the equationin the exterior of the computationaldomain(by the Laplace–method)which is thebasicideaof thederivationof theTBC. In Section3 we will seehow todroptheassumption(A1).

We presenta formal derivationof theTBC for smooth(i.e. C1) solutions.Afterwards,theobtainedTBC canbe regardedfor lessregular solutions. The first stepis to cut the originalwhole–spaceproblem(1.1.1)into threesubproblems,the interior problemon thedomain0 x L, anda left andright exteriorproblem.They arecoupledby theassumptionthatψ, ψx arecontinuousacrosstheartificial boundariesat x 0, x L. The interior problemreads

i ψt 2

2∆ψ V

x t ψ 0 x L t 0

ψx 0 ψI x

ψx0 t

T0ψ 0 t ψx

L t

TLψ L t (1.1.2)

5

6 1. THE SCHRODINGER EQUATION

T0 L denotetheDirichlet–to–Neumannmapsat theboundaries,andthey areobtainedby solvingthetwo exterior problems:

i vt 2

2∆v VLv x L t 0

vx 0 0

vL t Φ

t t 0 Φ

0 0

v∞ t 0

TLΦ t vxL t

(1.1.3)

andanalogouslyfor T0. Sincethepotentialis constantin the exterior problems,we cansolvethemexplicitly by theLaplacemethodandthusobtainthetwo boundaryoperatorsT0 L neededin (1.1.2). This ideais illustratedin Figure1.1.

problem

(explicitly solvable) right

input:

exterior

L

output: v (0,t)x

ψboundary data (0,t)

interior problem

x0

left exterior problem

(x,t)ψ

Iψ

ψ

FIGURE 1.1. Schrodingerequation: Constructionideafor transparentboundaryconditions

TheLaplacetransformationof v is givenby

vx s ∞

0vx t e st dt (1.1.4)

wherewe sets η iξ, ξ IR, andη 0 is fixed, with the ideato later performthe limitη 0. Now theright exteriorproblem(1.1.3)is transformedto

vxx i2

s iVL v 0 x L

vL s Φ

s (1.1.5)

Sinceits solutionshave to decreaseasx ∞ (sincewehave ψ t L2

IR ), weobtain

vx s e i 2 s i VL x L Φ

s (1.1.6)

1. TRANSPARENT BOUNDARY CONDITIONS 7

HencetheLaplace–transformedDirichlet–to–NeumannoperatorTL reads

TLΦs vx

L s 2

e i π4 s i

VL Φs(1.1.7)

andT0 is calculatedanalogously. Here, ! denotesthebranchof thesquareroot with nonneg-ativerealpart.

An inverseLaplacetransformationyieldstheright TBCat x L:

ψxL t 2

πe i π

4 e i VL t ddt

t

0

ψL τ ei VL τ!

t τdτ

(1.1.8)

Similarly, the left TBCatx 0 is obtainedas

ψx0 t 2

πe i π

4ddt

t

0

ψ0 τ !t τ

dτ

(1.1.9)

TheseBCs arenonlocal in t andof memory–type,thusrequiring the storageof all previoustime levels at the boundaryin a numericaldiscretization.A seconddifficulty in numericallyimplementing(1.1.8),(1.1.9)is thediscretizationof thesingularconvolution kernel.A simplecalculationshowsthat(1.1.8)is equivalentto the impedanceboundarycondition[T18]:

ψL t

2πei π

4t

0

ψxL t τ e i VL τ!

τdτ

(1.1.10)

Likewise,(1.1.9)is equivalentto

ψ0 t 2π

ei π4

t

0

ψx0 τ !

t τdτ

(1.1.11)

REMARK 1.2(InhomogeneousTBC). The(homogeneous)TBC(1.1.9)wasderivedfor mod-eling the situationwherean initial wave function is supportedin the computationaldomain 0 L , andit is leaving this domainwithout beingreflectedback.If anincomingwave functionψin

t is givenat theleft boundary(e.g.a right travelingplanewave), theinhomogeneousTBC

ψ0 t ψin

0 t

x 2

πe i π

4ddt

t

0

ψ0 τ ψin

τ !

t τdτ (1.1.12)

hasto beprescribedat x 0. This is theTBC (1.1.9)formulatedfor ψ0 t ψin

t sincethe

TBC wasonly derivedfor outgoingwavefunctions.TheinhomogeneousTBC is describedandanalyzedin detail in [T5].

REMARK 1.3(Factorization). It shouldbenotedthattheSchrodingerequationcanformallybefactorizedinto left andright travelling waves(cf. [T6]):

∂∂x

2e i π

4∂∂t i

VL ∂∂x 2

e i π4

∂∂t i

VL ψ 0(1.1.13)

andin thepotential–freecase:

∂∂x

2e i π

4∂∂t

∂∂x 2

e i π4

∂∂t

ψ 0(1.1.14)

wheretheterm

ddt

ψ : 1!π

ddt

t

0

ψτ !

t τdτ(1.1.15)


canbeinterpretedasa fractional(12) timederivative.

1.2. Well–posednessof the IBVP. We now turn to thediscussionof the well–posednessof (1.1.2). Theexistenceof a solutionto the1D Schrodingerequationwith theTBCs(1.1.8),(1.1.9)is clearfrom theusedconstruction.For regularenoughinitial data,e.g.ψI H1 0 L ,thewhole–spacesolutionψ

x t will satisfytheTBCsat leastin aweaksense.A moredetailed

discussionis presentedin [T9].It remainsto checktheuniquenessof thesolution,i.e. whethertheTBC givesrise to spu-

rious solutions. In order to prove uniform boundednessof " ψ t #" L2 0 L in t we will need

thefollowing simplelemmawhichstatesthatthekernelof theDirichlet–to–Neumannoperatoreiπ $ 4 d % dt is of positivetypein thesenseof memoryequations(see,e.g.[M2]).

UsingthePlancherel equalityfor theLaplacetransformation(L.4) thefollowing lemmacanbeshown:

LEMMA 1.1( [T1]). For anyT 0, let u H140 T with theextensionu

t 0 for t T.

Then

Re ei π4

∞

0ut d

dt

t

0

us!

t sds dt 0

(1.1.16)

With this lemma we shall now derive an estimatefor the L2–norm of solutionsto theSchrodingerequation(1.1.2).Wemultiply (1.1.2)by ψ :

ψψt i2

ψψxx i

Vx t '& ψ & 2 0 x L t 0

(1.1.17)

Integratingby partson 0 x L, andtakingtherealpartgives

∂t

L

0& ψ t (& 2 dx Re i ψ

x t ψx

x t x) L

x) 0

2π

Re ei π4 ψ

L t e i VL t d

dt

t

0

ψL τ ei VL τ!

t τdτ

2π

Re ei π4 ψ

0 t d

dt

t

0

ψ0 τ !t τ

dτ

(1.1.18)

Now integratingin timeandapplyingLemma1.1for thesecondtermandananalogouslemmafor thefirst termyieldstheestimate

" ψ t (" L2 0 L *" ψI " L2 0 L t 0

(1.1.19)

This impliesuniquenessof thesolutionto theSchrodingerIBVP. Equation(1.1.19)reflectsthefactthatsomeof theinitial massor particledensityn

x t & ψ

x t (& 2 leavesthecomputationaldomain 0 L duringtheevolution. In thewhole–spaceproblem,x IR, " ψ

t (" L2 IR is of courseconserved.

Finally, we addressthequestionof thewell–posednessof theSchrodingerequation(1.1.2)with inhomogeneousTBCs (cf. Remark1.2). We assumethat the incoming wave functionψin

t is givenat theleft boundaryby

ψinx t αe i ωt 2 ωx ω 0(1.1.20)

2. DISCRETETRANSPARENT BOUNDARY CONDITIONS 9

i.e.a right travelingplanewave. Thentheauxiliary function

ϕx t : ψ

x t 1 x

Lψin

0 t 0 x L t 0(1.1.21)

fulfils thefollowing inhomogeneousSchrodingerequation

i ϕt 2

2∆ϕ V

x t ϕ f

x t

fx t : 1 x

Lψin

0 t V

x t ω

ϕx 0 ϕI x ψI x 1 x

Lα

(1.1.22)

with theleft homogeneousTBC

∂∂x

2e i π

4∂∂t

ϕ0 t 0(1.1.23)

andtheright TBC (1.1.8). Proceedingasin thehomogeneouscaseweobtain

∂t " ϕ t (" 2

L2 0 L 1Im

L

0f ϕdx 1 " f

t #" L2 0 L " ϕ t (" L2 0 L (1.1.24)

If we furtherassume" f t (" L2 0 L F , t 0 thenit followseasilythat

" ψ t #" L2 0 L *" ϕI " L2 0 L α

L3 F

2t t 0(1.1.25)

andthis impliesthewell–posedness.

2. DiscreteTransparent Boundary Conditions

In this Sectionwe shall discusshow to discretizethe TBC (1.1.8) in conjunctionwith aCrank–Nicolsonfinitedifferenceschemeandreview thederivationof theDTBC from [T1].

With theuniformgrid pointsx j j∆x, tn n∆t, andtheapproximationsψnj + ψ

x j tn the

discretizedSchrodinger equation(1.1.1)reads:

i Dt ψnj 2

2D2

xψn 12

j Vn 1

2j ψn 1

2j V

n 12

j Vx j tn 1

2(1.2.1)

with the time averagingψn 1$ 2j

ψn 1j ψn

j % 2. Here Dt denotesthe forward differencequotientin timeandD2

x is thesecondorderdifferencequotientin space,i.e.

Dt ψnj ψn 1

j ψn

j

∆t D2

xψnj ψn

j 1 2ψn

j ψnj 1

∆x 2

REMARK 2.1. Mostexistingdiscretizationschemesfor theSchrodingerequationwith TBCs

arealsobasedon theCrank–Nicolsonfinite differences( [T6], [T16], [T19]).

For our analysis,oneof themainadvantagesof this secondorder (in ∆x and∆t) schemeis,thatit is unconditionallystable[F8] andaneasycalculationshowsthatit preservesthediscreteL2–norm: " ψn " 2

2 ∆x∑ j , ZZ & ψnj & 2, which is the discreteanalogueof the massconservation

propertyof (1.1.1).


In ordertoderivethisdiscretemassconservationpropertywemultiply (1.2.1)with iψn 1$ 2j % :

ψn 12

j Dt ψnj i

2ψn 1

2j D2

xψn 12

j i

Vn 1

2j ψn 1

2j

2 (1.2.2)

Summingit up for j ZZ (i.e. in absenceof boundaryconditions)giveswith summationbyparts

∑j , ZZ

ψn 12

j Dt ψnj i

2 ∑j , ZZ

Dx ψn 12

j

2 i ∑j , ZZ

Vn 1

2j ψn 1

2j

2 (1.2.3)

Finally, takingtherealpartby usingthesimpleidentity (“discreteproductrule”)

Dt ψn υn ψn 1

2Dt υn υn 1

2Dt ψn (1.2.4)

i.e. with υn ψn

Dt & ψn & 2 2Re- ψn 12Dt ψn . (1.2.5)

yieldstheconservationof themass:

Dt ∑j , ZZ

ψnj

2 0

(1.2.6)

REMARK 2.2. We remarkthatan arbitraryhigh (even)order conservativeschemefor theSchrodingerequation(1.1.1) can be obtainedby using the diagonalPade approximationstotheexponential[F1]. TheCrank–Nicolsonschemecorrespondsto secondorder, andthefourthorderis known in theODEliteratureasHammerandHollingsworthmethod[F2].

2.1. Discretization strategiesfor the TBC. We shallnow comparefour strategiesto dis-cretizethe TBC (1.1.8)with its mildly singularconvolution kernel. First we review a knowndiscretizationfrom the literature,wherethe analytic TBC in the equivalent form (1.1.10)atL J∆x wasdiscretizedin anad–hocfashion.

DiscretizedTBC of Mayfield. In [T16] Mayfield proposedtheapproximation

tn

0

ψxL tn τ e i VL τ!

τdτ / 1

∆x

n 1

∑m) 0

ψn m

J ψn m

J 1 e i VL m∆ttm 1

tm

dτ!τ

2!

∆t∆x

n 1

∑m) 0

ψn m

J ψn m

J 1 e i VL m∆t!m 1 !

m

(1.2.7)

wheresheusedtheleft–pointrectangularquadraturerule. Thisleadsto thefollowingdiscretizedTBC for theSchrodingerequation:

ψnJ ψn

J 1 ∆x

2B!

∆tψn

J n 1

∑m) 1

ψn mJ

ψn mJ 1

˜0 m(1.2.8)

with

B 2π

ei π4 ˜0 m e i VL m∆t!

m 1 !m

On thefully discretelevel this BC is no longerperfectlytransparent.For theresultingschemewith a homogeneousDirichlet BC at j 0 and(1.2.8),Mayfield obtainedthefollowing result:


THEOREM 1.2( [T16]). Thenumericalscheme(1.2.1), (1.2.8)is stable, if andonly if

4π∆t

∆x2 j , IN0

2 j 1 2 2 j 2

(1.2.9)

This shows that the chosendiscretizationof the TBC (1.2.8) destroys the unconditionalstability of theunderlyingCrank–Nicolsonscheme.Thestability regionsof Theorem1.2 areillustratedin Figure1.2aslight areas.Thedarkintervalsareregionsof instability.

1111

∆x2__C ∆t

______4916

FIGURE 1.2. DiscretizedTBC of Mayfield: Stability regions.

As thenumericalexamplein Section2.5will show this discretizationgivesriseto unphysi-cal reflectionsat theboundary.

REMARK 2.3(Approachof Baskakov andPopov). A similar strategy usinga higher-orderquadraturerule for the l.h.s.of (1.2.7)wasintroducedby Baskakov andPopov in [T6]. Thisapproachtypically induceslessnumericalreflectionscomparedto the resultswhenusingthediscretizedTBC of Mayfield.

Semi–DiscreteTBC of SchmidtandDeuflhard. In thesemi–discreteapproachof SchmidtandDeuflhard[T23] aTBC is derivedfor thesemi–discretized(in t) Schrodingerequation.Thismethodalsoappliesfor a nonuniformin t (e.g.adaptive) interior schemeandit admitsa time–dependentpotentialin theexteriordomain(i.e.VL VL

t ). While beingunconditionallystable

(in conjunctionwith aninterior finite elementscheme)[T24], it still exhibits smallresidualre-flectionsat the artificial boundary. In [T24] this approachis alsoappliedto uniform exteriorz–discretizations,andonethenrecovers— througha differentderivation— thediscreteTBCfrom [T1].

Approachof Lubich and Schadle. Thetime discretizationis doneby thetrapezoidalrulein theinteriorandby convolutionquadratureon theboundary. Thenumericalintegrationof theconvolution integral is donein thefollowing way (cf. [T14], [T15] [T22]). If f

s denotesthe

Laplacetransformof f , thenformally settings ∂t yields

f∂t g

t ∞

0fτ e τ∂tg

t dτ

∞

0fτ g

t τ dτ

f 1 g(1.2.10)

whereg is a function satisfyinggt 0 for t 0. Now f

∂t g

t is approximatedby the

discreteconvolution

f∂k

t gt : ∑

n2 0ωng

t nk(1.2.11)


with the stepsize k ∆t. The quadratureweightsωn are definedas the coefficients of thegeneratingpowerseries:

∑n2 0

ωn ξn : fδξ k

& ξ & small

(1.2.12)

Hereδξ is thequotientof thegeneratingpolynomialsof alinearmultistepmethod,e.g.δ

ξ

1 ξ for theimplicit Eulermethodandδξ 2

1 ξ %

1 ξ for thetrapezoidalrule. If onechoosesfor thequadraturethesamenumericalschemeasin theinterior thenoneobtainsalsoareflection–freediscreteTBC.

DiscreteTBC. Insteadof usingan ad–hocdiscretizationof the analyticTBC like (1.2.7)we will constructdiscreteTBCsof the fully discretizedwhole–spaceproblem.Our new strat-egy solvesbothproblemsof thediscretizedTBCatnoadditionalcomputationalcosts.With ourDTBC thenumericalsolutiononthecomputationaldomain0 j J exactlyequalsthediscretewhole–spacesolution(on j ZZ) restrictedto thecomputationaldomain.Therefore,ouroverallschemepreventsany numericalreflectionsattheboundaryandinheritstheunconditionalstabil-ity of thewhole–spaceCrank–Nicolsonscheme(seeTheorem1.7).Thesedifferentapproaches,discretizationof theanalyticTBC and(semi–)discreteTBC, aresketchedin Figure1.3.

Discrete Schrodinger Equation..

same computational effort

(Analytic) Transparent BC

Discrete TBCDiscretized TBCArnold & Ehrhardt

semi-discrete TBCMayfield

Baskakov & Popov

only conditionally stable

numerical reflections

unconditionally stable

small reflections

Lubich & Schadle

unconditionally stable

reflection-free

Schmidt & Deuflhard..

..Schrodinger Equation

FIGURE 1.3. Discretizationstrategiesfor theTBC

Consequently, whenconsideringthediscretizationof TBCs,it shouldbeastandardstrategyto derivethediscreteTBCsof thefully discretizedproblem,ratherthenattemptingto discretizethe differentialTBC whenever it is possible. A comparisonof thesetwo strategiesfor a 1Dwavepropagationproblemis givenin [T10].

2.2. Derivation of the DTBC. To derivethediscreteTBC wewill now mimick thederiva-tion of theanalyticTBC from Section1 onadiscretelevel. TheCrank–Nicolsonscheme(1.2.1)canbewritten in theform:

iRψn 1

j ψn

j ∆2xψn 1

j ∆2xψn

j wVn 1

2j

ψn 1

j ψnj (1.2.13)


with

R 4∆x 2

∆t w 2

∆x 2

2 Vn 1

2j V

x j tn 1

2

where∆2xψn

j ψnj 1

2ψnj ψn

j 1, andR is proportionalto theparabolicmeshratio.Again,wewill only considertheright BC. In analogyto thecontinuousproblemweassume

for thepotentialandinitial data:Vnj VL const, j J 1, ψ0

j 0, j J 2 andsolve thediscreteright exteriorproblemby usingthediscreteanalogueof theLaplacetransformation,the

–transform:

- ψnj. ψ j

z : ∞

∑n) 0

ψnj z n z IC & z&3 1(1.2.14)

which is describedin moredetail in theAppendix.Hence,the

–transformedCrank–Nicolsonfinite differencescheme(1.2.13)for j J 1 reads

z 1 ∆2xψ j

z iR z 1 iκ

z 1 ψ j

z κ ∆t

2VL

(1.2.15)

Thetwo linearly independentsolutionsof theresultingsecondorder differenceequation

ψ j 1z 2 1 iR

2z 1z 1 iκ ψ j

z ψ j 1

z 0 j J 1(1.2.16)

take theform ψ jz ν j

1 2 z , j J 1, whereν1 2 z solve

ν2 2 1 iR2

z 1z 1 iκ ν 1 0

(1.2.17)

For thedecreasingmode(as j ∞) wehaveto require & ν1

z(&4 1andobtain(usingν1

z ν2

z

1) the

–transformedright DTBCas

ψJ 1z ν2

z ψJ

z (1.2.18)

Analogously, the

–transformedleft DTBC reads:

ψ1z ν2

z ψ0

z(1.2.19)

whereν2

z with & ν2

z#&5 1 is obtainedfrom asolutionto theleft discreteproblem, i.e. (1.2.16)

on therangej 1.

REMARK 2.4(DiscreteFactorization). If Sdenotestheusualshiftoperator givenby Sψ j

z

ψ j 1

z , thenanalogouslyto thecontinuouscase(cf. Remark1.3)thediscretizedSchrodinger

equation(1.2.16)canformally befactorizedas:

S ν1

z S ν2

z ψ j 1

z 0 j J 1(1.2.20)

which leadsto thesameDTBCs(1.2.18), (1.2.19).

It remainsto inversetransform(1.2.18)usingtheinversionrulesof the

–transformgivenin theAppendix.By thefollowing tediouscalculationthis canbeachievedexplicitly.

CALCULATION (of 1 - ν2

z . ). First we rewrite ν1 2 z as:

ν1 2 z 1 iR2

z 1z 1 iκ 6 iR

2z 1z 1 iκ 2 iR

2z 1z 1 iκ

1 iR2 Rκ

2 iR

zz 1 7

iR2

1z 1 Az2 2Bz C

(1.2.21)


with theconstants

A 1 iκ 1 iκ i

4R

(1.2.22a)

B 1 κ2 4κR

(1.2.22b)

C 1 iκ 1 iκ i

4R

(1.2.22c)

For theinverse

–transformweuse

Az2 2Bz C 1

! A

Az2 2Bz Cz

8 A 8 Cz

ACz2 2B

Cz 1

(1.2.23)

With theabbreviations

Fz µ z

z2 2µz 1 λ ! A

! C µ B

! A ! C(1.2.24)

we obtainfrom equation(1.2.23)

1z 1 Az2 2Bz C 1

! A

Az2 2Bz Czz 1 F

λz µ

1

! AA C

z E

z 1F

λz µ

! C λ λ 11z E

! A ! C

1z 1

Fλz µ

(1.2.25)

with E A 2B C. Theinversionrulesnow yield

1 ! Az2 2Bz Cz 1 ! C λδ0

n λ 1δ1n E

! A ! C

1 n δ0n 1 Pn

µ

! C λPnµ λ 1Pn 1

µ E

! A ! C

n 1

∑k) 0

1 n kPkµ

where 1 denotesthediscreteconvolution. Finally, weobtain

1 - ν1 2 z . 1 iR2 Rκ

2δ0

n iR 1 n

7iR ! C

2λ n λPn

µ Pn 1

µ E

! A ! C

n 1

∑k) 0

λ n kPkµ (1.2.26)

SinceC A wehave & λ & 1 with

λ A

! A ! C R 4κ Rκ2 2i

Rκ 2

1 κ2 R2 Rκ 4 2

(1.2.27)

Thereforewewrite

λ eiϕ with ϕ arctan2Rκ 2

R 4κ Rκ2

(1.2.28)


Weobtainfor theparameterµ:

µ R1 κ2 4κ

1 κ2 R2 Rκ 4 2

IR(1.2.29)

andit caneasilybeseenthat 1 µ 1 is valid. TheconstantE simply equals4 andwe get

τ E

! A ! C 4R

1 κ2 R2 Rκ 4 2

IR

(1.2.30)

Thechoiceof thesignin (1.2.26)canbejustifiedanalyticallyor simplyby testingit numer-ically. We finally obtaintheconvolutioncoefficients

0 n 1 - ν2

z . as

0 n 1 iR2 σ

2δ0

n iR

1 n i2

4 R2 σ2 R2

σ 4 2 e iϕ $ 2 99 e inϕ λPn

µ Pn 1

µ τ

n 1

∑k) 0

λ n kPkµ (1.2.31)

with σ Rκ andPn denotestheLegendrepolynomials.TheresultingdiscreteTBCsat thegridpointsx j j 0 J read

ψn1 0 0

0 ψn0 n 1

∑k) 1

0 n k0 ψk

0 n 1(1.2.32a)

ψnJ 1

0 0J ψn

J n 1

∑k) 1

0 n kJ ψk

J n 1

(1.2.32b)

Thesubscriptj of thecoefficients0 n indicatesat which boundarythevaluesareto be taken.

In thesequelmany parameterswill besuppliedwith this subscript.

2.3. The Asymptotic Behaviour of the Convolution Coefficients. We studytheasymp-

totic behaviour of the0 n

0 ,0 n

J in (1.2.32).It will turnout thatit is advantageousto reformulatetheDTBC usingnew coefficients.Afterwardsweshallderivearecursionformulafor thesenewcoefficientsandcomparetheir decayratewith thedecayrateof thecontinuousintegral kernelin (1.1.8),(1.1.9).

The summedconvolution coeffcients. Firstwewantto studytheasymptoticbehaviourof

theconvolutioncoefficients0 n

j , j 0 J. With thenotationµj cosθ j , 0 θ j π, weusethefollowing classicalresulton theasymptoticpropertyof theLegendrepolynomials:

LEMMA 1.3(Theorem8.21.2(Formulaof Laplace),[S7]).

Pncosθ j

!2!

π sinθ j

cosn 1

2 θ j π

4!n O

n 3$ 2 0 θ j π

(1.2.33)

Theboundfor theerror termholdsuniformlyin theinterval ε θ j π ε.

From this lemmawe concludethat limn: ∞ Pnµj 0 holds. Consequently, the coefficients

have thefollowing asymptoticbehaviour for n ∞:

0 nj

iR 1 n + iτ j

24

R2 σ2

j R2 σ j 4 2 e iϕ j $ 2 1 n

n 1

∑k) 0

e iϕ j kPkµj (1.2.34)


Using

limn: ∞

n 1

∑k) 0

e iϕ j kPkµj 1

1 2µj λ j λ j2

1

2λ j

1

Reλ j µj

12R

4R2 σ2

j R2 σ j 4 2 eiϕ j $ 2

(1.2.35)

we finally obtain

0 nj + iR

1 n iτ j

4R

R2 σ2

j R2 σ j 4 2

1 n 2iR 1 n (1.2.36)

Thesequence0 n

j is asymptoticallyanalternating,purelyimaginarysequence,whichmaylead(on thenumericallevel) to subtractivecancellationin (1.2.32). To circumventthis problemweconsiderthesummedcoefficients

s nj : 0 n

j 0 n 1j n 1 s

0j : 0 0

j j 0 J (1.2.37)

andcompute:

0 0j 1 i

R2 σ j

2 i

24

R2 σ2

j R2 σ j 4 2 eiϕ j $ 2

1 iR2 σ j

2 i

2 R2 4σ j σ2

j 2iRσ j 2(1.2.38)

0 1j iR i

24

R2 σ2

j R2 σ j 4 2 e iϕ j $ 2 µj e iϕ j τ j (1.2.39)

andfor n 2 wecomputeusingthedefinitionof E:

s nj i

24

R2 σ2

j R2 σ j 4 2 eiϕ j $ 2e inϕ j

Pnµj λ j λ 1

j τ j

) 2µj

Pn 1µj Pn 2

µj (1.2.40)

With therecurrencerelationof theLegendrepolynomials

µjPn 1µj n

2n 1Pn

µj n 1

2n 1Pn 2

µj n 1

we finally get

s nj i

24

R2 σ2

j R2 σ j 4 2 eiϕ j $ 2e inϕ j

Pnµj Pn 2

µj

2n 1 n 2

(1.2.41)

REMARK 2.5. Thecoefficient0 0

j canalsobecalculatedwith [S3,Theorem39.1]by

0 0j lim

z: ∞ν2

z 1 i

R2 σ j

2 i

2 R iσ j R i

σ j 4 (1.2.42)

Wesummarizeour resultsin thefollowing theorem:


THEOREM 1.4( [T1]). Theleft (at j 0) andright (at j J) discreteTBCsfor theCrank–Nicolsondiscretization(1.2.1)of the1D Schrodinger equationare respectively

ψn1 s

00 ψn

0 n 1

∑k) 1

s n k0 ψk

0 ψn 1

1 n 1(1.2.43a)

ψnJ 1

s 0J ψn

J n 1

∑k) 1

s n kJ ψk

J ψn 1

J 1 n 1(1.2.43b)

with

s nj 1 i

R2 σ j

2δ0

n 1 iR2 σ j

2δ1

n α j e inϕ jPn

µj Pn 2

µj

2n 1(1.2.44)

ϕ j arctan2R

σ j 2

R2 4σ j σ2

j

µj R2 4σ j σ2j

R2 σ2j R2

σ j 4 2

σ j 2∆x2

2 Vj α j i2

4R2 σ2

j R2 σ j 4 2 eiϕ j $ 2 j 0 J

Pn denotestheLegendrepolynomials(P 1 ; P 2 ; 0) andδ jn theKronecker symbol.

ThePn only have to beevaluatedat the two valuesµ0 µJ 1 1 , andhencethenumeri-cally stablerecursionformulafor theLegendrepolynomialscanbeused[E2].

The recurrenceformula for the summedcoefficients. In this subsectionwe shall give

two different derivationsfor the recursionformula of the convolution coefficients s nj . The

first oneis basedon the explicit representation(1.2.41)of s nj by first calculatinga recursion

formulafor Pn 1µj Pn 1

µj . Thesecondderivationdoesnotrequiretheexplicit form of the

coefficientss nj but only thegrowthfunctionsν1 2 z from the

–transformedDTBCs(1.2.18)

and(1.2.19).

FIRST DERIVATION: Herewe startwith the standard recursion formula [S1] for the Le-gendrepolynomialsPn 1

µj , Pn 1

µj :

n 1 Pn 1µj

2n 1 µjPnµj nPn 1

µj n 0

n 1 Pn 1µj

2n 3 µjPn 2µj

n 2 Pn 3µj n 2(1.2.45)

and

Pn 1µj 2µjPn 2

µj Pn 3

µj Pn 3

µj Pn 1

µj

2n 3 n 2

(1.2.46)

TheexpressionPn 1µj Pn 1

µj is convertedin thefollowing way:

n 1 Pn 1

µj Pn 1

µj

2n 1 µjPn

µj

2n 1 2n 3n 1

µjPn 2µj

2n 1 n 2n 1

Pn 3µj

2n 1 µj Pn

µj Pn 2

µj

n 2 2n 12n 3

Pn 1µj Pn 3

µj


wherewehaveusedtherelation

Pn 3µj µjPn 2

µj 1

2Pn 3

µj Pn 1

µj

2n 3 Pn 1

µj Pn 3

µj

n 12n 3

Pn 1µj Pn 3

µj

Finally, weobtainfor n 2 thefollowing recursionformulafor Pn 1µj Pn 1

µj :

(1.2.47) Pn 1µj Pn 1

µj

2n 1n 1

µj Pnµj Pn 2

µj

n 2 2n 1n 1 2n 3 Pn 1

µj Pn 3

µj

Sincethes nj aredeterminedby

s nj α j

λ nj

2n 1Pn

µj Pn 2

µj n 2(1.2.48)

we getfrom (1.2.47)therecurrencerelation for thesummedconvolutioncoefficients:

s n 1j 2n 1

n 1µjλ 1

j s nj

n 2n 1

λ 2j s

n 1j n 2(1.2.49)

which canbeusedaftercalculatingthefirst valuess nj for n 0 1 2 by theformula(1.2.44).

SECOND DERIVATION: Next weshallpresentanalternativederivationof thefirst convolu-

tion coefficientss nj , n 0 1 2 andtherecurrencerelation(1.2.49). Theadvantageof thisalter-

nativeapproachis thatweshallonly needthegrowthfunctionsν1 2 z from the

–transformedDTBCs(1.2.18)and(1.2.19). Hencethis approachmight alsoapplyto a biggerclassof linearevolutionequations,whereit is notpossible(or too tedious)to deriveanexplicit representationof theconvolutioncoefficients.

We remarkthat an even moreadvantageousapproachmight be basedon the polynomialequation(1.2.17)for thegrowth functionν

z , ratherthanon its explicit solution. Thebenefit

of sucha strategy would lie in the possibility to obtain the convolution coefficients also forhigherorderdifferenceschemes,thatwould leadto quartic(or evenhigherorder)equationsforνz . To our knowledgethis has,however, notbeenaccomplishedyet.

In this secondapproachweshallfirst deriveafirst orderODEfor thegrowth functionνz ,

which is explicitly givenby (1.2.21). In fact,it is moreconvenientto consider

νz :

z 1 ν z ∞

∑n) 1

s n 1j z n (1.2.50)

Usingtheconstants(1.2.22)and(1.2.24)it hastheexplicit form

ν1 2 z z 1 iR2 σ j

2 1 iR2 σ j

26 i

2 R2 4σ j σ2j z2λ

µ 2z 1

λµ

(1.2.51)

Theindex j 0 J againdenotesthegrid pointwheretheDTBC is to beconstructed.Multiply-

ing ν< dνdz

by z2λµ

2z 1λµ

thenyieldsaninhomogeneousfirst orderODE for νz :

z2λµ

2z 1λµ

ν< z zλµ

1 νz β

z : β 1 z β 0 (1.2.52)


with

β 1 λµ

1 iR2 σ j

2 1 iR2

σ j

2

β 0 1λµ

1 iR2 σ j

2 1 iR2 σ j

2

(1.2.53)

Its generalsolutionincludesν1 2 z asdefinedin (1.2.51).UsingtheLaurentseries(1.2.50)of

ν andν< in (1.2.52)immediatelyyieldsthedesiredrecursionfor thecoefficientss nj :

λµ

s 1j

s 0j β 1

2λµ

s 2j s

1j 1

λµs 0j β 0

n 1 s n 1

j 2n 1 µ

λs nj

n 2 1

λ2s n 1j 0 n 2

(1.2.54)

whichcoincideswith (1.2.49).Thestartingcoefficientof therecursioncanbedeterminedasin (1.2.42):

s 0j lim

z: ∞

νzz 1 i

R2 σ j

26 i

2R2 4σ j σ2

j λµ

Here,thesignhasto befixedsuchthat & ν1

z(&= 1 for the right DTBC and & ν2

z(&= 1 for the

left DTBC. Thiscanbedonefor e.g.for z ∞.

STABILITY OF THE RECURRENCE RELATION: For proving that the recurrencerelation(1.2.49)is well–conditionedwe follow the notationin [E2] andwrite (1.2.49)as the secondorderdifferenceequation

s n 1j a

nj s

nj b

nj s

n 1j 0 n 2(1.2.55)

with

a nj 2n 1

n 1µjλ 1

j b nj n 2

n 1λ 2

j > 0

(1.2.56)

Therearetwo linearly independentsolutionss nj 1, s

nj 2 to (1.2.55).If they have theproperty

limn: ∞

s nj 2

s nj 1 0(1.2.57)

thens nj 2 is calleda minimalsolutionandseriousnumericalproblemsariseif onetriesto com-

putethe solutions nj 2 in a straightforward way by usingthe recursion(1.2.49). If thereis an

errorin theinitial databut recurringwith infinite precisionthentherelativeerrorof theintended

approximationto s nj 2 becomesarbitrarily large. Methodsof calculatingminimal solutionsof

three–termrecurrencerelationscanbefoundin [E2]. To provethat(1.2.49)is well–conditionedwehave to show thattheseekedsolutionis not a minimal solutionto (1.2.55). This typeof so-lution is calleddominant.


Sincethecoefficientsa nj , b

nj in (1.2.55)have thefinite limits

a j limn: ∞

a nj 2µjλ 1

j 2B j

A j b j lim

n: ∞b nj λ 2

j Cj

A j j 0 J (1.2.58)

onecalls(1.2.55)aPoincaredifferenceequationand

Φ jt t2 a j t b j(1.2.59)

thecharacteristicpolynomialof (1.2.55).Thecharacteristicpolynomialhasthecomplex conju-

gatezerost 1 2j

B j 6 i 4% R% A j . Thezeroshave thesamemoduli: & t 1 2j & 1, andthereforethe classicalTheorem of Poincare (formulatedbelow for the specialcaseof a second–orderdifferenceequation)cannotbe appliedto distinguishtwo solutionswith distinct asymptoticproperties.

THEOREM 1.5(Poincare Theorem,[E1]). Supposethat thezerost 1j , t

2j of thecharacter-

istic polynomial(1.2.59)havedistinctmoduli.Thenfor anynontrivial solutions nj of (1.2.55)

limn: ∞

s n 1j

s nj

t kj

for k 1 or k 2.

REMARK 2.6. If equation(1.2.55)hascharacteristicrootswith equalmodulithenPoincare’sTheoremmayfail (cf. exampleof Perron[E1]).

It is well–known that the LegendrepolynomialsPnµj andthe Legendre functionsof the

secondkind (of order zero) Qnµj satisfy the samethree–termrecurrencerelation (1.2.45).

Therefore,the two linearly independentsolutionsto (1.2.55)are the convolution coefficients

s nj 1 s

nj (1.2.48)ands

nj 2 givenby

s nj 2 β j

λ nj

2n 1Qn

µj Qn 2

µj n 2(1.2.60)

with someconstantβ j .Now we want to studytheasymptoticbehaviourof thesetwo solutions.With thenotation

µj cosθ j , 0 θ j π, weuseLemma1.3whichgives

Pncosθ j Pn 2

cosθ j 2

!2 sinθ j!

πsin

n 1

2 θ j π

4!n O

n 3$ 2 (1.2.61)

andfrom (1.2.48)weseethat

s nj + α j

!2 sinθ j!

πλ n

j

sinn 1

2 θ j π

4n 1

2 ! n n ∞

(1.2.62)

An analogousformulato (1.2.33)for theLegendrefunctionsof thesecondkind Qnµj is given

by thefollowing lemma:

LEMMA 1.6(Theorem8.21.14,[S7]). For 0 θ j π

Qncosθ j

!π!

2 sinθ j

cosn 1

2 θ j π4!

n On 3$ 2 (1.2.63)


Thisholdsuniformlyin theinterval ε π ε .As beforewecandeducefrom (1.2.60)that

s nj 2 + β j

!π sinθ j!

2λ n

j

sinn 1

2 θ j π4

n 12 ! n

n ∞(1.2.64)

holds.Thereforetheratioof thetwo solutionsbehavesasymptoticallyas

s nj 2

s nj

+ β j

α j

π2

sinn 1

2 θ j π4

sinn 1

2 θ j π

4 β j

α j

π2

tann 1

2 θ j π

4 n ∞

i.e.neithers nj nors

nj 2 canbeaminimalsolution.Consequently, theproblemof determiningthe

requiredvaluesof theconvolutioncoefficientsis well–conditioned[E2]: they canbecomputednumericallyfrom therecurrencerelation(1.2.49)in astablefashion.

REMARK 2.7. In thespecialcaseθ j π % 2 weobserve theasymptoticbehaviour

s nj 2

s n 1j

+ β j

α j

π2

n ∞

Decayrate of the convolution kernel. Since & λ j & 1 therelation(1.2.62)shows thats n0 ,

s nJ O

n 3$ 2 , whichagreeswith thedecayof theconvolutionkernelin thedifferentialTBCs

(1.1.8), (1.1.9). To show this propertywe considerthe left TBC (1.1.9) andobtainafter anintegrationby parts

ψx0 t c

ddt

t

0

ψ0 τ !t τ

dτ

cddt

t

t ε

ψ0 τ !t τ

dτ 12

t ε

0

ψ0 τ

t τ 3$ 2 dτ ψ0 t ε !

ε

(1.2.65)

with

c 2π

e i π4 1 i!

π

(1.2.66)

To comparethediscreteconvolution in theDTBCswith thecontinuousconvolution in thedif-ferentialTBCsweconsiderthefollowing discretization(with ε ∆t)

c2

t ε

0

ψ0 τ

t τ 3$ 2 dτ / c2

n 1

∑k) 1

ψk0

n k ∆t 3$ 2∆t

(1.2.67)

If wecomparethis with (1.2.43a)written in theform

ψn1 ψn

0

∆x 1∆x

n 1

∑k) 1

s n k0 ψk

0 ψn 1

1 s 00 ψn

0 ψn

0 n 1(1.2.68)

wewould roughlyexpect

s n0 + c∆x

2!

∆tn 3$ 2 i 1

2!

π∆x!∆t

n 3$ 2 n ∞

(1.2.69)

In Figure1.4 we comparethe s n0 for increasingtime levelsn with the r.h.s.of (1.2.62). The

usedparametersare 1, ∆t 10 6, ∆x 1% 160andthepotentialis setto zero. Figure1.4


displaysa quitegoodagreementof theasymptoticbehaviour of thes n0 with thepredictedone

in (1.2.62). Thevaluesof s n0 oscillatearoundtherate(1.2.69).

1 1.5 2 2.5 3 3.5 4−9

−8

−7

−6

−5

−4

−3

−2

−1

0

Asymptotic Behaviour of Convolution Coefficients s0(n)

log10

(n)

log 10

(s0(n

) )

s0(n)

s0(n) asympt.

n−3/2 rate

FIGURE 1.4. New discreteTBC: convolution coefficientss n0 comparedto the

r.h.s.of (1.2.62)andthedecayingrate(1.2.69).

2.4. Stability of the resulting Scheme.We shall now discussthestability of theCrank–Nicolsonfinite differenceschemein connectionwith theDTBCs(1.2.43).

Sincethediscretewhole–spacesolutionsatisfiesthediscreteTBCs(1.2.43), it is trivial thattheimplicit scheme(1.2.1),(1.2.43)for theIBVP canbesolvedat eachtime level n. To proveuniquesolvability andstability of thescheme,a discreteanalogueof (1.1.19)canbe derived.We sumup (1.2.2)for thefinite interior range j 1 2 ? J 1, usethesummationby partsrule:

∆xJ 1

∑j ) 1

g jD @x f j ∆xJ 1

∑j ) 0

f jDxg j fJ 1gJ f0 g0(1.2.70)

andobtain(notethatD2x D @x Dx )

∆xJ 1

∑j ) 1

ψn 12

j Dt ψnj i ∆x

2

J 1

∑j ) 0

Dx ψn 12

j

2 i∆x J 1

∑j ) 1

Vn 1

2j ψn 1

2j

2

i2

ψn 12

J Dx ψn 12

J 1 ψn 1

20 Dx ψn 1

20

(1.2.71)

Finally, takingtherealpartusing(1.2.5)yields

Dt " ψn " 22 Re i ψn 1

2J D @x ψn 1

2J

Re i ψn 12

0 Dx ψn 12

0 (1.2.72)


with thediscreteL2–normdefinedby " ψn " 22 : ∆x∑J 1

j ) 1 & ψnj & 2. After summationwith respectto

thetime index wegetfrom (1.2.72):

" ψN 1 " 22 " ψ0 " 2

2 ∆t Re iN

∑n) 0

ψn 12

J D @x ψn 12

J Re i

N

∑n) 0

ψn 12

0 Dx ψn 12

0

" ψ0 " 22 ∆t

∆xRe i

N

∑n) 0

ψn 12

J

ψn 1

2J 1 ˜0 n

J ∆t∆x

Re iN

∑n) 0

ψn 12

0

ψn 1

20 1 ˜0 n

0

(1.2.73)

where ˜0 nj : 0 n

j δ0

n, j 0 J. Again, asin thecontinuouscase,it remainsto show that theboundary–memory–termsin (1.2.73)areof positivetype. Weconcentrateon theboundarytermat j J anddefinethefinite sequences

fn ψn 12

J 1 ˜0 nJ gn ψn 1

2J n 0 1 N (1.2.74)

with fn gn 0 for n N, i.e.Re- i ∑Nn) 0 fn gn

. 0 is to show. A

–transformationusingthetransformedDTBC (1.2.18)yields

- fn. f

z z 1

2ψN

Jz ν2

z 1

iR4

ψNJz z 1 iκ

z 1 7 Az2 2Bz C (1.2.75)

whereψNJ

z ∑N

n) 0 ψnJz n is analyticon & z&A 0. Theexpressionabove in thecurly bracketsis

analyticfor & z&= 1 andcontinuousfor & z&B 1, sincethezerosz1 2 of thesquareroot aregivenby z1 2

B 6 i 4% R% A with & z1 2 & 1. Thereforefz is analyticon 1 *& z&3 ∞. Notethatwe

have to choosethesignin (1.2.75)suchthatit matcheswith ν2

z for & z& sufficiently large. For

thesecondsequencegn weobtain

- gn. g

z z 1

2ψN

Jz(1.2.76)

i.e. gz is analyticon0 *& z&3 ∞.

Now thebasicideais to usePlancherel’s theoremin theform (Z.7) whichgives

Re iN

∑n) 0

fn gn 18π

Re i2π

0& z 1 & 2 ψN

Jz 2 ν2

z 1

z) eiϕdϕ

18π

Im2π

0ν2

eiϕ dϕ

(1.2.77)

Weremarkthatthepoleof ν2

z atz 1 is “cancelled”by & z 1 & 2. From(1.2.77)weconclude

thatthediscreteL2–norm(1.2.73)is non–increasingin time if

Im ν2

eiϕ C 0 D ϕ E 0 2π F(1.2.78)

holds.Thispropertyof ν2 canbeshown in thefollowing way. If wedefine

y iR2

z 1z 1 iκ(1.2.79)

then(1.2.21)simply reads

ν2

y 1 y y

2 y (1.2.80)


On theunit circlez eiϕ, 0 ϕ 2π, wehavez 1%

z 1 i tanϕ % 2 andtherefore

y R2

tanϕ2 Rκ

2 ∆x 2 2

∆ttan

ϕ2 VL 0 ϕ 2π (1.2.81)

is real. Consequently, ν2

eiϕ becomescomplex only in the interval ϕa ϕ ϕb, whereϕa,

ϕb G 0 2π solve

tanϕa

2 ∆tVL

2 ∆x 2 tan

ϕb

2 ∆t2

VL (1.2.82)

andwehave therequestedresult

Im ν2

eiϕ Im y

2 yH 0 0 ϕ 2π

(1.2.83)

0 π/2 π 3/2π 2π −1.5

−1

−0.5

0

0.5

φ

Imag ν2(φ)

FIGURE 1.5. Imaginarypartof ν2

z on theunit circlez eiϕ, 0 ϕ 2π.

The situationis illustratedin Figure1.5, wherewe have set 1, VL 2 9 104 andusedthe parameters∆x 1% 500, ∆t 2 9 10 5. This givesthe following valuesfor ϕa, ϕb: ϕa 2π 2atan

52/ 3

5216,ϕb 2π 2atan

02/ 5

8884.

We thenhave themainresultof this Section:

THEOREM 1.7( [T1]). Thesolutionof thediscretizedSchrodingerequation(1.2.1)with thediscreteTBCs(1.2.43)is uniformlybounded

" ψn " 22 : ∆x

J 1

∑j ) 1

ψnj

2 I" ψ0 " 22 n 1(1.2.84)

andtheschemeis thusunconditionallystable.

REMARK 2.8. It canalsobeshown that(1.2.43)is a consistentdiscretizationof thediffer-entialBCs(1.1.8),(1.1.9).


Simplified DTBC. Thedecayof thes nj shown in (1.2.62)motivatesto considera simpli-

fiedversionof theDTBC (1.2.43)with theconvolution coefficientscut off at anindex M. Thismeansthatonly the“recentpast”(i.e.M time levels)is takeninto accountin theconvolution in(1.2.43):

ψn1 s

00 ψn

0 n 1

∑k) n M

s n k0 ψk

0 ψn 1

1 n 1(1.2.85a)

ψnJ 1

s 0J ψn

J n 1

∑k) n M

s n kJ ψk

J ψn 1

J 1 n 1

(1.2.85b)

This,of course,reducestheperfectaccuracy of theDTBC (1.2.43),but it is numericallycheaperwhile still yielding reasonableresultsfor moderatevaluesof M. We remarkthatthenumericalstability of the schemewith simplified DTBC dependingon the valueof M is not anymoreobtainedautomatically. This issueis currentlyunderinvestigation[D1].

2.5. Numerical Results. In this Sectionwe presentanexampleto comparethenumericalresultsfrom usingour new discreteTBC to thesolutionusingotherdiscretizationstrategiesoftheTBC for theSchrodingerequation(1.1.1). We alsoshow thenumericaleffect if theDTBCis simplified by (1.2.85). Due to its construction,our DTBC yields exactly (up to round–offerrors)thenumericalwhole–spacesolutionrestrictedto thecomputationalinterval 0 L . Thecalculationwith discretizedTBCs requiresthe samenumericaleffort. However, the solutionmay(on coarsegrids)stronglydeviatefrom thenumericalwhole–spacesolution.

Example. This exampleshows a simulationof a right travelling Gaussianbeam ψI x exp

i100x 30

x 0

5 2 at four consecutive time stepsevolving underthe freeSchrodinger

equation( 1)with therathercoarsespacediscretization∆x 1% 160andthetimestep∆t 2 910 5. DiscretizingtheanalyticTBCsvia (1.2.7)(schemeof Mayfield [T16]) or asin BaskakovandPopov [T6] inducesstrongnumericalreflections. Our discreteTBCs (1.2.43),however,yield thesmoothnumericalsolutionto thewhole–spaceproblem,restrictedto thecomputationalinterval 0 1 (up to round–off errors).

Weobservein Figure1.6theartificial reflectionstravelling to theleft inducedbydiscretizingtheanalyticTBC while thesolutionwith thenew discreteTBC leavethecomputationaldomainwithout any numericalreflections. At time t 0

01 the solutionwith the DTBC hasalmost

completelyleft the domain 0 1 andthe solutionswith discretizedTBCs containa reflectedwavepacket with themaximummodulus(whichcorrespondsto themaximumerror)of around0.17for theapproachof Mayfieldandaround0.025in caseof thediscretizedTBC of BaskakovandPopov.

Now we presenttheresultswhenusingthesimplifiedDTBC (1.2.85)andwantto comparethe outcomewith the discretizedTBCs at time t 0

01. All theseboundaryconditionsneed

a comparablecomputationaleffort. Thecut–off valueM is chosenappropriately, suchthat thesimplified DTBC yields similar resultswith respectto the numericalreflectionsat the rightboundaryx 1. As a referencewealsoplot thesolutionwith thediscreteTBCs( 99?9 ).

Weseein Figure1.7thatthesolutionwith thesimplifieddiscreteTBCs(1.2.85)with M 5is alreadybetterthanthesolutionwith thediscretizedanalyticTBCs(1.2.7)from [T16].

We observe in Figure1.8 that the error of the solutionwith the discretizedanalyticTBCfrom [T6]. lies betweentheerrorsof thesolutionswith thesimplifieddiscreteTBCs (1.2.85)usingthecut–off valueM 30,M 35.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

x

|ψ|

Schroedinger: t=0.004

new discrete TBCdiscretized TBC (Mayfield)discretized TBC (Baskakov & Popov)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

|ψ|



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

x

|ψ|



0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

x

|ψ|



FIGURE 1.6. Solution & ψ x t (& at time t 0

004, t 0

006, t 0

008, t

001: the solutionwith the new discreteTBCs (—) coincideswith the whole–

spacesolution,while thesolutionwith thediscretizedanalyticTBCs(1.2.7)from[T16] ( @J@J@ ) or from [T6] ( 999 ) introducesstrongnumericalreflections.

3. DTBC for non–compactlysupported Initial Data

In this Sectionwe show how to dropassumption(A1), i.e. herethe initial dataψI x neednot becompactlysupportedinsidethecomputationaldomain.We only assumethat the initialfunctionψI x is continuous.Firstwereview thederivationof theTBC on thecontinuouslevelandmimick thisderivationstrategy afterwardsfor thediscretescheme.

3.1. The Transparent Boundary Condition. Herewe review thederivationof the (con-tinuous)TBC from [T13]. In the caseof the free Schrodingerequationwith non–compactlysupportedinitial dataψI the Laplacetransformed(using(L.2)) right exterior problem(1.1.5)now reads

vxx c2sv c2ψI x x L (1.3.1a)

vL s Φ

s(1.3.1b)

3. DTBC FOR NON–COMPACTLY SUPPORTED INITIAL DATA 27

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

x

|ψ|


discretized TBC (Mayfield)discrete TBCdiscrete TBC: M=5discrete TBC: M=4

FIGURE 1.7. Solution & ψ x t #& at time t 0

01: the solution with the dis-

cretizedTBCs of Mayfield [T16] (—) in comparisonto the solutionusingthesimplifiedDTBC (1.2.85)with M 4 ( @JK @JK ) andM 5 ( @J@J@ ).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03

x

|ψ|


discretized TBC (Baskakov & Popov)discrete TBCdiscrete TBC: M=35discrete TBC: M=30

FIGURE 1.8. Solution & ψ x t #& at time t 0

01: the solution with the dis-

cretizedTBCsof Baskakov andPopov [T6] (—) in comparisonto thesolutionusingthesimplifiedDTBC (1.2.85)with M 30 ( @JK @JK ) andM 35( @J@J@ ).

wherewe setc 1 i % ! . Again, the idea is to solve this inhomogeneoussecondorder

differentialequation(1.3.1)explicitly. Thehomogeneoussolutionis

vhomx s C1

s eic 8 s x L C2

s e ic 8 s x L x L (1.3.2)


andaccordingto [M3, (14.31)]aparticularsolutionof (1.3.1a)is givenby

vparx s c

2i!

s

x

Leic 8 s x xL ψI x< dx< x

Leic 8 s xL x ψI x< dx< x L (1.3.3)

i.e. thegeneralsolutionis

vx s vhom

x s vpar

x s

C1s e ic 8 sL c

2i!

s

x

Le ic 8 sxL ψI x< dx< eic 8 sx

C2s eic 8 sL c

2i!

s

∞

Leic 8 sxL ψI x< dx< c

2i!

s

∞

xeic 8 sxL ψI x< dx< e ic 8 sx

(1.3.4)

We notethatthelasttermin (1.3.4)is boundedfor fixeds andx ∞. Sincethesolutionshaveto decreaseasx ∞, the ideais to eliminatethegrowing factore ic 8 sx e

1 @ i 8 sx by simplychoosing

C2s c

2i!

s

∞

Leic 8 s xL L ψI x< dx< (1.3.5)

Consequently, weobtainC1s from theboundarycondition(1.3.1b):

C1s Φ

s c

2i!

s

∞

Leic 8 s xL L ψI x< dx< (1.3.6)

Fromthis wegetthefollowing representationof thetransformedright TBC:

vxL s ic

!sC1

s c2

2

∞

Leic 8 s xL L ψI x< dx<

ic!

sΦs c2

∞

Leic 8 s xL L ψI x< dx<

(1.3.7)

It remainsto inversetransform(1.3.7). If we furtherassumethat ψI is continuouslydifferen-tiable,thenintegrationby partsyields:

vxL s ic!

ssΦ

s ψI L ic!

s

∞

Leic 8 s xL L ψI

xx< dx< (1.3.8)

TheinverseLaplacetransformationusingtheconvolution theoremgives

ψxL t ic!

π

t

0

ψtL τ !

t τdτ ic M 1 1!

s

∞

Leic 8 s xL L ψI

xx< dx<

(1.3.9)

Finally, if ψIx is integrablefor x L, Levy proved( [T13, Theorem3.1]) thattheintegrationand

the inverseLaplacetransformcanbe interchangedin (1.3.9)to obtain(with (IL.8)) the rightTBC

ψxL t ic!

π

t

0

ψtL τ !

t τdτ ic!

πt

∞

LψI

xx e

i N x @ LO 22

t dx

(1.3.10)

REMARK 3.1. Clearly, if ψI x 0 for x L then(1.3.10)reducesto the previously ob-tainedright TBC (1.1.8)in thepotential–freecase(notethat !

2e i π4 i 1).

As motivatedin thepreviousSectionwewill not discretizethis TBC. Insteadwewill shownow how to derive theTBC on a fully discretelevel by mimicking thederivationof thecontin-uousTBC.


3.2. The DiscreteTransparent Boundary Condition. First we show how to solve anin-homogeneoussecondorderdifferenceequationwith constantcoefficientsof theform

U j 1 aU j bU j 1 γ j j J 1

(1.3.11)

Wealreadyknow from Section2 thatthetwo linearly independenthomogeneoussolutionstaketheform α j , β j , j J with αβ b. A particularsolutionVj of (1.3.11)canbefoundwith theansatzof “variation of constants” [E3], [E1] :

Vj 1 c jα j 1 d jβ j 1 j J

(1.3.12)

It follows that

Vj c j 1α j d j 1β j c jα j d jβ j j J 1(1.3.13)

if we forcethecondition ∆ c j α j

∆ d j β j 0(1.3.14)

to hold. Here∆ denotestheusualforwarddifferenceoperator, i.e. ∆ c j c j 1 c j . Analo-

gously, againassuming(1.3.14),we obtainfor Vj 1

Vj 1 c jα j 1 d jβ j 1 ∆ c j α j 1

∆ d j β j 1 (1.3.15)

Inserting(1.3.12), (1.3.13), (1.3.15)into thedifferenceequation(1.3.11)gives∆ c j α j 1

∆ d j β j 1 γ j (1.3.16)

togetherwith thecondition(1.3.14). Thiscaneasilybesolvedto obtain

∆ c j 1α β

α jγ j ∆ d j 1α β

β jγ j (1.3.17)

i.e.wegetthecoefficients

c j cJ j 1

∑m) J

∆ cm cJ 1α β

j 1

∑m) J

α mγm j J (1.3.18)

d j dJ j 1

∑m) J

∆ dm dJ 1

α β

j 1

∑m) J

β mγm j J

(1.3.19)

Consequently, theparticular solutionreads

Vj c j 1α j d j 1β j

cJ 1α β

j

∑m) J

α mγm α j dJ 1

α β

j

∑m) J

β mγm β j j J 1(1.3.20)

andthegeneral solutionof (1.3.11)is of theform

U j cα j dβ j 1α β

j

∑m) J

α j mγm j

∑m) J

β j mγm j J 1(1.3.21)

which is thediscreteanalogueto thesolutionformula(1.3.3)in thecontinuouscase.Now we use(1.3.21)to designa boundarycondition at j J. For that purposewe as-

sume & α &J 1, & β &J 1 (recall thatb αβ 1 for theCrank–Nicolsonschemefor solving the


Schrodingerequation).Proceedinganalogouslyto thecontinuouscasewehaveto eliminatethegrowing factorβ j by choosingd appropriatelyas

d 1α β

∞

∑m) J

β mγm

(1.3.22)

We obtainfrom (1.3.21)

U j c 1α β

j

∑m) J

α mγm α j 1α β

∞

∑m) j 1

β j mγm j J 1

(1.3.23)

Thevalueof c canbeexpressedwith UJ 1:

c UJ 1

αJ 1 β

α

J 1 1α β

∞

∑m) J

β mγm(1.3.24)

andinsertingthis into (1.3.23)with j J:

UJ cαJ 1α β

∞

∑m) J

βJ mγm(1.3.25)

yields

UJ αUJ 1 1 αβ

1α β

∞

∑m) J

βJ mγm

αUJ 1 β 1

∞

∑m) 0

β mγJ m(1.3.26)

or equivalently

bUJ 1 βUJ ∞

∑m) 0

b mαmγJ m

(1.3.27)

Finally, we wantto applytheseresultsto thediscretizedSchrodingerequation(1.2.13)andderivetheDTBC at j J in thesituation,whentheinitial dataψ0

j doesnotvanishfor j J 1.In this casethe

–transformedright exterior Crank–Nicolsonschemereads:

ψ j 1z 2 iR

z 1z 1 iκ ψ j

z ψ j 1

z z

z 1ϕ j j J 1(1.3.28)

wheretheinhomogeneityϕ j is givenby

ϕ j ∆2xψ0

j iRψ0j Rκψ0

j j J 1

(1.3.29)

We canuse(1.3.27)to obtainthe transformedright DTBC:

ψJ 1z ν2

z ψJ

z z

z 1

∞

∑m) 0

νm1

z ϕJ m(1.3.30)

whereν1, ν2 arethetwo solutionsof thequadraticequation(1.2.17).

REMARK 3.2. Again,(1.3.30)reducesto theDTBC (1.2.18)for ϕ j ; 0.


In orderto formulatetheDTBCwedefinep nm : 1 - νm

1

z . andset

0 nJ : 1 - ν2

z . .

Using(IZ.7) weobtainby inversetransforming(1.3.30)

ψnJ 1

0 0J ψn

J n 1

∑k) 0

0 n kJ ψk

J 1 nϕJ ∞

∑m) 1

n

∑k) 0

1 n kp km ϕJ m n 1

(1.3.31)

Sincethe coefficients0 n

J asymptoticallyalternatein time, this formulationcanbe improved

andshortenedby regardingoncemores nJ : 0 n

J 0 n 1J , which givesfinally theDTBC for

non–compactlysupportedinitial data:

ψnJ 1

s 0J ψn

J n 1

∑k) 0

s n kJ ψk

J ψn 1

J 1 ∞

∑m) 1

p nm ϕJ m n 1

(1.3.32)

REMARK 3.3. Notethatin contrastto theDTBC in Section2 ther.h.s.(1.3.32)for n 1 isnot zerobut

ψ1J 1

0 0 ψ1J s

1 ψ0J ψ0

J 1 ∞

∑m) 1

p 1m ϕJ m

(1.3.33)

In practicalsituationsthesum(over m) in (1.3.32)of coursehasto befinite (e.g. up to anindex m M). This meansthat the initial conditionis still compactlysupported,but possibly

outsideof thecomputationalinterval. Thecoefficientsp nm , m 1 2 M, canbecalculated

recursively by “continuedconvolution”, i.e.

p n1 1 - ν1

z . p

n2 n

∑k) 0

p n k1 p

k1 p

n3 n

∑k) 0

p n k2 p

k1 etc.

(1.3.34)

Sincethis computationis rathercostly (evenwhenusingfastconvolution algorithmswith

FFTs[S5,Chapter4]) weseekfor anotherwayto calculate∑Mm) 1 p

nm ϕJ m, n 1. Thekey idea

is to usethequadraticequation(1.2.17)for ν1

z in orderto constructa recurrencerelationfor

the p nm (w.r.t. m). Equation(1.2.17)for ν1

z gives

νm 11

z 2 1 iR

2z 1z 1 iκ νm

1

z νm 1

1

z

c1νm1

z c2

zz 1

νm1

z νm 1

1

z m 1

(1.3.35)

with c1 2 iR Rκ andc2 2iR. An inverse

–transformationgives

p nm 1 c1p

nm c2

n

∑k) 0

1 kp n km p

nm 1 m 1(1.3.36)

with thestartingsequencesp n0 δ0

n andp n1 1 - ν1

z . , n 0. To circumventtheconvo-

lution in (1.3.36)weconsiderq nm : p

nm p

n 1m , p

1m 0 andobtain

q nm 1 c1q

nm c2p

nm q

nm 1 m 1(1.3.37)

to usein theDTBCof theform

ψnJ 1

s 0J ψn

J n 1

∑k) 0

t n kJ ψk

J 2ψn 1

J 1 ψn 2

J 1 S nM n 1(1.3.38)


wheret nJ : s

nJ s

n 1J , n 1 and

S nM : M

∑m) 1

q nm ϕJ m n 1

(1.3.39)

REMARK 3.4. For n 1 (1.3.32)reads:

ψ1J 1

s 0J ψ1

J t 1J ψ0

J 2ψ0

J 1 S 1M

(1.3.40)

The calculationof (1.3.39)with the aid of the recursionformula (1.3.37)is doneby thefollowing algorithm

1. q n0 δ0

n δ1n

2. q n1 p

n1 p

n 11 s

nJ

3. S n1 q

n1 ϕJ 1

4. for m 1 ? M 1 do

q nm 1 c1q

nm c2p

nm q

nm 1

S nm 1 S

nm q

nm 1ϕJ m 1

p 0m 1 q

0m 1

for n 1 ? N do

p nm 1 q

nm 1

p n 1m 1

endend

HereN denotesthemaximumtime index and s nJ the summedcoefficientsbut with theother

sign. Thecomputationaleffort of theabove implementationof theDTBC is OM 9 N , i.e. the

sameeffort aswhenenlarging thecomputationaldomainsufficiently. Theusageof theDTBCis especiallybeneficialwhenoneneedsseveralcomputationswith thesameinitial data.Thenthe calculationof the additionalterm hasonly to be doneonce. The sameapplieswhentheinitial field is concentratedfar outsidethecomputationaldomain.This is thecasein radiowavepropagationwhencomputingcoveragediagramsof airborneantennas.

Alternatively, asecondpossibleimplementationis to considerthetransformedDTBC(1.3.30)andto calculatenumericallytheinverse

–transformof thefinite sumonce:

Fn 1 zz 1

M

∑m) 0

νm1

z ϕJ m

(1.3.41)

TheDTBC thenreads

ψnJ 1

0 0J ψn

J n 1

∑k) 0

0 n kJ ψk

J Fn n 1

(1.3.42)

Thenumericalinverse

–transformationwill bethetopic of thenext section.

REMARK 3.5. While the DTBC (1.3.32)solves the problemof initial datathat are sup-portedoutsideof thecomputationaldomain,the resultingnumericaleffort of this approachis

4. NUMERICAL INVERSE P –TRANSFORMATIONS 33

not completelysettledyet andsubjectto further investigations.In particularonehasto com-

parean“optimal” computationalgorithmfor thecoefficientsp mn or q

mn with simulationson a

sufficiently enlargedcomputationaldomain.

3.3. Numerical Results. Herewe presentthe numericalresultswhenusingour new dis-creteTBC (1.3.32)for the Schrodingerequation(1.1.1). We usethe sameinitial dataas inSection2.5, but shiftedsuchthat it is partially outsidethe computationaldomain0 x L.Again,dueto its construction,our DTBC yieldsexactly (up to round–off errors)thenumericalwhole–spacesolutionrestrictedto thecomputationalinterval 0 L .

Example. This exampleshows a simulationof a right travelling Gaussianbeam ψI x exp

i100x 30

x 0

8 2? at threeconsecutivetimesevolving underthefreeSchrodingerequa-

tion ( 1) with the rathercoarsediscretizationof 161grid pointsfor the interval 0 x 1(i.e.∆x 1% 160)andthetimestep∆t 2 9 10 5. For theright exterior (computational)domainwe choosethesamespacestep∆x anduse60 grid pointswhich resultsin theexterior interval1 x 1

38125.

In thefollowing Figure1.9we plottedtheabsolutevalueof theinitial dataandthesolutionobtainedwith thediscreteTBCs(1.3.32)at thetimestepst 0

002,t 0

004,t 0

006.One

clearly seesin Figure1.9 that the solution is solely propagatedto the right andno artificialreflectionsarecaused.

In this examplethecomputationusingthe inhomogeneousDTBCs(1.3.32)needsapprox-imatelythesameCPU–timethanjust enlarging thedomainto theinterval 0 x 1

8 usinga

simpleNeumannboundaryconditionat x 0 andx 18. FromFigure1.9onecanguessthat

thesolutionat t 0004hasalreadyreachedtheright boundaryat x 1

8. Henceit is worth-

while in thisexampleto usetheinhomogeneousDTBCswhenever thesolutionfor t 0004is

needed.

4. Numerical Inverse

–Transformations

Thecrucialpoint in thederivationof theDTBC in Section2 wasto find theexact inverse–transformations.If it is not possibleto calculatethe convolution coefficientsanalytically

thentheinverse

–transformationcanbeperformednumerically.Thenumericalinversionof the

–transformationis basedonthesimpleobservationthatthe

–transformationof thesequence- fn. , n 0 1 ? .

- fn. f

z : ∞

∑n) 0

fnz n z IC & z&3 R 1 (1.4.1)

is nothingelsebut a Taylor seriesin z 1% z, i.e. the problemof calculatingthe inverse

–transformationof f

z is the numericalevaluationof the Taylor coefficients of the function

fz : f

1% z . For thatpurposeweusedheretheFORTRAN subroutineENTCAF(Evaluation

of NormalizedTaylorCoefficientsof anAnalytic Function)from LynessandSande[C5].First we want to outline themethod.The (normalized)Taylor coefficientsrn fn canbeob-

tainedby Cauchy’s integral representation:

rn fn rn

2πi Q fz z n 1 dz r R(1.4.2)

where R denotesacirclearoundtheorigin with radiusr smallerthantheradiusof convergenceR of theTaylor series.Theapproximationto fn basedon usinganN–pointtrapezoidalrule for


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

|ψ|

Schroedinger: t=0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

|ψ|


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

|ψ|


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

|ψ|


FIGURE 1.9. Solution & ψ x t (& at time t 0, t 0

002,t 0

004,t 0

006:

thesolutionwith thenew discreteTBCs(1.3.32)coincideswith thewhole–spacesolutionanddoesnot introduceany numericalreflections.

thecontourintegral is f N n givenby

rn f N n 1

N

N 1

∑k) 0

e in 2πkN f rei 2πk

N n 0 1 ? N 1

(1.4.3)

Theapproximationrn f N n is obtainedby an iterative process.First approximationsf

N 0 , N

1 2 4 8 arecomputedusing(1.4.3). Theconvergencecriterion is basedon theknowledge

of theexactvalueof the limit of thesequence:limN : ∞ f N 0 f0 f

0 (cf. the Initial Value

TheoremZ.6 in theAppendix).After converging thesecondpartconsistsin evaluating(1.4.3)for n 0 1 ? N 1 usingthestoredfunction valuesobtainedin the first part. SinceN is apowerof 2 it is particularlyappropriateto usea fastFouriertransformtechniquefor this part.

Theuserhasto specifytherequiredabsoluteaccuracyεreq andtheradiusof computationr(theonly restrictionis that r mustbe lessthanR). ENTCAF returnsanaccuracyestimateεest


togetherwith approximationsrn f N n andanumberN, which aresupposedto satisfy

rn f N n rn fn εest n 0 1 2 N 1(1.4.4a)

& rn fn & εest n N N 1 ?4(1.4.4b)

Weseefrom(1.4.4a)thatthisalgorithmnaturallydeliversapproximationsrn f N n with auniform

boundon thediscretizationerror. An outputstatusparameterindicatesto theuserwhetherornot convergenceor roundoff errorshave occurred. Exploiting the informationof this outputparameteronecouldconstructadriverprogramwhichfindstheappropriatevalueof r by itself.

Due to the asymptoticbehaviour (1.2.36)it is not advisableto calculatethe0 n

j . Instead

we show how to computethesummedconvolution coefficientss nj numerically. Thes

nj were

definedby:

s nj : 0 n

j 0 n 1j n 1 s

0j 0 0

j j 0 J (1.4.5)

Weconcentrateon theright BC at j J. If weassumel 1j 0 wehave:

- s nJ

. ν2

z z 1ν2

z

1 z ν2

z(1.4.6)

with ν2

z ν2

z andν2

z givenby formula(1.2.21).

4.1. Numerical Results. Herewepresentthenumericalresultswhenusingthesubroutine

ENTCAF to computetheconvolution coefficients0 n

J , s nJ . In eachexamplewe choseεreq

10 6 andsetthemachineaccuracyparameterεM to 10 15.

Example1. Thevaluefor thepotentialVL wassetto 2 9 104 andthediscretizationparameterweretaken from the exampleof Subsection2.5, i.e. ∆x 1% 160,∆t 2 9 10 5. We usedthecomputationalradiusr 0

92. For thatparameterchoiceENTCAF returneda numberof N

256nontrivial calculatedcoefficientsandanestimateduniformabsoluteaccuracy εest 983029

10 7 in caseof thecoefficients0 n

J . For thesummedcoefficientss nJ weobtainedN 128and

εest 476849 10 7. In thefollowing Figure1.10wepresenttherealandimaginarypartof the

numericallyobtainedcoefficientsin comparisonto theexactvalues.

0 20 40 60 80 100 120 140 160 180−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Re lJ(n)

n0 20 40 60 80 100 120 140 160 180

−20

−15

−10

−5

0

5

10

15

20

Im lJ(n)

n

FIGURE 1.10. Example1: Valuesfor Re0 n

J , Im0 n

J


One seriousproblemis the rescalingof the computedcoefficients. Sincethe algorithm

yieldsapproximationsrn f N n with auniformaccuracy (1.4.4a)thecomputationis only reliable

to a limited numberof n whencalculating f N n from rn f

N n for r 1. Thereforesomevisible

numericalerrorsoccurin thecalculationof Re0 n

J for n 130andIm0 n

J for n 170.On theleft sideof Figure1.10we observe thealternatingbehaviour shown in (1.2.36)

0 nj + 2iR

1 n i8

∆x 2

∆t

1 n i1258

1 n (1.4.7)

0 10 20 30 40 50 60 70 80 90−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Re sJ(n)

n0 10 20 30 40 50 60 70 80 90

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Im sJ(n)

n

FIGURE 1.11. Example1: Valuesfor Res nJ , Im s

nJ .

0 10 20 30 40 50 60 70 80 900

1

2

3

x 10−4 Error in Re l

J(n), Re s

J(n)

n

Error in Re lJ(n)

Error in Re sJ(n)

0 10 20 30 40 50 60 70 80 900

1

2x 10

−3 Error in Im lJ(n), Im s

J(n)

n

Error in Im lJ(n)

Error in Im sJ(n)

FIGURE 1.12. Example1: Dif ferencebetweenthenumericalandexactvalue

of therealandimaginarypartof0 n

J (—) ands nJ ( 999 ).

Example 2. In this secondexamplewe setthe potentialto zeroandchangedthe compu-tationalradiusto r 0

95 andthestepsizes∆x 1% 1600,∆t 2 9 10 4. For thecoefficients0 n

J ENTCAFreturnedanumberof N 256nontrivial calculatedcoefficientsandanestimated

uniform absoluteaccuracy εest 298029 10 6. In caseof the summedcoefficients s

nJ we


obtainedN 128andεest 122889 10 7. As beforewe show therealandimaginarypartof

thenumericallyobtainedcoefficientsin comparisonto theexactvalues.Again,on theleft side

0 20 40 60 80 100 120 140 160 180−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

Re lJ(n)

n0 20 40 60 80 100 120 140 160 180

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Im lJ(n)

n

FIGURE 1.13. Example2: Inverse

–transformationusingENTCAF:

Numericallyobtainedvaluesfor Re0 n

J , Im0 n

J comparedto theexactones.

of Figure1.13onecanseethealternatingbehaviour of thecoefficients0 n

j andthenumericalerrorsdueto therescalingof thecomputedcoefficientsfor approximatelyn 30.

0 10 20 30 40 50 60 70 80 90−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

Re sJ(n)

n0 10 20 30 40 50 60 70 80 90

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Im sJ(n)

n



Numericallyobtainedvaluesfor Res nJ , Im s

nJ comparedto theexactones.

Example 3. Finally we usethesettingsof thesecondexampleandintendto usea compu-

tationalradiusto r 1 to circumventtheproblemof therescaling.For thecoefficients0 n

J thiscannotbe donedueto the singularityof ν2

z at z 1. On the otherhand,this singularity is


0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2x 10

−6 Error in Re lJ(n), Re s

J(n)

n0 10 20 30 40 50 60 70 80 90

0

0.2

0.4

0.6

0.8

1

1.2x 10

−6 Error in Im lJ(n), Im s

J(n)

n


–transformationusingENTCAF:Differencebetweenthenumericalandexactvalueof therealandimaginarypart

of0 n

J (—) ands nJ ( 999 ).

removedin (1.4.6): - s nJ

. 1 z ν2

z

1 z iR2

y iR2 4i

R

1 z y y(1.4.8)

with theabbreviation y 1 z iκ

z 1 .

For thesummedcoefficientss nJ ENTCAFreturnedanumberof N 1024nontrivial calcu-

latedcoefficientsandanestimateduniformabsoluteaccuracy εest 224929 10 6. Themaximal

absoluteerrorfor Res nJ is 2

86099 10 6 and2

14559 10 6 for calculatingIm s

nJ .


0 100 200 300 400 500 600 700 800−1

0

1x 10

−3 Re sJ(n)

n0 100 200 300 400 500 600 700 800

0

1

2

3

4

5

6

7

8

9

10x 10

−4 Im sJ(n)

n



Numericallyobtainedvaluesfor Res nJ , Im s

nJ comparedto theexactones.

CHAPTER 2

The Convection–DiffusionEquation

Herewewill demonstratehow thederivationof theDTBC of theprecedingChaptercarriesoverto thecaseof alinearconvection–diffusionequation.Thistypeof parabolicequationarisesfor examplein thestudyof heatconduction,reactiondiffusionproblemsandwhenlinearizingtheNavier–Stokesequation.Theresultswill begeneralizedto theθ–schemewith the implicit-nessparameter0 θ 1 andto aflexible discretizationof theconvectionterm.A shortversionof this Chaptercanbefoundin [P2]. This approachcanbegeneralizedto a vector–valuedpar-abolicequationdescribing(secondorder) fluid stochasticPetri nets[P7].


In this Sectionwe shortly presentthe derivation of the (analytic)TBC. For simplicity ofthenotationwerestrictourselveshereto a constantcoefficientparabolicequationin onespacedimension:

ut auxx bux

cu x IR t 0ux 0 uI

x (2.1.1)

Of course,in generalthecoefficientsin (2.1.1)maydependon x andt but for our purposesweshall assumethat they remainconstantoutsideof the computationaldomain. Without lossofgeneralitywe assumefor the reactionratec 0 (otherwisethis propertycanbeachievedby achangeof variablesv uexp

ct ).1.1. Derivation of the TBC. Herewe determinetheTBCs at x 0 andx L, suchthat

the solution of the resulting IBVP coincideswith the solution of the whole–spaceproblem(2.1.1)restrictedto

0 L . Oncemorewehaveto assumethattheinitial datauI

x is compactly

supportedin thecomputationaldomain0 L . Proceedinganalogouslyto theformalderivation

of theTBCsin Chapter1 weconsiderthe interior problem

ut auxx bux

cu 0 x L t 0ux 0 uI

x

ux0 t

T0u 0 t ux

L t

TLu L t (2.1.2)

andobtaintheDirichlet–to–NeumannmapT0 by solvingthe left exterior problem:

vt avxx bvx

cv x 0 t 0vx 0 0

v0 t Φ

t t 0 Φ

0 0

v ∞ t 0T0Φ t vx

0 t

(2.1.3)

41

42 2. THE CONVECTION–DIFFUSIONEQUATION

(2.1.3)is coupledwith (2.1.1)by theassumptionthatu, ux arecontinuousacrosstheartificialboundaryat x 0. Sincethe initial datavanishesfor x 0, we cansolve (2.1.3)explicitly bytheLaplace–methodandobtainthetransformedleft exterior problem:

avxx bvx

c s v 0 x 0

v0 s Φ

s (2.1.4)

Thesolutionwhich decaysasx ∞ is simply vx s Φ

s eλ1 s x, x 0, with

λ1 2 s b2a

6 1!a ! η s(2.1.5)

andtheparameter

η b2

4a c 0

(2.1.6)

Consequently, thetransformedleft TBC reads:

ux0 s λ1

s u

0 s (2.1.7)

Weremindthereaderthatin (2.1.5) ! denotesthebranchof thesquarerootwith nonnegativerealpart.After theinverseLaplacetransformationthe left TBC reads:

ux0 t b

2au0 t 1!

aπe ηt d

dt

t

0

u0 τ eητ!

t τdτ

(2.1.8)

We observe againthat (2.1.8) is nonlocalin t (of memory–type),i.e. the computationof thesolutionat sometimeusesthesolutionat all previoustimes.Theright TBCat x L is derivedsimilarly:

uxL t b

2auL t 1!

aπe ηt d

dt

t

0

uL τ eητ!

t τdτ

(2.1.9)

REMARK 1.1(InhomogeneousEquation). We note that it is possibleto consider(2.1.1)with an inhomogeneityf

x t provided that f

x t f is constantoutsideof

0 L . In this

casethetransformedleft exterior problemis aninhomogeneousODE in x:

avxx bvx

c s v s 1 f x 0

v0 s Φ

s (2.1.10)

anda (constantin x) particular solutionreads

vpart x s fsc s x 0(2.1.11)

i.e. thegeneralsolutionto (2.1.10)is

vx s Φ

s f

sc s eλ1 s x f

sc s x 0(2.1.12)

with λ1

s givenin (2.1.5).Thereforethe transformedleft TBC is givenby:

vx0 s λ1

s Φ

s λ1

s f

c1s 1

c s(2.1.13)

andaftertheinverseLaplacetransformationthe left TBC reads:

ux0 t b

2au0 t ψ

t 1!

aπe ηt d

dt

t

0

u0 t ψ

t eητ!

t τdτ (2.1.14)


whereψt is obtainedusing(IL.3):

ψt f

c

1 e ct (2.1.15)

1.2. Well–posednessof the IBVP. It is known thattheIVP (2.1.1)is well–posed:

THEOREM 2.1(Theorem6.2.1,[F8]). Theinitial valueproblemfor theequation(2.1.1)iswell–posed,i.e. for anytimeT 0 there is a constantCT such thatanysolutionu

x t satisfies

" u t (" L2 IR t

0" ux

τ (" L2 IR dτ CT " uI " L2 IR(2.1.16)

for 0 t T.

However, thewell–posednessof theassociatedIBVP is not cleara-priori. While theexis-tenceof a solutionto the1D parabolicequation(2.1.2)with theTBCs(2.1.9), (2.1.8)at x 0andx L is clearfrom theusedconstructionit remainsto provetheuniquenessof thesolution.A straightforwardcalculationusingtheenergy method,i.e.multiplying (2.1.1)with u

x t and

integratingby partsin x, yields

12

ddt

" u t (" 2L2 0 L a " u S t (" 2

L2 0 L bL

0ux t ux

x t dx c " u t #" 2

L2 0 L au

x t ux

x t (& x) L

x) 0

aux t ux

x t (& x) L

x) 0 (2.1.17)

if c b2 % 4a (which canalwaysbe achieved by a changeof variables).Finally, integrating

in time usingPlancherel’s Theoremfor theLaplacetransformation(TheoremL.3) (u0 t and

uL t areextendedby 0 for t T) gives

" u t (" L2 0 L I" uI " L2 0 L 2aT

0 u

L t uxL t u

0 t ux

0 t dt

" uI " L2 0 L 2aIR

λ2

iξ '& u

L iξ (& 2 λ1

iξ '& u

0 iξ (& 2 dξ

" uI " L2 0 L 4a∞

0Reλ2

iξ '& u

L iξ (& 2 Reλ1

iξ '& u

0 iξ (& 2 dξ

(2.1.18)

sinceu0 iξ ¯u

0 iξ , u

L iξ ¯u

L iξ , λ1 2 L iξ λ1 2 L iξ . Now it caneasilybe

checkedthattheconditionsfor well–posedness

Reλ1

iξ 0 for ξ IR(2.1.19a)

Reλ2

iξ 0 for ξ IR(2.1.19b)

(λ1 2 givenby (2.1.5))arefulfilled andwecanstatethefollowing theorem.

THEOREM 2.2. Theresultingparabolic IBVP is well–posed

" u t #" L2 0 L *" uI " L2 0 L t 0(2.1.20)

andthis impliesuniquenessof thesolutionto theparabolicIBVP.



Next weshalladdressthequestionhow to adequatelydiscretizetheanalyticTBC (2.1.8)fora chosenfull discretizationof (2.1.1). Insteadof discretizingtheanalyticTBC (2.1.8)with itssingularitywe derive thediscreteTBCsof thefully discretizedproblembasedon theso–calledweightedaverageor θ–schemefor 0 θ 1. With theuniform grid pointsx j j∆x tn n∆tandtheapproximationun

j / ux j tn theθ–schemefor solving(2.1.1)reads:

Dt unj aD2

xun θj

bDσxun θ

j cun θ

j (2.2.1)

with

un θj θun 1

j 1 θ un

j 0 θ 1

(2.2.2)

REMARK 2.1. While a uniform grid in x is necessaryin the exterior domain,the interiorgrid maybenonuniformin x.

In the scheme(2.2.1) Dt denotesthe usualforward and D2x the secondorder difference

quotient:

Dt unj un 1

j un

j

∆t D2

xun θj un θ

j 1 2un θ

j un θj 1

∆x 2

(2.2.3)

The convectionterm is discretizedby the parametrizeddifferencequotientDσx σDx

1 σ D @x , i.e.

Dσxun

j σunj 1

1 2σ unj

1 σ unj 1

∆t 0 σ 1(2.2.4)

which allows to switchbetweenacentereddifferencequotient(σ 1% 2) or aone–sideddiffer-encequotient(σ 0 or σ 1).

REMARK 2.2(“upwind differencing”). Onetypicalpropertyof theparabolicequation(2.1.1)is that

supx, IR

& u x t (& sup

x , IRux t < if t t <

Thatis, themaximumvalueof & u x t (& will not increaseast increases(“maximumprinciple” ).

In [D10, Chapter4.1.6] it wasproven that the differencescheme(2.2.1) will have a similarpropertyif andonly if thetwo conditions

σb∆xa

1 and1 σ b∆x

a 1(2.2.5)

aresatisfied.Wenotethatfor areasonablechoiceof σ (”upwind differencing”),i.e.0 σ 1% 2for b 0 and1% 2 σ 1 for b 0 theseconditionscanalwaysbefulfilled.

REMARK 2.3(stabilityof thescheme). Secondlyonecaneasilyshow usingthevon–Neu-mannanalysis[F8] that the differencescheme(2.2.1) for solving the initial value problem(2.1.1)is stable(in thediscreteL2–norm)providedthatthestability condition

1 2θ a

∆t∆x 2 1(2.2.6)

holdsandupwinddifferencingis used.I.e. for a ratherimplicit scheme(θ 1% 2) noadditionalconditionhasto beimposedto ensurestability.


2.1. Discretization strategiesfor the TBC. Herewe want to comparethreestrategiestodiscretizethe TBC (2.1.8) which is a ratherdelicatequestion. First we review two knowndiscretizationtechniquesfrom Mayfield [T16] andHalpern[P3].

DiscretizedTBC of Mayfield. To compareourresultswefirst repeatthead-hocdiscretiza-tion strategy of Mayfield appliedto the convection–diffusion equation. Accordingto the ap-proachof Mayfield [T16] for theSchrodingerequationoneway to discretizetheanalyticTBC(2.1.8)at x 0 in theequivalentform

u0 t a

π

t

0

ux0 τ e η t τ !

t τdτ b

2!

aπ

t

0

u0 τ e η t τ !

t τdτ (2.2.7)

is for thefirst integral

t

0

ux0 t τ e ητ!

τdτ / 1

∆x

n 1

∑m) 0

un m

1 un m

0 e ηm∆ttm 1

tm

dτ!τ

2!

∆t∆x

n 1

∑m) 0

un m

1 un m

0 e ηm∆t!m 1 !

m

(2.2.8)

Discretizingthe secondintegral in (2.2.7)analogouslyleadsto the following discretizedTBCfor theconvection–diffusionequation:

un1 1 b

2aun

0 !

π∆x

2!

a∆tun

0 n 1

∑m) 1

un m

1 un m

0 ˜0 m b2a

n 1

∑m) 1

un m0

˜0 m(2.2.9)

with theconvolutioncoefficientsgivenby

˜0 m e ηm∆t!m 1 !

m

andη definedby (2.1.6).Onthefully discretelevel (2.2.9)is notperfectlytransparentany moreandmaypossiblyleadto anunstablenumericalschemeasshown in Theorem1.2in caseof theSchrodingerequation.

REMARK 2.4(AlternativeDiscretizationof theTBC). In thecontext of underwateracous-ticsThomsonandMayfield [T26] proposedanalternativediscretizationof (2.1.8)to handletheexponentialtermsmoreaccuratelywhenη

t τ is large. This approachwill bepresentedin

Chapter3.

Artificial BC of Halpern. Secondlywe presenttheapproach of Halpern [P3] which wasgeneralizedby Loheacin [P5], [P6] to the casethat the diffusion coefficient a in (2.1.1)candependon x. HalpernandRauchderivedin [P4] absorbingboundaryconditionswith variablecoefficients,curved artificial boundaryandarbitraryconvection. The numericalstudyof thisconditionswerecarriedout in [P1] by Dubach.

In [P3]Halperndevelopedafamilyof artificial boundaryconditionsfor thelinearconvection–diffusion equationwith small diffusion a. To startwith we rewrite the transformedleft TBC(2.1.7)as

ux0 s 1

2ab b2 4

c s a u

0 s (2.2.10)


Now Halpern’sapproachconsistsof usingTaylor or Padeapproximationsof thetermin paren-thesesin (2.2.10)with respectto a small valueof a in order to obtaina local in t boundarycondition.A first orderTaylorapproximationgives

b b2 4c s a / b & b & 2a

c s& b & (2.2.11)

which leadsto thetransformedboundarycondition

ux0 s b & b &

2a c s& b & u

0 s (2.2.12)

Finally an inverseLaplacetransformationyields (dependingon the sign of b) the first orderartificial boundaryconditions:

inflow BCb 0 : ut

0 t bux

0 t b2

a c u0 t 0(2.2.13a)

outflow BCb 0 : ut

0 t bux

0 t cu

0 t 0

(2.2.13b)

To discretize(2.2.13)we follow thesuggestionin [P3] andgeneralizeit for theθ–scheme:

inflow BCb 0 : Dt un

0 bDxun θ

0 b2

a c un θ0 0(2.2.14a)

outflow BCb 0 : Dt un

0 bDxun θ0 cun θ

0 0(2.2.14b)

and analogouslyat the right boundaryat L J∆x. While Halpernshowed that the interior(Crank–Nicolson)schemetogetherwith theabsorbingboundarycondition(2.2.14)(for c 0) isstableandhasordertwo in timeandspace,theresultingschemesuffersfrom reducedaccuracyaswewill seelaterin thenumericalexamplesof Section3.

Discrete TBC. In orderto avoid any numericalreflectionsat the boundaryandto ensureunconditionalstabilityof theresultingschemewewill constructdiscreteTBCsinsteadof choos-ing anad–hocdiscretizationof theanalyticTBC (2.1.8)like(1.2.7)or theapproachof Halpern.ThediscreteTBCscompletelyavoid any numericalreflectionsat theboundaryat noadditionalcomputationalcosts(comparedto (2.2.9)).

2.2. Derivation of the DTBC. Wemimic thederivationfrom Section1 onadiscretelevel:we obtaintheDTBC by solvingthediscreteleft exterior problem, i.e. (2.2.1)for j 1:

∆t unj b

∆t∆x

σ∆x 1 σ ∆ @x un θ

j 2κun θj r∆2

xun θj (2.2.15)

wherer a∆t % ∆x 2 denotesthe(parabolic)meshratio, κ c∆t % 2 andσ denotestheparam-

eterexplainedin (2.2.4). In thescheme(2.2.15)we usedthedifferenceoperators∆t ∆t Dt ,∆2

x ∆x 2D2

x, etc.Oncemoreweapplythe

–transformation: - un

j. u j

z : ∑∞

n) 0unj z n,

j fixed,to solve (2.2.15)explicitly. We assumefor theinitial datau0j 0 j 2 andobtainthe

transformedscheme

1r

z 1θz 1 θ

u jz τ ∆x τ @ ∆ @x 2κ

ru j

z ∆2

xu jz j 1(2.2.16)

with

τ σb∆xa

τ @ 1 σ b∆x

a

(2.2.17)


Thetwo linearlyindependentsolutionsof theresultingsecondorderdifferenceequation(2.2.16)take theform

u jz ν j 1

1 2 z j 1(2.2.18)

whereν1 2 z arethesolutionsof thequadraticequation

ν2 21 τ 1 1

rz 1

2θz 1 θ κ τ τ @

2ν 1 τ @

1 τ 0

(2.2.19)

Sincewe areseekingdecreasingmodesas j ∞ we have to require & ν1

z(&B 1 andobtain

the

–transformedleft discreteTBCas

u1

z ν1

z u0

z (2.2.20a)

Analogously, the

–transformedright discreteTBC reads

uJ 1z 1 τ

1 τ @ ν1

z uJ

z(2.2.20b)

wherewehaveusedthepropertyν1

z ν2

z

1 τ @ % 1 τ .

REMARK 2.5. It wasshown by Lill [D4, Theorem3.11] that for the solutionsto thequa-draticequation(2.2.19) & ν1

z#&A 1, & ν2

z(&A 1 holdsfor & z&A 1.

It only remainsto inverse

–transformν1

z in order to obtain the discreteTBCs from

(2.2.20)andin a tediouscalculationthis canbeperformedexplicitly:

CALCULATION (of1 τ 1 - ν1

z . ). Firstwereformulate

1 τ ν1 2 z appropriately:

1 τ ν1 2 z 1 τ τ @

2 κr 1

rz 1

2θz 1 θ

6 1 1r

z 12θz 1 θ κ τ τ @

2

2 1 τ @ 1 τ

H κr 1

21 θ

1r

1θ

z

z 1 θθ

1 6 1r

! Az2 2Bz C2θz 1 θ

HerePe b∆x%

2a denotesthecell Pecletnumber. Theconstantscanbedeterminedas

A κ 2 4rθHκ

β 2 (2.2.21a)

B κ κ @ 2rH1 θ κ θκ @ β β @((2.2.21b)

C κ @ 2 4r

1 θ Hκ @

β @ 2 (2.2.21c)

with theabbreviations

κ 1 2θκ 1 θc∆t (2.2.22a)

κ @ 1 21 θ κ 1

1 θ c∆t (2.2.22b)

β θb∆t∆x

β @ 1 θ b∆t

∆x(2.2.22c)

H 1 1 2σ Pe

(2.2.22d)

We notethat for a reasonablechoiceof σ (”upwind differencing”),i.e. 0 σ 1% 2 for b 0and1% 2 σ 1 for b 0,

1 2σ Pe 0 holds andconsequentlyH 1 and A is always

positive. Thiswill beassumedfrom now on.


For theinverse

–transformationweuseagain(1.2.23)andobtain

! Az2 2Bz C2θz 1 θ 1!

A

Az2 2Bz C2z

θz 1 θ F

λz µ

1!A

A2θ C

21 θ

1z

E2θz 1 θ F

λz µ

with E givenby

E 1 θθ

A 2B θ1 θ

C

(2.2.23)

Now applyingtheinversionrules(especially(IZ.9)) givenin theAppendixyieldsfor 0 θ 1:

1 ! Az2 2Bz C2θz 1 θ 1!

A

A2θ

δ0n C

21 θ δ1

n

E21 θ 1 θ

θn δ0

n 1 Pnµ

1!A

A2θ

Pnµ C

21 θ Pn 1

µ

E21 θ

n 1

∑k) 0

1 θθ

n kPk

µ

where 1 denotesthediscreteconvolution. Finally, weobtainwith (IZ.10)1 τ 1 - ν1 2 z . H κ

r 1

21 θ r δ0

n 1 n

2θ1 θ r

1 θθ

n

6 1

r!

A

A2θ

Pnµ C

21 θ Pn 1

µ

12θ

1 θ 2

n 1

∑k) 0

1 θθ

n kPk

µ

(2.2.24)

sincetheconstantE canbesimplydeterminedfrom (2.2.23)as

E 1θ1 θ

(2.2.25)

We recallfrom Chapter1 thattheparametersλ, µ aregivenby

λ !

A

! C µ B!

A ! C

(2.2.26)

It caneasilyseenfrom (2.2.21)thatthefollowing relationshold:

C A 4rH 2c∆t 1 2θ 4rHc∆t

c∆t 2 b∆t∆x

2 (2.2.27)

B2 AC 2r 2 H2 Pe

2 (2.2.28)

For C 0, i.e. if f

H2 Pe2 H 1

21 θ r c

∆x 2

2a

2 (2.2.29)


we concludefrom (2.2.27)thatfor a ratherimplicit scheme(θ 1% 2) we have λ 1 andfrom(2.2.28)weseethatin thetypical case2σ & Pe &3 1 wehaveµ 1, i.e. theLegendrepolynomialsPn

µ in (2.2.24)haveto beevaluatedoutsidethestandarddomain 1 1 . Wenotethatspecial

routinesfor thatpurposeexist [C4].

REMARK 2.6(“damped”Legendrepolynomials). In contrastto thecalculationof thecon-volutioncoefficients

0 n in caseof theSchrodingerequationin Chapter1 wehaveµ 1 whichmakesit essentialto usethedampingfactorλ n to circumventthe“blowingup” of theLegendrepolynomialsPn

µ asn ∞. Thisbehaviour canbeseenfrom thefollowing Lemma:

LEMMA 2.3(Theorem8.21.1(Formulaof Laplace–Heine),[S7]). Letµbeanarbitraryrealor complex numberwhich doesnotbelongto theclosedsegment 1 1 . Thenasn ∞,

Pnµ+ 1!

2πnµ2 1 1

4 µ µ2 1n 1

2 (2.2.30)

This formula holdsuniformly in the exterior of an arbitrary closedcurvewhich enclosesthesegment 1 1 , in thesensethat theratio tendsuniformlyto 1.

The “damped” Legendre polynomialsPnµ : λ nPn

µ arecomputeddirectly by the re-

cursionformula

Pn 1µ 2n 1

n 1µλ 1Pn

µ n

n 1λ 2Pn 1

µ n 0(2.2.31)

with the startingvaluesP0 ; 1, P 1 ; 0. The asymptoticbehaviour of the Pnµ will be the

topicof thenext subsection.

REMARK 2.7(negativeC). If C 0 thenλ, µ becomecomplex, but Pnµ λ nPn

µ re-

mainsa realquantity. This caneasilybeseenfrom the recursionformula (2.2.31). Thereforethis caseneedsnospecialtreatment.

REMARK 2.8(Implicit Eulerscheme). Note that the convolution coefficients for the im-plicit Euler scheme(θ 1) arenot includedasa specialcase.For θ 1 the calculationhasto bedoneseparately, sincein this caseweobtain

1 z 12θz 1 θ 1

2δ0

n 1

2δ1

n

(2.2.32)

Theresultsaregivenin (2.2.35)below.

To summarizeour calculationsdonesofar wewrite theobtaineddiscreteTBCs:

1 τ un

1 0 n 1 un0 n

∑k) 1

0 n k uk0 (2.2.33a)

1 τ @J un

J 1 0 n 1 unJ n

∑k) 1

0 n k ukJ n 1(2.2.33b)


with convolutioncoefficients0 n for 0 θ 1 givenby

0 0 H κr 1 !

A2θr0 n

1 n

2θ1 θ r

1 θθ

n 1

r!

A9

9 A2θ

Pnµ C

21 θ Pn 1

µ 1

2θ1 θ 2

n 1

∑k) 0

1 θθ

n kPk

µ

(2.2.34)

for n 1, andin thecaseθ 1:

0 0 H κr 1 !

A2r

0 1 1

2r 1

2r!

AAP1

µ 2B 1

2r1 B!

A

0 n 1

2r!

AAPn

µ 2BPn 1

µ CPn 2

µ n 2

(2.2.35)

wherePnµ : λ nPn

µ denotesthe“damped” Legendrepolynomialsandδ0

n is theKroneckersymbol.

REMARK 2.9(Explicit Eulerscheme). Thecalculationabovedoesnot work directly in thecaseof the explicit Euler scheme(θ 0) sincethe inverse

–transformationof ν1

z does

not exist. In this situationwe useinsteadof (2.2.20)the

–transformeddiscreteTBCs in theformulation

z 1u1

z z 1ν1

z u0

z(2.2.36a)

z 1uJ 1z 1 τ

1 τ @ z 1ν1

z uJ

z (2.2.36b)

To derive theconvolutioncoefficientsof theDTBC weconsider

1 τ z 1ν1

z H κ

rz 1 1

2rz 1

z 12r

! Az2 2Bz Cz

12r H κ

r 1

2rz 1 A 2Bz 1 Cz 2

2r!

AF

λz µ

(2.2.37)

andcalculate 1 τ 1 z 1ν1

z 1

2rδ0

n H κr

12r

δ1n

1

2r!

AAPn

µ 2BPn 1

µ CPn 2

µ (2.2.38)

Finally, weobtainthefollowing discreteTBCsfor theexplicit Eulerscheme:

0 0 un0

1 τ un 11

n 1

∑k) 1

0 n k uk0 (2.2.39a)

0 0 unJ

1 τ @J un 1J 1

n 1

∑k) 1

0 n k ukJ n 1(2.2.39b)


with convolutioncoefficients0 n givenby

0 0 1 !A

2r

0 1 H κr

12r

1 B!A

0 n 1

2r!

AAPn

µ 2BPn 1

µ CPn 2

µ n 2

(2.2.40)

2.3. The Asymptotic Behaviour of the Convolution Coefficients. In this subsectionwewantto investigatetheasymptoticbehaviour of theconvolutioncoefficients

0 n givenby (2.2.33).As in Chapter1 wewill seethatit is beneficialto reformulatetheDTBC (2.2.33).

Thesummedconvolution coeffcients. It followsimmediatelyfromLemma2.3thatlimn: ∞Pn

µ 0 holds.Thusthecoefficientshave thefollowing asymptoticbehaviour for n ∞:

0 n 1 n

2θ1 θ r

1 θθ

n + 1 n

2rθ1 θ 2

!A

1 θθ

n n 1

∑k) 0

θ1 θ

kPk

µ (2.2.41)

Using

limn: ∞

1!A

n 1

∑k) 0

θ1 θ

kPk

µ 1!

A

1

1 2 θ1 θµλ 1 θ

1 θ 2λ 2

1

A 2 θ1 θB θ

1 θ 2C

1

θ1 θ E

1 θ

(2.2.42)

in (2.2.41)wefinally obtain

0 n + 1 n

θ1 θ r

1 θθ

n 0 θ 1 as n ∞

(2.2.43)

In the specialcaseof the Crank–Nicolsonscheme(θ 1% 2) we get0 n + 4

1 n % r. Thisalternatingbehaviour (2.2.43)may leadto subtractive cancellationin (2.2.33). Thereforewepreferto usethefollowing (weighted)summedcoefficientsin theimplementation

s n : θ

0 n 1 θ 0 n 1 n 1 s

0 : θ0 0 0 θ 1(2.2.44)

andcomputewith thehelpof (2.2.23)

s n 1

r!

A

A2

Pnµ 1 θ

θA2 θ

1 θC2

12θ

1 θ Pn 1

µ C

2Pn 2

µ

1

2r!

AAPn

µ 2BPn 1

µ CPn 2

µ

!

A2r

Pnµ 2µλ 1Pn 1

µ λ 2Pn 2

µ

(2.2.45)


Using

µλ 1Pn 1µ n

2n 1Pn

µ n 1

2n 1λ 2Pn 2

µ

we finally get

s n !

A2r

Pnµ λ 2Pn 2

µ

2n 1 n 2(2.2.46)

to usein theDTBC:

θ1 τ un

1 θs

0 un0 n 1

∑k) 1

s n k uk

0

1 θ 1 τ un 11 (2.2.47a)

θ1 τ @ un

J 1 θs

0 unJ n 1

∑k) 1

s n k uk

J

1 θ 1 τ @ un 1J 1 n 1

(2.2.47b)

REMARK 2.10. We seefrom (2.2.45)thatfor n 2 thesummedcoefficientss n coincideswith theconvolution coefficients(2.2.35)for theimplicit Eulerscheme,i.e. (2.2.44)alsoholdsfor θ 1.

The recurrenceformula for the summedcoefficients. In this subsectionwe shall givetwo different derivationsfor the recursionformula of the convolution coefficients s n . Thefirst oneis basedon theexplicit representation(2.2.46)of s n by first calculatinga recursionformulafor Pn 1

µ Pn 1

µ . Thesecondderivationdoesnot requiretheexplicit form of the

coefficientss n but only thegrowthfunctionν1

z from the

–transformedDTBCs(2.2.20).

FIRST DERIVATION: We considerthe recursion formula (1.2.47)adaptedfor the scaledLegendrepolynomialsPn 1

µ Pn 1

µ :

(2.2.48)Pn 1

µ λ 2Pn 1

µ

2n 1 µλ 1Pn

µ λ 2Pn 2

µ

n 1 n 2

n 1λ 2Pn 1

µ λ 2Pn 3

µ

2n 3 n 2

From(2.2.46)weseetherecurrencerelation for thesummedconvolutioncoefficients:

s n 1 2n 1

n 1µλ 1s

n n 2n 1

λ 2s n 1 n 2(2.2.49)

which canbeusedaftercalculatings n , n 0 1 2 by theformula(2.2.45).

SECOND DERIVATION: Now we presentan alternative derivation of the first convolutioncoefficientss n , n 0 1 2andtherecurrencerelation(2.2.49). Theadvantageof thisalternativeapproachis thatwe shallonly needthegrowthfunctionν1

z from the

–transformedDTBCs

(2.2.20).

In thissecondapproachweshallfirst deriveafirst orderODEfor thegrowth functionν1

z ,

which is givenexplicitly in thecalculationonpage47. In fact,it is moreconvenientto consider

νz :

θz 1 θ 1 τ ν z ∞

∑n) 1

s n 1 z n (2.2.50)


Usingtheconstants(2.2.21)and(2.2.22d)it hastheexplicit form

ν1 2 z z θ H κr 1

2r 1 θ H κ

r 1

2r6 1

2r Az2 2Bz C

(2.2.51)

Here,thesignhasto befixedsuchthat & ν1

z(&= 1 holds. This canbedonefor e.g.for z ∞.

Multiplying ν< dνdz

by Az2 2Bz C thenyields an inhomogeneousfirst orderODE for

νz :

AB

z2 2z CB

ν< z AB

z 1 νz β

z : β 1 z β 0 (2.2.52)

with

β 1 AB

1 θ H κ

r 1

2r θ H κr 1

2r

β 0 1 θ H κ

r 1

2r CB

θ H κr 1

2r

(2.2.53)

Its generalsolutionincludesν1 2 z asdefinedin (2.2.51). Recallingfrom (2.2.26)thatA% B λ % µ, C % B 1%

λµ holdsandusingtheLaurentseries(2.2.50)of ν andν< in (2.2.52)immedi-atelyyieldsthedesiredrecursionfor thecoefficientss n :

λµ

s 1 s

0 β 1 2

λµ

s 2 s

1 1λµ

s 0 β 0

n 1 s n 1

2n 1 µλ

s n

n 2 1λ2s

n 1 0 n 2(2.2.54)

whichcoincideswith (2.2.49).We remarkthat the startingcoefficient of the recursion(cf. (2.2.34))can be determined

with [S3,Theorem39.1]:

s 0 θ

0 0 limz: ∞

ν1

z

z θ H κr 1 !

A2r

0 θ 1

STABILITY OF THE RECURRENCE RELATION: For investigatingwhetherthe recurrencerelation (2.2.49)is well–conditionedwe proceedsimilar to Subsection2.3 in Chapter1 andwrite (2.2.49)asthesecondorderdifferenceequation

s n 1 a

n s n b n s n 1 0 n 3(2.2.55)

with thecoefficients

a n 2n 1

n 1µλ 1 b

n n 2n 1

λ 2 > 0

(2.2.56)

Therearetwo linearlyindependentsolutionss n1 , s

n2 to (2.2.55). To provethat(2.2.49)is well–

conditionedwe have to show that theseekedsolutionis not a minimalsolutionto (2.2.55),i.e.is adominantsolution.

Sincethecoefficientsa n , b n in (2.2.55)have thefinite limits

a limn: ∞

a n 2µλ 1 2

BA

b limn: ∞

b n λ 2 C

A > 0(2.2.57)


(2.2.55)is aPoincare differenceequationwith thecharacteristicpolynomial

Φt t2 at b

(2.2.58)

Thecharacteristicpolynomialhasthetwo zerost 1 2 µ 6 µ2 1% λ andsinceµ 1 it is

readilyverifiedthatthezeroshavedistinctmoduli

t 1 λ 1 t

2 0

ThereforetheclassicalTheoremof Perronwhichis animprovementof theTheoremof Poincare1.5canbeappliedto distinguishtwo solutionswith distinctasymptoticproperties:

THEOREM 2.4(PerronTheorem,[E2]). If thecharacteristicpolynomial(2.2.58)haszerost 1 , t 2 of distinctmoduli,

t 1 t

2 thenthereexist two linearly independentsolutionss

n1 ands

n2 of (2.2.55)such that

limn: ∞

s n 1k

s nk

t k k 1 2

(2.2.59)

REMARK 2.11(Minimal Solution). Theorem2.4 implies thats n2 is a minimal solutionof

(2.2.55)[E2].

Fromtheaboveweconcludethat(2.2.55)hasaminimalsolutions n2 for whichlimn: ∞ s

n 12% s

n2 t 2 , while the limit is t 1 for every othersolution. Now it is known that theLegendre

polynomialsPnµ andtheLegendre functionsof thesecondkind (of orderzero)Qn

µ satisfy

theidenticalthree–termrecurrencerelation.Thusasecondsolutionto (2.2.55)is simply

s n2 β

λ nj

2n 1Qn

µ Qn 2

µ n 2(2.2.60)

with someconstantβ. For Qnµ wehave thefollowing asymptoticbehaviour [E2, p. 60] :

Qnµ + π

2n 1$ 2 µ2 1 1$ 4 µ µ2 1

n 1$ 2 n ∞ (2.2.61)

for µ noton therealaxisbetween ∞ and1, thusin particularfor thoseµ whichweareconsid-eringhere.From(2.2.61)we obtainimmediately

(2.2.62) λ n Qnµ Qn 2

µ +

π2

µ µ2 11$ 2

µ2 1 1$ 4 1 n 2n

µ µ2 1 2 n 1$ 2 t 2 n n ∞

Sinceonecaneasilyverify that limn: ∞ s n 12 % s

n2 t 2 it follows directly that the minimal

solutionis s n2 , andthats

n1 s n givenin (2.2.46)is now adominantsolution,i.e. theproblem

of determiningtherequiredvaluesof theconvolutioncoefficientsis well–conditioned[E2].


2.4. Stability of the resulting Scheme.To prove stability of the numericalschemewiththediscreteTBCs(2.2.47)weusethediscreteenergymethod. Weconcentrateonthecaseof theCrank–Nicolsonscheme(θ 1% 2) with centraldifferencingof theconvectionterm(σ 1% 2),i.e.weconsider(2.2.15)in theform

∆t unj r∆2

xun 1

2j

rPe∆0xu

n 12

j 2κu

n 12

j (2.2.63)

where∆0x ∆x ∆ @x denotesthe centereddifferenceoperator. We multiply theequationwith

un 1$ 2j andsumit up for the finite interior range j 1 2 ? J 1, using the summationby

partsrule (1.2.70):

J 1

∑j ) 1

un 1

j 2 un

j 2 2rJ 1

∑j ) 1

un 1

2j ∆2

xun 1

2j

2rPe

J 1

∑j ) 1

un 1

2j ∆0u

n 12

j 4κ

J 1

∑j ) 1

u

n 12

j 2

2rJ 1

∑j ) 0

∆xun 1

2j

2 2run 1

2J ∆ @xu

n 12

J 2ru

n 12

0 ∆xun 1

20

2rPeun 1

2J u

n 12

J 1 2rPeun 1

20 u

n 12

1 4κ

J 1

∑j ) 1

u

n 12

j 2

2run 1

2J 1 Pe un 1

2J 1

un 1

2J 2ru

n 12

0 1 Pe un 12

1 u

n 12

0

(2.2.64)

Finally asummationwith respectto thetimeindex yieldsthefollowing estimatefor thediscreteL2–norm(definedby " un " 2

2 : ∆x∑J 1j ) 1 & un

j & 2)

" uN 1 " 22 I" uI " 2

2 2a∆t

∆x

N

∑n) 0

un 1

2J

1 Pe un 1

2J 1

un 1

2J

N

∑n) 0

un 1

20

1 Pe un 1

21

un 1

20

I" uI " 22 2a∆t

∆x

N

∑n) 0

un 1

2J u

n 12

J 1 ˜0 n N

∑n) 0

un 1

20 u

n 12

0 1 ˜0 n

(2.2.65)

where ˜0 n : 0 n δ0n, j 0 J, givenin (2.2.33).

Again, asin the continuouscase,it remainsto show that the boundary–memory–termsin(2.2.65)areof positivetype. Weconcentrateon theboundarytermat j J anddefinethefinitesequences

fn un 1

2J 1 ˜0 n gn u

n 12

J n 0 1 N (2.2.66)

with fn gn 0 for n N, i.e. ∑Nn) 0 fngn 0 is to show. A

–transformationusing the

transformedDTBC (2.2.20)yields

- fn. f

z z 1

2uN

Jz

1 Pe ν1

z 1

12r

uNJz z 1 κ

z 1 6 Az2 2Bz C (2.2.67)


whereuNJ

z ∑N

n) 0unJz n is analyticon & z&# 0. The zerosz1 2 of the squareroot above are

givenby z1 2 λ 1 µ 6 µ2 1 with λ, µ definedin (2.2.26). As remarkedon page48 in thetypicalcasewehaveλ 1, µ 1 andit canbeshown that0 z2 z1 1 holds.Theexpressionabove in thecurly bracketsis analyticfor & z&A z1 andcontinuousfor & z&A z1 andthereforef

z

is analyticon & z& z1. Notethatwehaveto choosethesignin (2.2.67)suchthatit matcheswithν1

z for & z& sufficiently large.For thesecondsequencegn weobtain

- gn. g

z z 1

2uN

Jz(2.2.68)

i.e. gz is analyticon & z&T 0. Now the basicideais to usePlancherel’s TheoremZ.4 which

gives:

N

∑n) 0

fngn 12π

2π

0fz g

z

z) eiϕdϕ

12π

π

0fz g

z f

z g

z

z) eiϕdϕ

1π

π

0Re f

z g

z

z) eiϕdϕ

(2.2.69)

wherewe have usedthe fact that fz f

z , g

z g

z , since fn gn IR. Using (2.2.67),

(2.2.68)weobtainN

∑n) 0

fngn 14π

π

0& z 1 & 2 uN

Jz 2

1 Pe Re- ν1

z . 1

z) eiϕdϕ

(2.2.70)

Weremarkthatthepoleof ν1

z atz 1 is “cancelled”by & z 1 & 2. From(2.2.70)weconclude

thatthediscreteL2–norm(2.2.65)is non–increasingin time if1 Pe Re- ν1

eiϕ . 1 0 D ϕ G 0 2π F(2.2.71)

holds. This propertyof ν1 can be shown in the following way. On the unit circle z eiϕ,0 ϕ 2π, wehave

z 1%

z 1 i tanϕ % 2 andthereby

yz : 1

rz 1z 1 κ 1

rκ i tan

ϕ2

0 ϕ 2π

(2.2.72)

Now ν1

z fulfils simply

1 Pe ν1

z 1 y

zU6 y

z 2 y

z Pe2(2.2.73)

andthereforeweobtaintherequestedproperty1 Pe Re- ν1

z . 1 κ

r Re yz 2 y

z Pe2 0(2.2.74)

for z eiϕ, 0 ϕ 2π.We thenhave thefollowing mainresultof thisSection:

THEOREM 2.5. Thenumericalschemewith theDTBCsis stablewith theproperty:

" un 1 " 2h : h

J 1

∑j ) 1

un 1j

2 *" u0 " 2h n 0

(2.2.75)

3. NUMERICAL RESULTS 57

Simplified DTBC. Therapiddecayof thes n On 3

2 motivatesto restrict(2.2.47)to aconvolutionover the“recentpast”(lastM time levels)asanoption:

θ1 τ un

1 θs

0 un0 n 1

∑k) n M

s n k uk

0

1 θ 1 τ un 11 (2.2.76a)

θ1 τ @B un

J 1 θs

0 unJ n 1

∑k) n M

s n k uk

J

1 θ 1 τ @J un 1J 1 n 1

(2.2.76b)

Wenotethatthestabilityof theresultingschemeis still notprovenyet.

3. Numerical Results

In theexamplesof thisSectionwewantto comparethenumericalresultsfromusingournewdiscreteTBC (2.2.47)to thesolutionusingeitherthediscretizedTBCof Mayfield[T26] or theapproximativeabsorbingBCof Halpern[P3]. Dueto its construction,ourDTBC yieldsexactly(up to round–off errors)the numericalwhole–spacesolution restrictedto the computationalinterval 0 L .

To measuretheinducederror(especiallyat theboundary)wecalculatea referencesolutionon a muchbiggerdomain(with DTBCs). The differencebetweenthe referencesolutionandthe computedsolution is called the reflectedpart. In eachexamplewe choose 0 1 as thecomputationaldomainandusethe initial data

uIx cos2 π

2x 0V 50V 45 for & x 0

5 &3 0

45

0 else(2.3.1)

Example 1. We usedthe parametersa 9, b 100, c 0, the time step∆t 10 4

andtherathercoarsespacegrid ∆x 1% 20 (which makesthedifferencein theaccuracy moreclear).Figure2.1shows thetime evolution of thesolutioncomputedwith theCrank–Nicolsonschemewith DTBCs at x 0, x 1 and σ 1% 2. Figure 2.2 shows the discreteL2–norm" un " 2 :

∆x∑J 1j ) 1 & un

j & 2 1$ 2 of the reflectedpart computedwith the Crank–Nicolsonschemewith the new DTBCs. and in Figure 2.3 one can seethe resultswhen using the boundaryconditionsof Mayfield and Halpernat x 0, x 1. This illustratesthe superiorityof ourdiscreteTBCs(2.2.47)(observe thedifferentscales!).

Thesimulationwith thediscretizedTBCs(2.2.9)from Mayfieldrequiresthesamenumericaleffort but thecomputedsolutionstronglydeviates(especiallyoncoarsegrids)from thewhole–spacesolution.Thediscretizationof Mayfield is convergent(if astabilityconditionis fulfilled).Thenumericalresultswith thefirst order boundarycondition(2.2.14)from Halpernarebetterthan the onewith (2.2.9) but worsethan the resultswith the simplifiedDTBC (2.2.76)withM 10,whichhasacomparablecomputationaleffort.

Example 2. In this examplewe usedagaintheCrank–Nicolsonschemeandcentraldiffer-encingof theconvectionterm: σ 1% 2. Now we useda finer grid thanbefore: the time step∆t 2 9 10 5 andthespacegrid ∆x 1% 160. We reducedthediffusivity: a 0

9, b 100,

c 0 andcomputedthesolutionuntil T 0014,i.e. 700time steps.Figure2.5demonstrates

oncemorethat usingour DTBC we obtain(up to round–off errors)the discretewhole–spacesolutionrestrictedto thecomputationalinterval.

Now we observe that the numericalresultswith the first orderBC (2.2.14)from Halpernarebetterthanbefore:weobtainapproximatelythesameerrorasthesimplifiedDTBC (2.2.76)


t = 0

t = 0.002

t = 0.004

t = 0.006

t = 0.008

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Evolution of Solution u

FIGURE 2.1. Example1: Temporalevolutionof thesolutionto (2.1.1)with theparametersa W 9, b WYX 100,c W 0.

discrete TBC

0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

0.5

1

1.5

2

2.5x 10

−16

time t

L^2−Norm of reflected Part

FIGURE 2.2. Example1: new DTBC: L2–norm of the reflectedpart (onlyroundoff–errorsoccur).


discretized TBC

L. Halpern BC

discrete TBC: M=10

discrete TBC: M=20

0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

0.02

0.04

0.06

0.08

0.1

0.12

time t

L^2−Norm of reflected Part

FIGURE 2.3. Example1: OtherApproaches:L2–normof thereflectedpart.

t = 0

t = 0.002

t = 0.004

t = 0.006

t = 0.008

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Evolution of Solution u

FIGURE 2.4. Example2: Temporalevolution of the solution to (2.1.1) witha W 0Z 9, b WYX 100,c W 0.


0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

1

2x 10

−16

time t

DTBC: L^2−Norm of reflected Part


with M W 50 (seeFigure2.6). This is dueto thefiner discretizationandthesmallervalueof a.Notethat theBC of HalpernneedslesscomputationalcoststhanthesimplifiedDTBC (2.2.76)with M W 50. Nevertheless,to obtainasmallererroronehasto usea (simplified)DTBC.

Example 3. In the third examplewe usedthe parametersfrom Example2, but now wecomputethesolutionwith thefully implicit Eulerscheme.Figure2.8provesthatwe obtainedthe whole–spacesolutionup to round-off errors. In Figure2.9 onecanseethat usingthe BCof Mayfield inducesbig numericalreflections. The error lies betweenthe error of using thesimplifiedDTBC with M W 7 andM W 10. NotethatFigure2.10looksthesameasFigure2.6,i.e. theinnerschemeseemsto havenearlyno influenceon thereflectedpart.

Example 4. Herewe againusedthe parametersfrom Example2, but now we setthe re-actionratec W 300. Figure2.11shows the time evolution of the solutioncomputedwith theθ–schemeθ W 0Z 8, σ W 0Z 9 with DTBCsat x W 0, x W 1.

Again Figure2.12shows that the computedsolutionequalsthe whole–spacesolution(upto round-off errors).In Figure2.13oneobservesthatthenumericalresultswith thediscretizedTBCs(2.2.9)of Mayfield arecomparableto theresultswith thesimplifiedDTBC (2.2.76)withM between7 and10. Theresultsusingthefirstorderboundarycondition(2.2.14)from Halpernarebetter:they arecomparableto theresultswith thesimplifiedDTBC with M between30and40.


0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

0.5

1

1.5

2

2.5x 10

−4 Halpern BC vs. simplified DTBC: L2−Norm of reflected Part

time t

Halpern BCcut = 40cut = 50cut = 60

FIGURE 2.6. Example2: HalpernBC andsimplified DTBC: L2–normof thereflectedpart.

cut = 30

cut = 40

cut = 50

cut = 60

cut = 70

0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

1

2

3

4

5

6x 10

−4

time t

TBC with cut : L^2−Norm of reflected Part

FIGURE 2.7. Example2: simplifiedDTBC: L2–normof thereflectedpart.


0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−16 DTBC: L2−Norm of reflected Part

time t


0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

1

2

3

4

5

6

7

8x 10

−3 Mayfield BC vs. simplified DTBC: L2−Norm of reflected Part

time t

Mayfield BCcut = 7cut = 10

FIGURE 2.9. Example3: Mayfield BC andsimplifiedDTBC: L2–normof thereflectedpart.


0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

0.5

1

1.5

2

2.5x 10


time t

Halpern BCcut = 40cut = 50cut = 60

FIGURE 2.10. Example3: HalpernBC andsimplifiedDTBC: L2–normof thereflectedpart.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Evolution uf Solution u

x

t = 0t = 0.001t = 0.002t = 0.003t = 0.004

FIGURE 2.11. Example4: Temporalevolution of the solutionto (2.1.1)witha W 0Z 9, b WYX 100,c W 300.


0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−16 DTBC: L2−Norm of reflected Part

time t


0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

0.5

1

1.5

2

2.5

3x 10

−3

time t

Mayfield BC vs. simplified DTBC: L2−Norm of reflected Part

Mayfield BCcut = 7cut = 10

FIGURE 2.13. Example4: Mayfield BC andsimplifiedDTBC: L2–normof thereflectedpart.


0 0.002 0.004 0.006 0.008 0.01 0.012 0.0140

0.5

1

1.5

2

2.5x 10


time t

Halpern BCcut = 30cut = 40cut = 50cut = 60

FIGURE 2.14. Example4: HalpernBC andsimplifiedDTBC: L2–normof thereflectedpart.

CHAPTER 3

The Wide–AngleEquation of Underwater Acoustics

This Chapteris concernedwith a finite differencediscretizationof so–calledwide–angle“par abolic” equations(WAPEs).Thesemodelsappearasone–way wave propagationapprox-imationsto the Helmholtz equationin cylindrical coordinateswith azimuthalsymmetryandincludeasa specialcasetheSchrodingerequation.In particularwe will discussthediscretiza-tion of transparentboundaryconditionsusingtheideasof Chapter1.

In the past two decades“par abolic” equation(PE) modelshave beenwidely usedforwavepropagationproblemsin variousapplicationareas,e.g.seismology[U4], [U5], opticsandplasmaphysics(cf. thereferencesin [U3]). Furtherapplicationsto wavepropagationproblemscanbefound in radio frequency technology[U21]. Herewe will bemainly interestedin theirapplicationto underwateracoustics,wherePEshave beenintroducedby Tappert[U22]. Anaccounton thevastrecentliteratureis givenin thesurvey article[U18] andin thebook[U16].

1. Intr oduction to Underwater Acoustics

Onestandardtaskin oceanographyis to calculatetheunderwateracousticpressure p [ z\ r ]emergingfrom atime–harmonicpointsourcelocatedin thewaterat [ zs\ 0] . Here,r ^ 0 denotestheradial rangevariableand0 _ z _ zb thedepthvariable.Thewatersurfaceis at z W 0, andthe seabottomat z W zb. In our numericaltestsof discretetransparentboundaryconditions(in Section6) we will only dealwith horizontalbottoms(for thetreatmentof a slopingbottomsee[T21]). However, irregular bottom surfacesand sub–bottomlayerscan be includedbysimply extendingthe rangeof z. We denotethe local soundspeedby c [ z\ r ] , the densitybyρ [ z\ r ] , andthe attenuationby α [ z\ r ]a` 0. n [ z\ r ]bW c0 c c [ z\ r ] is the refractive index, with areferencesoundspeedc0 (usuallythesmallestsoundspeedin themodel). Thenthe referencewavenumberis k0 W 2π f c c0, where f denotesthe(usuallylow) frequencyof theemittedsound.Thesituationis shown in Figure3.1.

ThepressuresatisfiestheHelmholtzequation

1r

∂∂r

r∂p∂r

d ρ∂∂z

ρ e 1 ∂p∂z

dk2

0 N2p W 0\ r ^ 0\(3.1.1)

with thecomplex refractiveindex

N [ z\ r ]W n [ z\ r ] d iα [ z\ r ] c k0 Z(3.1.2)

In thefar field approximation(k0r f 1) the(complex valued)outgoingacousticfield

ψ [ z\ r ]W k0r p [ z\ r ] ee ik0r(3.1.3)

satisfiestheone–waywaveequation:

ψr W ik0 g 1 X L X 1 ψ \ r ^ 0Z(3.1.4)

67

68 3. THE WIDE–ANGLE EQUATION OF UNDERWATER ACOUSTICS

= 0 : reflection0

ρ (z,r) : density

c(z,r) : sound speed

r

z

TBC

z

future : elastic model

s

zb

z r

ocean

sea bottom : homogenous

ψ

FIGURE 3.1. OceanModel

Here, g 1 X L is apseudo–differentialoperator, andL theSchrodinger operator

L WYX k e 20 ρ∂z [ ρ e 1∂z] d V [ z\ r ](3.1.5)

with thecomplex valued“potential” V [ z\ r ]W 1 X N2 [ z\ r ] .Theevolutionequation(3.1.4)is mucheasierto solvenumericallythantheelliptic Helmholtz

equation(3.1.1). Hence,(3.1.4) forms the basisfor all standardlinear modelsin underwateracoustics(normalmode,ray representation,“parabolic” equation)[U1], [U22]. Strictly speak-ing, (3.1.4)is only valid for horizontallystratifiedoceans,i.e.for range–independentparametersc, ρ, andα. In practice,however, it is still usedin situationswith weakrangedependence,andbackscatteris neglected.

“Parabolic” approximationsof (3.1.4)consistof formally approximatingthepseudo–diffe-rential operatorg 1 X L by rational functionsof L, which yields a PDE that is easierto dis-cretize than the pseudo–differential equation(3.1.4). The name“parabolic” approximationarisesfrom the circumstancethat planewave solutionsof (3.1.1) in a homogeneousmediumsatisfy a dispersionrelation, which is quadraticfor the linear approximationof the squareroot operator(cf. [U11]). For a detaileddescriptionandmotivation of this procedurewe re-fer to [U6], [U10], [U11], [U18], [U22], [U23]. Thelinearapproximationof g 1 X λ by 1 X λ c 2givesthenarrow angleor standard “par abolic” equation(SPE)of Tappert[U22]

ψr WhX ik0

2Lψ \ r ^ 0Z(3.1.6)

This Schrodinger equationis a reasonabledescriptionof waveswith a propagationdirectionwithin about15i of thehorizontal.Rationalapproximationsof theform

1 X λ j f [ λ ]W p0 X p1λ1 X q1λ

(3.1.7)

1. INTRODUCTION TO UNDERWATER ACOUSTICS 69

with real p0, p1, q1 yield thewide–angle“par abolic” equations(WAPE)

ψr W ik0

p0 X p1L1 X q1L

X 1 ψ \ r ^ 0Z(3.1.8)

In thesequelwe will repeatedlyrequirethat theslopeof theapproximatingfunction f shouldbenegative:

f k [ 0]W p0q1 X p1 _ 0Z(3.1.9)

Figure3.2illustratesthatthecondition(3.1.9)is quitenatural.With thechoicep0 W 1, p1 W 3c 4,

−0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2Approximation of Square Root Operator

λ

(1−λ)1/2

SPE: 1 − λ/2WAPE: (1 − 3λ/4)/(1 − λ/4)

FIGURE 3.2. Approximation f [ λ ] of thesquareroot g 1 X λ

q1 W 1c 4 ((1,1)–Pade approximantof g 1 X λ) oneobtainsthe WAPE of Claerbout [U4]. In[U11] Greenedeterminesthesecoefficientsby minimizing theapproximationerrorof g 1 X λover suitableλ–intervals. TheseWAPE modelsfurnisha muchbetterdescriptionof thewavepropagationup to anglesof about40i . To theauthor’s knowledge,no estimateof theapproxi-mationerrorfor suchparabolic/ wide–angle“parabolic” approximationexistsin theliterature.Also,higherorderanaloguesof (3.1.7),(3.1.8)[U8], [U13] andsplit–stepPadealgorithms[U9]havebeensuccessfullyusedfor acousticproblems.While wewill restrictourselveshereto theWAPE(3.1.8),weremarkthattheconstructionof discretetransparentboundaryconditions(seeSection5) couldbegeneralizedto higherorderPEsandeven3D–problems.

In this Chapterwe shall focuson boundaryconditions(BC) for theWAPE (3.1.8). At thewatersurfaceoneusuallyemploys a Dirichlet (“pressurerelease”)BC: ψ [ z W 0\ r ]lW 0. At theseabottomthe wave propagationin waterhasto be coupledto the wave propagationin thesedimentsof thebottom. Thebottomwill bemodeledasthehomogeneoushalf–spaceregionz ^ zb with constantparameterscb, ρb, andαb. Throughoutmostof this Chapterwe will usea fluid model for the bottomby assumingthat (3.1.8)alsoholds for z ^ zb, possiblywith adifferentrationalapproximation(3.1.7)(subjectto thecouplingcondition(3.3.9)).


REMARK 1.1. Now wewantto explainshortlywhy therationalapproximationsof theform(3.1.7)areespeciallyappropriatefor far–field simulations.

If weassumeV, ρ, ρ e 1 m L∞ [ IRn] thentheSchrodingeroperator

L WYX k e 20 ρ∂z [ ρ e 1∂z] d V [ z](3.1.10)

with a homogeneousDirichlet BC at z W 0 is self–adjointin L2 [ IRn ;ρ e 1dz] with the densedomainD [ L ]bW H1

0 [ IRn ]'o ϕ p ρ e 1ϕzm H1 [ IRn ] . Thereforewe have the following spectral

representationfor all ψI m L2 [ IRn ;ρ e 1dz] :ψI [ z]lW M

∑j q 1

a jϕ j [ z] d ∞

Vb

a [ λ ] ϕλ [ z] dλ \ z ^ 0\(3.1.11)

wheretheϕ j aretheeigenfunctionsof L correspondingto theeigenvaluesλ j andtheϕλ denotethegeneralizedeigenfunctionsof L to λ.

In applicationsof underwateracousticsthe soundspeedc [ z] is typically larger in the seabottom than in the water. ThereforeV [ z] forms a “potential well” in the water region 0 _z _ zb, which typically givesrise to boundstatesof L that representthepropagatingmodesof(3.1.4)and(3.1.8). All of thecorrespondingeigenvaluessatisfy0 _ λ j _ Vb W 1 X c2

0 c c2b _ 1,

if c0 W minzr 0c [ z] . This is illustratedin Figure3.3whereσc WtsVb \ ∞ ] denotesthecontinuousundσp WYu λ j p j W 1\ZZZv\ M wyxz[ 0\ Vb ] thediscretespectrum.

z

|V [ z]\ σ [ L ]

Vb

1

σp

σc

0Dirichlet

BC

zb

FIGURE 3.3. PotentialV [ z] andspectrumσ [ L ] of theSchrodingeroperator

Thenthesolutionat ranger ^ 0 for theone–waywaveequation(3.1.4)is simply

ψ [ z\ r ]W M

∑j q 1

a j eik0 g 1e λ j e 1~ rϕ j [ z] d ∞

Vb

a [ λ ] eik0 1e λ e 1~ rϕλ [ z] dλ Z(3.1.12)

For λ ^ 1 we have only “evanescentmodes”, i.e. exponentialdampingfor the generalizedmodesin σc [ L ]o[ 1\ ∞ ] . If weassumefor a functiona thatsuppa xhs a1 \ a2 xsVb \ 1 holdsthen

we obtainwith ϕλ [ z]W ei λz for theintegral in (3.1.12)

limr ∞

a2

a1

a [ λ ] eik0 1e λ e 1~ r ei λzdλ W 0\ [ pointwisein z]\(3.1.13)

which follows from theLemmaof Riemann–Lebesgue.Fromtheabovewe concludethatonlypropagatingmodescontribute in the far field both for g 1 X λ and rational approximations.

1. INTRODUCTION TO UNDERWATER ACOUSTICS 71

Thereforea goodapproximationfor λ j is importantfor the far field. Figure3.4 illustratesthesituationshowing theSchrodingermodes(λ 1) andtheparabolicmodes(λ ^ 1).

0 1 2 30

1

Vb

Approximation of Square Root Operator

λ

Schroed. modes

real

parabol. modes

imag.

| (1 − λ)1/2 |

σp

σc

traveling. modes↑

1 − λ / 2

(4 − 3 λ) / (4 − λ)

FIGURE 3.4. Approximation f [ λ ] of thesquareroot g 1 X λ

In practicalsimulationsoneis only interestedin theacousticfield ψ [ z\ r ] in thewater, i.e.for0 _ z _ zb. While thephysicalproblemis posedontheunboundedz–interval [ 0\ ∞ ] , onewishesto restrict the computationaldomainin the z–directionby introducingan artificial boundaryat or below the seabottom. This artificial BC shouldof coursechangethe modelas little aspossible.Until recently, the standardstrategy wasto introduceratherthick absorbinglayersbelow the seabottomand then to limit the z–rangeby againimposinga Dirichlet BC [U6],[U8], [U19], [U20], [U23]. With acarefullydesignedabsorptionprofileandlayerthicknessthisstrategy hasbeenvery successful.But without a comparisonto theexacthalf–spacesolutionitis hardto estimatehow muchanabsorbinglayermodifiestheoriginalproblem.Also,absorbinglayersincreasethe computationalcosts,for SPE–or WAPE–simulationstypically by a factoraround2 [U18], [U26]. However, in simulationswithout attenuation(“f alseabsorbinglayermethod”[U18], [U26]) or overanelasticseabottom[U8], muchthickerabsorbinglayershavebeenusedto ensureaccuracy and,respectively, numericalstability.

In [T18] and[T20] Papadakisderived impedanceBCsor transparentboundaryconditions(TBC) for the SPEandthe WAPE, which completelysolvesthe problemof restrictingthe z–domainwithout changingthe physicalmodel: complementingthe WAPE (3.1.8)with a TBCat zb allows to recover — on the finite computationaldomain [ 0\ zb] — the exact half–spacesolution on 0 _ z _ ∞. As the SPEis a Schrodingerequation,similar strategies have beendevelopedindependentlyfor quantummechanicalapplications[T1], [T6], [T12] (cf. Chapter1).

Towardsthe end of this introductionwe shall now turn to the main motivation. WhileTBCs fully solve the problemof cutting off the z–domainfor the analyticalequation,their


numericaldiscretizationis far from trivial. Indeed,all availablediscretizationsarelessaccuratethan the discretizedhalf–spaceproblemand they renderthe overall numericalschemeonlyconditionallystable[T6], [T16], [T19], [T26]. Theobjectof this work is to constructdiscretetransparentboundaryconditions(DTBC) for a Crank–Nicolsonfinite differencediscretizationof the WAPE suchthat the overall schemeis unconditionallystableand as accurateas thediscretizedhalf–spaceproblem.

In the literaturedifferentWAPEs(or WAPE andthe standard“parabolic” equation)havebeencoupledin the water and the seabottom. We also analyzein Section3 underwhichconditionsthe resultinghybrid model is conservative for the acousticenergy (weightedL2–norm).

While weshallassumeauniformdiscretizationin ranger, theinteriordepthzdiscretization(i.e. in thewatercolumn)maybenonuniform,andweshalldiscussin Subsection5.5strategiesfor the‘bestexteriordiscretization’(i.e. in theseabottom).

This Chapteris organizedasfollows: In Section2 we review theTBCs for theWAPE. InSection3 wediscussthecouplingof theWAPEto theSPEandin Section4 wederive theTBCfor theelasticPE.In Section5 discreteTBCsarederivedandanalyzed.Their superiorityoverexistingdiscretizationsis illustratedonseveralbenchmarkproblemsin Section6.


In this Sectionwe first discussthewell–posednessof theevolution problemfor theWAPE(in the non–dissipative caseα W 0). Thenwe presentthe matching conditionsat the water–bottominterfacewhich hasto be taken into accountwhenwe derive afterwardstheconserva-tion/decayof theL2–norm. Finally we shortly review the transparentboundaryconditionfortheSPEandsketchthederivationtheTBC for WAPE.

2.1. The Well–Posednessof the Evolution Problem for the WAPE. Herewe shall dis-cussthewell–posednessof theevolution problemfor theWAPE in thecritical non–dissipativecase, i.e. for α W 0:

ψr W ik0 f [ L ]UX 1 ψ \ z ^ 0\ r ^ 0\(3.2.1)

subjectto the homogeneousDirichlet BC ψ [ 0\ r ]bW 0, andwith the rationalfunction f givenin (3.1.7).For simplicity of theanalysiswe only considertherange–independentsituation,i.e.α W α [ z] , ρ W ρ [ z] , V W V [ z] .

THEOREM 3.1(well–posednessof theWAPE). We assumethat the refractive index n [ z] ,thedensityρ [ z]^ 0, andρ e 1 [ z] areboundedfor z ^ 0. Then,theWAPEhasa uniquesolutionfor all initial data in the weightedL2–spaceL2 [ IRn ;ρ e 1dz] if and only if the pole of f [ λ ] atλ W qe 1

1 is not aneigenvalueof theoperator L with DirichletBCsat z W 0.

PROOF. In Theorem3.1 we assumedthat V, ρ, ρ e 1 m L∞ [ IRn ] . Then, the Schrodingeroperator

L WYX k e 20 ρ∂z [ ρ e 1∂z] d V [ z](3.2.2)

with a homogeneousDirichlet BC at z W 0 is self–adjointin L2 [ IRn ;ρ e 1dz] with the densedomain

D [ L ]lW H10 [ IRn ]o ϕ p ρ e 1ϕz

m H1 [ IRn ] Z(3.2.3)


Wenow considertheoperatorf [ L ]lW p0 e p1L1e q1L definedas

f [ L ]lW ∞

e ∞f [ λ ] dPλ \(3.2.4)

with dPλ denotingtheprojectionvaluedspectralmeasureof theoperatorL (cf. [M1], [M7]).Accordingto [M1, TheoremXII.2.6] thedomainof f [ L ] is densein L2 [ IRn ;ρ e 1dz] if andonlyif λ W qe 1

1 , thepoleof f [ λ ] , is not aneigenvalueof L. In this casef [ L ] is self–adjointand,byStone’s Theorem[M7], ik0 f [ L ] generatesa unitaryC0–groupon L2 [ IRn ;ρ e 1dz] , which yieldstheuniquesolutionto (3.2.1).

REMARK 2.1. If λ coincideswith aneigenvalueλ j of L, then(3.2.1)still admitsa uniquemild solutionfor all initial datain theorthogonalcomplementof ϕ j , theuniqueeigenfunctioncorrespondingto λ j . Theorem3.1 generalizesthe well–posednessanalysisfor theWAPE onfinite intervalsgivenin [U2]. There,however, λ caneasilylie in the(pureeigenvalue)spectrumof L, whatthenrestrictstheclassof admissibleinitial conditions.

As q1 is muchsmallerthan1 in all practicalsimulations(14 in theWAPEof Claerbout;also

cf. [U11]), λ usuallylies in sVb \ ∞ ] , thecontinuousspectrumof L. Theorem3.1thenguaranteestheuniquesolvability of theevolution equation(3.2.1)for any initial data.Let uscomparethesituationathand(i.e. theWAPEontheoriginalunboundedinterval — andlateralsotheWAPEwith a TBC) to theWAPE restrictedto thez–interval s 0\ zmax with a homogeneousRobinBCatzmaxasasimplemodelfor anabsorbinglayer: there,L hasapureeigenvaluespectrumwhichinhibits thesolvability of (3.2.1)in severalcasesof practicalrelevance[U2].

2.2. The Matching Conditions and the Conservation of the Acoustic Power. Now weturn to thematchingconditionsandlater theTBCsat thewater–bottominterface(z W zb). Asthe densityis typically discontinuousthere,onerequirescontinuity of the (acoustic)pressureandthenormalparticlevelocity:

ψ [ zb e \ r ]lW ψ [ zb n \ r ]\(3.2.5a)

ψz [ zb e \ r ]ρw

W ψz [ zb n \ r ]ρb

\(3.2.5b)

whereρw W ρ [ zb e \ r ] is the waterdensityjust above the bottomandρb denotesthe constantdensityof thebottom.

With thesematchingconditions(3.2.5)weshallnow deriveanestimatefor theL2–decayofsolutionsto the WAPE (3.1.8),z ^ 0. We assumeρ W ρ [ z] andapply the operator1 X q1L to(3.1.8):

(3.2.6) 1 X q1Vd

q1k e 20 ρ∂z [ ρ e 1∂z] ψr

W ik0 p0 X 1 X[ p1 X q1 ] V d [ p1 X q1 ] k e 20 ρ∂z [ ρ e 1∂z] ψ Z


Multiplying (3.2.6)by ψρ e 1, integratingby partson 0 _ z _ zb, andtakingtherealpartgives

∂r

zb

0p ψ p 2ρ e 1dz

W 2 [ p1 X q1 ] k0

zb

0Im sV p ψ p 2 ρ e 1dz X k e 2

0 ρ e 1w Im s ψzψ p zq zb e

dq1

zb

0Re sV ∂r p ψ p 2 X 2Im sV Im s ψrψ ρ e 1dz

dk e 2

0 ∂r

zb

0p ψz p 2 ρ e 1dz X 2k e 2

0 ρ e 1w Re s ψzrψ p zq zb e Z

(3.2.7)

Analogously, multiplying (3.2.6)by ψrρ e 1, andtakingtheimaginarypartwe get

[ p0 X 1] ∂r

zb

0p ψ p 2 ρ e 1dz

WX 2q1k e 10

zb

0Im sV p ψr p 2ρ e 1dz

d2q1k e 3

0 ρ e 1w Im s ψzrψr p zq zb e

d [ p1 X q1 ] zb

0Re sV ∂r p ψ p 2 d

2Im sV Im s ψrψ ρ e 1dz

dk e 2

0 ∂r

zb

0p ψz p 2 ρ e 1dz X 2k e 2

0 ρ e 1w Im s ψzψr p zq zb e Z

(3.2.8)

After aneasyalgebraicmanipulationweobtainfrom (3.2.7), (3.2.8)

(3.2.9) ∂r

zb

0p ψ p 2 ρ e 1dz

WYX 2C1

zb

0α

c0

c∂rψ

2ρ e 1dz X C1k e 1

0 ρ e 1w Im ∂rψz∂rψ

zq zb e \with

C1 W 2 [ p1 X q1 ] 2

p1 X p0q1

\ ∂r W Id iq1

p1 X q1

k e 10 ∂r Z

In thesamewayasimilarequationcanbederivedfor thebottomregionz ^ zb:

(3.2.10) ∂r

∞

zb

p ψ p 2 ρ e 1dz

WX 2C1αbc0

cb

∞

zb

∂rψ2

ρ e 1dzd

C1k e 10 ρ e 1

b Im ∂rψz∂rψzq zb n Z

Adding thetwo aboveequationswith (3.2.5)gives

∂r ψ [ Z\ r ] 2 WhX 2C1

∞

0α

c0

c∂rψ

2ρ e 1dz(3.2.11)

for theweightedL2–norm

ψ [ Z\ r ] 2 W ∞

0p ψ [ z\ r ](p 2 ρ e 1 [ z] dzZ(3.2.12)

In thedissipation–freecase(α 0) ψ [SZ\ r ] is conservedandfor α ^ 0 and p0q1 X p1 _ 0 itdecays.Thediscreteanalogueof this “energy”–conservation(or –decayfor α ^ 0) will bethemainingredientfor showing unconditionalstabilityof thefinite differenceschemein Section5.


REMARK 2.2(L2–estimatefor SPE). For p1 W 1c 2, q1 W 0, (3.2.9)simplifiesto:

∂r

zb

0p ψ p 2 ρ e 1dz WhX 2

zb

0α

c0

cp ψ p 2 ρ e 1dz X 1

k0ρwIm s ψzψ p zq zb e Z(3.2.13)

2.3. The Transparent Boundary Condition for the SPEand WAPE. Now we shall re-view the transparentbottomboundaryconditionfor the SPEandsketchthe derivation of theTBC for theWAPE.We assumethattheinitial dataψI W ψ [ z\ 0] , which modelsa point sourcelocatedat [ zs\ 0] , is supportedin the interior domain0 _ z _ zb (otherwise,if the sourceiscloseto thebottomtheapproachof Section3 in Chapter1 canbeapplied).Also, let thebottomregion behomogeneous, i.e. let all physicalparametersbeconstantfor z ^ zb. Thebasicideaof thederivation is to explicitly solve theequationin thebottomregion, which is theexteriorof thecomputationaldomain [ 0\ zb ] . TheTBCfor theSPE(or Schrodingerequation)(1.1.10)readsnow:

ψ [ zb \ r ]WYX 1

g 2πk0

ei π4

ρb

ρw

r

0

ψz [ zb \ r X τ ] eibτ

g τdτ \(3.2.14)

with b W k0 [ N2b X 1] c 2. ThisBC is nonlocalin therangevariabler andinvolvesamildly singular

convolutionkernel.Equivalently, it canbewritten as

ψz [ zb \ r ]WYX 2k0

πee iπ

4 eibr ρw

ρb

ddr

r

0

ψ [ zb \ τ ] ee ibτ

g r X τdτ \(3.2.15)

andther.h.s.canbeexpressedformally asa fractional(12) derivative[T1], [T6], [T7]:

ψz [ zb \ r ]lWhX 2k0ee π4 i eibr ρw

ρb∂r ψ [ zb \ r ] ee ibr Z(3.2.16)

In [T7] thissquarerootoperatoris approximatedby rationalfunctionswhichleadsto ahierarchyof highly absorbing(but not any moreperfectlytransparent)BCsfor theSPE.By introducingauxiliary boundaryvariablestheseBCscanbeexpressedthroughlocal–in–r operators.Hence,thisallowsfor a “local” (2–level in r) discretizationscheme[T9]. Thisscheme,however, intro-ducesnumericalreflectionsat theartificial boundary, whoseamplitudedependson thechosenapproximationorderof theabovesquareroot operator.

In orderto derive theTBCfor theWAPEweconsider(3.2.6)in thebottomregion:

[ δbd

q1k e 20 ∂2

z ] ψr W i υbd [ p1 X q1 ] k e 1

0 ∂2z ψ \ z ^ zb \(3.2.17)

with

δb W 1 X q1 [ 1 X N2b ]\ υb W k0 p0 X 1 X[ p1 X q1 ]B[ 1 X N2

b ] ZAfter aLaplacetransformationof (3.2.17)in r weget

q1s X i [ p1 X q1 ] k0 ψzz[ z\ s]W k20 [ iυb X δbs] ψ [ z\ s]Z(3.2.18)

Sinceits solutionhasto decayasz ∞ weobtain

ψ [ z\ s]'W ψ [ zb n \ s] exp X k0 iυb X δbsq1s X i [ p1 X q1 ] k0

[ z X zb] \ z ^ zb \(3.2.19)

andwith thematchingconditions(3.2.5)this gives

ψz [ zb e \ s]WYX k0

ρw

ρb iυb X δbs

q1s X i [ p1 X q1 ] k0

ψ [ zb e \ s]Z(3.2.20)


Here,again g denotesthebranchof thesquareroot with a nonnegative realpart. An inverseLaplacetransformation[S2] yieldstheTBCat thebottomfor theWAPE:

(3.2.21) ψ [ zb \ r ]WX iηρb

ρwψz [ zb \ r ]

d βηρb

ρw

r

0ψz [ zb \ r X τ ] eiθτeiβτ J0 [ βτ ] d iJ1 [ βτ ] dτ \

η W 1k0

q1

δb\ β WhX p1 X p0q1

2q1

k0

δb\ θ W p1 X q1

q1

k0 \whereJ0, J1 denotetheBesselfunctionsof order0 and1, respectively. This is a slight gener-alizationof the TBC derived in [T20] wherep0 wasequalto 1. Equivalently, (3.2.21)canbewritten as

(3.2.22) ψz [ zb \ r ]lW iη e 1ρw

ρbψ [ zb \ r ]

d βη e 1ρw

ρb

r

0ψ [ zb \ r X τ ] eiθτeiβτ J0 [ βτ ]UX iJ1 [ βτ ] dτ Z

BothTBCsarenonlocalin r; in rangemarchingalgorithmsthey thusrequirestoringthebottomboundarydataof all previousrangelevels.

Weremarkthattheasymptoticbehaviour(for r ∞) of theconvolutionkernelin theTBC(3.2.15)is O r e 3 2 , which canbe seenafter an integrationby parts. Using the asymptoticbehaviour of theBesselfunctions(see(3.5.5)) onefindsthattheconvolution kernelof (3.2.22)alsodecayslikeO r e 3 2 .

3. Coupled Models for Underwater Acoustics

We shall now briefly commenton coupledmodelsfor underwateracoustics,asproposedin [T19], [T20]. In [T20] theWAPEfor theocean[ 0 _ z _ zb ] is coupledto theSPEfor theseabottom [ z ^ zb ] . In fact,thesemodelsarecoupledvia aTBC correspondingto theSPE,but thisis equivalentto thehalf–spaceproblem.Herewewantto pointoutamathematicalambiguityofthis couplingthatmay stronglyinfluencethenumericalstability of thediscretizationscheme.To this endwe discussthis coupledWAPE–SPEmodel in the simplecaseof constantsoundspeedc anddensityρ, which is ratherunrealistic,but it illustratesthesituation.

3.1. The coupled WAPE–SPE model. Let us first review the WAPE (3.2.1) with theSchrodingeroperatorL WX k e 2

0 ∂2z. Whendiscretizing(3.2.1)oneusuallyappliesthe opera-

tor 1 X q1L to (3.2.1)which givesthefollowing PDEof “Sobolev type” [M6]

[ 1 X q1L ψr W ik0 p0 X 1 X[ p1 X q1 ] L ψ \ z ^ 0\ r ^ 0Z(3.3.1)

Since the operatorsin the numeratorand denominatorof (3.1.8) commute(even for non–constantc and ρ) this stepis mathematicallyrigorous,and (3.3.1) is easyto discretize(seeSection5).

Disregardingfor the momentthenonlocalityof the involvedpseudo–differentialoperator,onewould formally want to write the evolution equationfor the coupledmodel (WAPE andSPE)as

ψr W ik0Aψ(3.3.2)

3. COUPLEDMODELS FOR UNDERWATER ACOUSTICS 77

with

(3.3.3a)

Aψ Wp0

dp1k e 2

0 ∂2z

1d

q1k e 20 ∂2

z

X 1 ψ \ 0 _ z _ zb \

k e 20

2∂2

zψ \ z ^ zb Z(3.3.3b)

However, theright handsideof (3.3.2)is notwell–defined,dueto thenonlocalityof thepseudo–differentialoperatorin (3.3.3a).Also, its reformulationasin (3.3.1)is not any longerjustifiedin thecoupledcase.Evenin thedissipation–freecaseit wouldresultin anon–conservativeevo-lution equationandhencein a non–conservative numericalscheme(neverthelessthis strategyis usedin [T20]). This is illustratedin Example3 of Section6.

Usingmoreinvolvedpseudo–differentialoperatorswe will find a correctandconservativeinterpretationof (3.3.2),(3.3.3). Wefirst considerthepseudo–differentialoperatorf [ L ] appear-ing in theWAPE(3.2.1)with L WYX k e 2

0 ∂2z. Dueto theBC at z W 0 it canbeexpressedin terms

of Fourier–sinetransformsas

f [ L ] ψ [ z]W 2π

∞

0

∞

0Φ [ ξ ] ψ [ y] sin[ ξy] sin[ ξz] dydξ \(3.3.4)

with thesymbol

Φ [ ξ ]W p0 X p1k e 20 ξ2

1 X q1k e 20 ξ2

Z(3.3.5)

In thecoupledWAPE–SPEmodelonewould formally wantto write theevolutionequationas(3.3.2),(3.3.3). However, asthepseudo–differentialoperatorin (3.3.3a)is nonlocal,actingon L2 [ IRn] , it cannotbesimply restrictedto theinterval 0 _ z _ zb. It is thereforeappropriateto definethe coupledevolution equationon the symbol level of the two involved operators(cf. [U10], [M4]). Without attenuationboth the SPEand the WAPE conserve the L2–normandthediscreteanalogueof this conservationis themainingredientfor showing unconditionalstability of thefinite differenceschemein Section5. Thereforewe postulatethat thecoupledmodelalsohasto conserve the L2–norm. This canbe achieved if the operatorA on the righthandsideof (3.3.2)is interpretedastheWeyl operator (see[U10])

Aψ [ z]W 2π

∞

0

∞

0a

yd

z2

\ ξ ψ [ y] sin[ ξy] sin[ ξz] dydz(3.3.6)

to thesymbol

a [ z\ ξ ]W Φ [ ξ ]UX 1\ 0 _ z _ zb \X k 2

02 ξ2 \ z ^ zb Z(3.3.7)

As a [ z\ ξ ] is real,onereadilyverifiesthat theevolution equation(3.3.2),(3.3.6)conservestheL2–norm.

Due to the pole of the symbolΦ [ ξ ] it would be quite difficult to appropriatelydiscretize(3.3.2), (3.3.6),andit is beyond our scopehere. We remarkthat finite differenceschemesofpseudo–differentialequationswith smoothsymbolshaverecentlybeenstudiedin [F4].


However, its discretizationusingthe ideasof Markowich andPoupaud[F5] would beverydifficult andwill bea topicof futureresearch.

3.2. The coupling of different WAPEs. Fromtheabove we concludethat it is not advis-able to couplethe WAPE andthe SPEnumerically. As an alternative we shall now analyzecouplingsof WAPEswith differentparametersp0, p1, q1 that canbe reformulatedasa PDE,like in (3.3.1).Thecoupledmodel

ψr W ik0

p0 [ z]UX p1 [ z] L1 X q1 [ z] L X 1 ψ(3.3.8)

is well–definedandcanbe transformedto (3.3.1)if thenumeratoranddenominatorin (3.3.8)commute.Underthecondition

p1 [ z] c q1 [ z]W : µ W const(3.3.9)

we canrewrite thepseudo–differentialoperatorin (3.3.8)as

p0 [ z]UX p1 [ z] L1 X q1 [ z] L W µ τ [ z]UX µe 2σ [ z] L X 1

τ [ z]X µe 2σ [ z] L \(3.3.10)

with

τ [ z]W µ X p0 [ z] e 1 \ σ [ z]W p21 [ z] p1 [ z]UX p0 [ z] q1 [ z] e 1 Z(3.3.11)

Herethe numeratoranddenominatorcommute,andhence(3.3.8)canbe written in the formof (3.3.1). Theresultingevolution equationis conservative in L2 [ IRn ; [ σρ ] e 1dz] andit allowsfor a conservative andunconditionallystablediscretization(seeSection5 andExample3 inSection6).

If theparametersp0, p1, q1 arefixedin onemedium,condition(3.3.9)still leavestwo freeparametersto chooseadifferentrationalapproximationmodelof g 1 X λ for thesecondmedium(cp. [U11]). Hence,onecanin factobtaina betterapproximationin thesecondmediumthanwith theoriginally intended“parabolicapproximation”.

4. The Transparent Boundary Condition for an Elastic Bottom

In [T3] wepresentedthefollowing smallremarkon thecouplingof theSPEwith anelastic“par abolic” equation(EPE)for theseabottom[U7], [U12], [U24]. In [T19], [T20] a TBC forthis couplingwasderived.It readsfor theLaplacetransformedwavefield:

(3.4.1) ψ [ zb \ s]lWhX ρb

ρw

1k0N4

s

1

Mp [ s]

2Ms [ s] d N2s

2 X 4 Mp [ s] Ms [ s] Ms [ s] d N2s ψz [ zb \ s]\

with thenotation

Mp [ s]W 1 X N2p X i

2k0

s\ Ms [ s]W 1 X N2s X i

2k0

sZ(3.4.2)

Here, Np W npd

iαp c k0 and Ns W nsd

iαsc k0 denotethe complex refractive indicesfor thecompressionalandshearwavesin thebottom. In a tediouscalculationthis BC canindeedbeinverseLaplacetransformed(using[S2]) andit reads:

ψ [ zb \ r ]lW Cr

0ψz [ zb \ r X τ ] eiωτg [ τ ] dτ X 2iϕ

r

0ψzr [ zb \ r X τ ] eiωττ e 1

2 dτ \(3.4.3)

4. THE TRANSPARENT BOUNDARY CONDITION FOR AN ELASTIC BOTTOM 79

with

C WYX ρb

ρw

2

k5 20 N4

s

2π

eπ4 i \ ω W k0

2N2

p X 1 \ ϕ WhX k0

2N2

p X N2s \

g [ τ ]WhX 3 1 X eiϕτ τ e 52d

ik0

23N2

p X N2s X 2N2

s eiϕτ τ e 32

d k20

2N4

p X N2pN2

sd 1

2N4sd

N2p X N2

s τ e 12 W O τ e 1

2 ZWhile this inversetransformationwascarriedout numericallyin [T17], [T19], [T20], our an-alytical TBC maysimplify thediscretizationof this coupledmodel. However, a discretizationof theTBC for a numericalimplementationis possiblebut againstthespirit of our approachtoderive theBCsonadiscretelevel. Also thestabilityof theresultingschemeis unclear. DTBCsfor theSPE–EPEcoupling(in thespirit of Section5) arenot derivedyet, mainly sincewe donothavea conservativeformulationof thecoupledelastic–fluidPE.

Note that the TBC (3.4.3) for the elasticPE hasthe sameform as the TBC for the SPE(3.2.14)but thesingularityin theintegral kernelis stronger(of order3c 2).

DERIVATION (of (3.4.3)). With theabbreviation ψ [ s] : W ψ [ zb \ s] andthenotation

mp [ s]W k0

2Mp [ s]WYX i s s X i

k0

2[ N2

p X 1] \(3.4.4a)

ms [ s]W k0

2Ms [ s]WYX i s s X i

k0

2[ N2

s X 1] \(3.4.4b)

(3.4.1)reads

ψ [ s]WYX ρb

ρw

1k0N4

s

k0

2

4k0

ms [ s] d N2s

2

mp [ s] X 42k0

ms [ s] 2k0

ms [ s] d N2s ψz [ s]

WYX ρb

ρw

8g 2

k5 20 N4

s

ms [ s] d k04 N2

s2

mp [ s] X 4 ms [ s] ms [ s] d k0

2N2

s ψz [ s]\(3.4.5)

wherewedenotethecontentof thesquarebracketsby f [ s X σ ] with

σ W ik0

2[ N2

p X 1]Z(3.4.6)

Weobserve thatwe canwrite

mp [ s]WYX i s s X σ \ ms [ s]WX i s s X σ dik0

2[ N2

p X N2s ] Z(3.4.7)

Thenext stepis ashift in theargumentof ψz [ s] in (3.4.5)by σ:

ψ [ zb \ s d σ ]WYX ρb

ρw

8g 2

k5 20 N4

s

f [ s] ψz [ zb \ s d σ ]\(3.4.8)


With g X i W ee π4 i weobtainfor f [ s]f [ s]lW X is

d k02 [ N2

p X N2s ] d k0

4 N2s

2

ee π4 i g s

X ee π4 i s

dik0

2[ N2

p X N2s ]B[ X i ] s

dik0

2[ N2

p X N2s ]=s s d

ik0

2N2

p \(3.4.9)

where

s? 2 WYXs s dik0

2N2

p X ik0

4N2

s 2WYXs s d

ik0

2[ N2

p X N2s ] s s d

ik0

2N2

p d k20

16N4

s \(3.4.10)

f [ s]lW eπ4 i 1

g sk2

0

16N4

s X sd

ik0

2[ N2

p X N2s ] s

dik0

2N2

p

d sd

ik0

2[ N2

p X N2s ]s s d

ik0

2N2

p W e

π4 i s

dik0

2[ N2

p X N2s ]UX g s X i

k0

2[ N2

p X N2s ] 1

g s

sd

ik0

2N2

pd k2

0

16N4

s1

g s

W eπ4 i g s X γ X g s

d γ1

g ssd

ik0

2N2

pd k2

0

16N4

s1

g s\

(3.4.11)

with

γ WX ik0

2[ N2

p X N2s ]Z(3.4.12)

Now thetransformedtransparentboundaryconditionreads

ψ [ s d σ ]W C ik0

2N2

p g s X γ X g sd γ

1

g sd k2

0

16N4

s1

g sψz [ s d σ ]

dC g s X γ X g s

d γ1

g su sψz [ s d σ ]w\

(3.4.13)

with theconstant

C WhX ρb

ρw

8g 2

k5 20 N4

s

eπ4 i Z(3.4.14)

An inverseLaplacetransformationof (3.4.13)using(IL.6), (IL.7) yields

ψ [ r ] ee σr W Cr

0ψz [ r X τ ] ee σ r e τ ~ g1 [ τ ] dτ

dC

r

0X ∂

∂τψz [ r X τ ] ee σ r e τ ~ g2 [ τ ] dτ \(3.4.15)

4. THE TRANSPARENT BOUNDARY CONDITION FOR AN ELASTIC BOTTOM 81

wherewecomputewith thehelpof (IL.4), (IL.5)

g1 [ τ ]We 1 ik0

2N2

p g s X γ X g sd γ

1

g sd k2

0

16N4

s1

g s

W ik0

2N2

p e 1 g s X γ X g sd k2

0

4[ N2

p X 12N2

s ] 2e 1 1

g s

W ik0

4g πN2

p [ 1 X eγτ ] τ e 32d k2

0

4g π[ N2

p X 12N2

s ] 2 τ e 12 \

(3.4.16a)

g2 [ τ ]W e 1 g s X γ X g sd γ

1

g sW 1

2g π[ 1 X eγτ ] τ e 3

2d γ

g πτ e 1

2

W 12g π

[ 1 X eγτ ] τ e 32d γτ e 1

2

q g3 τ ~d γ

2g πτ e 1

2

q g4 τ ~\(3.4.16b)

I3 W r

0X ∂

∂τψz [ r X τ ] ee σ r e τ ~ g3 [ τ ] dτ

W r

0ψz [ r X τ ] ee σ r e τ ~ gk3 [ τ ] dτ X ψz [ r X τ ] ee σ r e τ ~ g3 [ τ ] τ q r

τ q 0

q 0 with ψz 0~q 0~Z(3.4.16c)

Thustheboundarycondition(3.4.15)readsnow

ψ [ r ] ee σr W Cr

0ψz [ r X τ ] ee σ r e τ ~ g1 [ τ ] d gk3 [ τ ] dτ

dC

r

0X ∂

∂rψz [ r X τ ] ee σ r e τ ~ g4 [ τ ] dτ

W Cr

0ψz [ r X τ ] ee σ r e τ ~ g1 [ τ ] d gk3 [ τ ] dτ

dC

r

0ψzr [ r X τ ] ee σ r e τ ~ X σψz [ r X τ ] ee σ r e τ ~ g4 [ τ ] dτ \

(3.4.17)

i.e.

ψ [ r ]lW Cr

0ψz [ r X τ ] eστg [ τ ] dτ d r

0ψzr [ r X τ ] eστg4 [ τ ] dτ \(3.4.18)

where

g [ τ ]W g1 [ τ ] d gk3 [ τ ]X σg4 [ τ ] Z(3.4.19)

Wecalculate

gk3 [ τ ]W 12g π

X 32[ 1 X eγτ ] τ e 5

2 X γeγτ τ e 32 X γ

2τ e 3

2

W 14g π

X 3 [ 1 X eγτ ] τ e 52 X 2γ [ 1

2d

eγτ ] τ e 32 \

(3.4.20a)

σg4 [ τ ]W σγ2g π

τ e 12 W 1

4g πk2

0

2[ N2

p X 1]B[ N2p X N2

s ] τ e 12 \(3.4.20b)


andtherefore

g [ τ ]W 14g π

ik0N2p 1 X eγτ τ e 3

2d

k20 N2

p X 12N2

s2τ e 1

2 X 3 1 X eγτ τ e 52

dik0 N2

p X N2s

12d

eγτ τ e 32 X k2

0

2N2

p X 1 N2p X N2

s τ e 12

W 14g π

k20

2N4

p X N2pN2

sd 1

2N4sd

N2p X N2

s τ e 12

dik0

23N2

p X N2s X 2N2

s eγτ τ e 32 X 3 1 X eγτ τ e 5

2

W O [ τ e 12 ]\ τ 0Z

(3.4.21)

Finally, wedefineϕ, ω by γ W : iϕ, σ W : iω, andset

g [ τ ]W 4g πg [ τ ]\ C W C

4g π(3.4.22)

We remarkthat thewell–posednessof thecoupledanalyticproblemis not provenyet andwill bea topic of futureresearch.


In thisSectionweshalldiscusshow to discretizetheTBCs(3.2.14),(3.2.21)in conjunctionwith aCrank–Nicolsonfinite differenceschemefor theSPEandtheWAPE.Mostof thetimeweshallonly consideruniformgridsin z andr. While auniformrangediscretizationis crucialforourconstructionof discreteTBCs,thisconstructionis independentof the(possiblynonuniform)z–discretizationon theinteriordomain.

For simplicity we first considertheuniform grid zj W jh, rn W nk (h W ∆z, k W ∆r) andtheapproximationψn

j ψ [ zj \ rn] . ThediscretizedWAPE (3.2.6)thenreads:

(3.5.1) 1 X q1Vnn 1

2j

dq1k e 2

0 ρ jD0h2[ ρ e 1

j D0h2] Dk ψn

j

W ik0 p0 X 1 X[ p1 X q1 ] Vnn 12

jd [ p1 X q1 ] k e 2

0 ρ jD0h2[ ρ e 1

j D0h2] ψn

jd ψnn 1

j

2\

with Vnn 1

2j : W V [ zj \ rnn 1

2] andtheusualdifferenceoperators

Dk ψnj W ψnn 1

j X ψnj

k\ D0

h2ψn

j W ψnj n 1

2X ψn

j e 12

hZ

It is well known thatthisschemeis secondorderin h andk andunconditionallystable[U2].Proceedingsimilarly to thederivationof (1.2.6)onecanshow

Dk ∑j ZZ

ψnj

2

ρ jWYX C1k e 1

0 ∑j ZZ

Im Vnn 1

2j ψnn 1

2j

d iq1

p1 X q1

k e 10 Dk ψn

j

2

ρ e 1j \(3.5.2)

with C1 W 2 [ p1 X q1 ] 2 c [ p1 X p0q1 ] . Hence,thescheme(3.5.1)preservesthediscreteweightedL2–norm ψn 2

2 W h∑ j ZZ p ψnj p 2 c ρ j in thedissipation–freecase(V real). This alsoholdswhen

usingahomogeneousDirichlet BC at j W 0.


5.1. Discretization strategiesfor the TBC. In theliteraturethreedifferentstrategieshavebeenproposedto discretizeTBCs,mostly, however, just for theSchrodingerequation.Firstwereview (asin Chapter1) a known ad–hocdiscretizationof theanalyticTBC.

Discretized TBC of Thomson and Mayfield. In [T26] ThomsonandMayfield usedthefollowing discretizedTBC for theSPE:

ψnJ X ψn

J e 1 W h

2Bk12

ψnJ X Bk ne 1

∑mq 1

ψne mJ X ψne m

J e 1˜ m\(3.5.3)

with

B WhX [ 2πk0 ]Ae 12 e

π4 i ρb

ρw\ Bk¡W e

i2bk sin[ 1

2bk]12bk

\ ˜ m W eibmk

2 md 1

2

ZOn the fully discretelevel this BC is not perfectlytransparentany moreandit mayalsoyieldanunstablenumericalscheme.In analogyto theanalyticTBC (3.2.14)it requirestheboundarydatafrom thewhole“pastrange” s 0\ rne 1 .

To comparenumericallyour discreteTBC for theWAPE to thediscretizedTBC we useadiscretizationof theTBC (3.2.21)for theWAPEthatis analogousto (3.5.3):

r

0ψz [ zb \ rn X τ ] eiθτeiβτ J0 [ βτ ] d iJ1 [ βτ ] dτ

W ne 1

∑mq 0

rm 1

rm

ψz [ zb \ rn X τ ] eiθτ J0 [ βτ ] d iJ1 [ βτ ] dτ

j ne 1

∑mq 0

ψne mJ X ψne m

J e 1

hJ0 [ βrmn 1

2] d iJ1 [ βrmn 1

2] rm 1

rm

eiθτ dτ \with thedampedBesselfunctionsJν [ z] : W eizJν [ z] , z m IC. This yieldsthefollowing discretizedTBC:

ψnJ X ψn

J e 1 W ihη

ρw

ρbψn

J X Bk ne 1

∑mq 0

ψne mJ X ψne m

J e 1˜ m\(3.5.4)

with

Bk¢W iβei2θk sin[ 1

2θk]12θ

\ ˜ m W eiθmk J0 [ βrmn 12] d iJ1 [ βrmn 1

2] Z

REMARK 5.1. In far field simulationsonehasto evaluateJν [ z] for largecomplex z, whennumericallycalculatingtheseconvolution coefficients ˜ n. This, however, is a ratherdelicateproblem,andmany standardsoftwareroutinesarenotableto evaluateJν [ z] for largecomplex z.This is dueto theexponentialgrowth of theBesselfunctionsfor fixedν and p zp ∞ (see[S1]):

Jν [ z]W 2πz

12

cos z X νπ2

X π4

de£ Im z£ O p zp e 1 \ X π _ argz _ π Z(3.5.5)

For this reasonweusedasubroutineof Amos[C1] (seealso[C2], [C3]) to evaluatethedampedBesselfunctionsJν [ z] , Im z ` 0 (note that Im β ` 0 for the standardparameterchoicesin(3.1.7): p1 X p0q1 ^ 0 andq1 ^ 0).


In [T16] Mayfield showed for the attenuation–freecasethat the discretizedTBC for theSPE(3.5.3)destroys theunconditionalstability of theunderlyingCrank–Nicolsonscheme(cf.Theorem1.2)andonecanexpecta similar behaviour for theWAPE.Theseexisting discretiza-tions alsoinducenumericalreflectionsat the boundary, particularlywhenusingcoarsegrids.Hence,theexisting discretizedTBC [T16], [T26] exhibits bothstability problemsandreducedaccuracy, which mayrequiretheusageof unnecessarilyfinegrids.

Semi–DiscreteTBC of Schmidt andDeuflhard. In thesemi–discreteapproachof SchmidtandDeuflhard[T23] a TBC is derivedfor thesemi–discretized(in r) SPE,which alsoappliesfor nonuniformr–discretizationsandrange–dependentcoefficientsin theexteriordomain.ThisTBC yields an unconditionallystablemethod(in conjunctionwith an interior finite elementscheme)[T24]. While thesemi–discreteapproachstill exhibits smallresidualreflectionsat theartificial boundary, thediscreteTBC is reflection–free[T24] (in Subsection5.5we shallreturnto thiscomparisonwhendiscussingthe‘bestexteriordiscretization’).In therecentarticle[T25]themethodsof [T24] areextendedto nonuniformr–discretizationsandrange–dependentpoten-tials.

DiscreteTBC. Insteadof usinganad–hocdiscretizationof theanalyticTBCslike (3.5.3)or (3.5.4)wewill constructdiscreteTBCsfor thefully discretizedhalf–spaceproblem,asdonein Chapter1. Our new strategy solvesboth problemsof the discretizedTBC at no additionalcomputationalcosts.With our DTBC thenumericalsolutionon thecomputationaldomain0 j J exactlyequalsthediscretehalf–spacesolution(on j m IN0) restrictedto thecomputationaldomain0 j J. Therefore,our overall schemeinherits the unconditionalstability of thehalf–spacesolutionthatis impliedby thediscreteL2–estimate(3.5.2).

5.2. Derivation of the DTBC. To derive theDTBC we will now mimic thederivationoftheanalyticTBCsfrom Section2 onadiscretelevel. For theinitial dataweassumeψ0

j W 0\ j `J X 2 andsolvethediscreteexteriorproblemin thebottomregion,i.e. theCrank–Nicolsonfinitedifferencescheme(3.5.1)for j ` J X 1:

Rδbd

q∆2h [ ψnn 1

j X ψnj ]lW i Rκb

d ∆2h [ ψnn 1

jd ψn

j ]\(3.5.6)

with

δb W 1 X q1 [ 1 X N2b ]\ R W 2k0

p1 X q1

h2

k\ q W k

2q1

p1 X q1

k e 10 \

κb W k2

k0 p0 X 1 X[ p1 X q1 ]B[ 1 X N2b ] \

where∆2hψn

j W ψnj n 1 X 2ψn

jd ψn

j e 1, andR is proportionalto theparabolicmeshratio. By usingthe ¤ –transform(for moredetailsseeAppendix):

¤yu ψnj w¥W ψ j [ z] : W ∞

∑nq 0

ψnj ze n \ z m IC\ p zp3^ 1\(3.5.7)

(3.5.6)is transformedto

zd

1d

iq [ z X 1] ∆2hψ j [ z]WhX iR δb [ z X 1]UX iκb [ z d

1] ψ j [ z]Z(3.5.8)


The solutionof the resultingsecondorder differenceequationtakes the form ψ j [ z]¦W ν j1 [ z] ,

j ` J X 1, whereν1 [ z] solves

ν2 X 2 1 X iR2

δb [ z X 1]X iκb [ z d1]

zd

1d

iq [ z X 1] ν d1 W 0Z(3.5.9)

For the decreasingmode(as j ∞) we require p ν1 [ z](p(_ 1. We obtain the ¤ –transformedDTBCas

ψJ e 1 [ z]W ν e 11 [ z] ψJ [ z]\(3.5.10)

andin a tediouscalculation(analogousto the onein Chapter1, page13) this canbe inversetransformedexplicitly:

CALCULATION (of [ 1 diq ]3¤ e 1 ν e 1

1 [ z] ). First wewrite [ 1 diq ] ν e 1

1 [ z] appropriately:

[ 1 diq ] ν e 1

1 [ z]W 1d

iq X iR2


zd 1e iq

1n iq

d X iR2


zd 1e iq

1n iq

2 [ 1 diq ]UX iR

2δb [ z X 1]UX iκb [ z d

1]zd 1e iq

1n iq

W 1d

iq X iR2

[ δb X iκb ] zzd

eiξd iR

2δb

diκb

zd

eiξ X iR2

1

zd

eiξ δb [ z X 1]UX iκb [ z d

1] δb X q4R

[ z X 1] d i4R

X κb [ z d1]

W 1d

iq X iR2

[ δb X iκb ] zzd

eiξd iR

2δb

diκb

zd

eiξ § iR2

g Az2 X 2Bzd

C

zd

eiξ \

(3.5.11)

with ξ W arg 1e iq1n iq andtheconstantsgivenby

A WI[ δb X iκb ] δb X iκbd

i4R

[ 1 diq ] \(3.5.12a)

B W δ2bd κ2

b X 4R

[ κbd δbq]\(3.5.12b)

C WI[ δbd

iκb ] δbd

iκb X i4R

[ 1 X iq ] Z(3.5.12c)

For theinverse¤ –transformweuse(1.2.23)andtheabbreviations(1.2.24)andobtain

g Az2 X 2Bzd

C

zd

eiξ W 1

g A

Az2 X 2Bzd

C

z[ z deiξ ] F [ λz\ µ]

W 1

g AA

d ee iξCz

X eiξE

zd

eiξ F [ λz\ µ]\wheretheconstantE canbedeterminedas:

E : W Ad

2ee iξBd

ee 2iξC W δb X iκbd [ δb

diκb ] ee iξ 2 Z(3.5.13)


Theinversionrulesgivenin theAppendixnow yield

¤e 1 g Az2 X 2Bzd

C

zd

eiξ W 1

g AAδ0

nd

Cee iξδ1nd

E [ X 1] neinξ X δ0n ¨ Pn [ µ]

W 1

g AAPn [ µ] d Cee iξPne 1 [ µ] d E

ne 1

∑mq 0

X eiξ ] ne kPk [ µ] \where ¨ denotesthediscreteconvolution. Finally, weobtain

¤e 1 ν e 11 [ z] W 1

diq

d iR2

[ δbd

iκb ] δ0n X iR

2δb X iκb

d [ δbd

iκb ] ee iξ [ X eiξ ] n

X iR2

1

g AAPn [ µ] d Cee iξPne 1 [ µ] d E

ne 1

∑mq 0

X eiξ ] ne kPk [ µ] Z(3.5.14)

Now introducingthenotationA W R2A, B W R2B, C W R2C with thesettings

γ W Rδb \ σ WYX Rκb \we recallfrom Chapter1 thattheparametersλ, µ aregivenby

λ W g A

g C\ µ W B

g A g C\(3.5.15)

with

A W©[ γ diσ ] γ X 4q

di [ σ d

4] \(3.5.16a)

B W γ [ γ X 4q] d σ [ σ d4]\(3.5.16b)

C W©[ γ X iσ ] γ X 4q X i [ σ d4] Z(3.5.16c)

Wesummarizeour calculationandformulatetheDTBCfor theSPEandtheWAPE:

[ 1 diq ] ψn

J e 1 W ψnJ ¨ n~ W n

∑mq 1

ψmJ ne m~ \ n ` 1\(3.5.17)

with theconvolutioncoefficients n~ WI[ 1 d

iq ]3¤ e 1 ν e 11 [ z] givenby

n~ W 1d

iqd i

2[ γ X iσ ] ee iξ δ0

n X i2

H [ X 1] neinξ

X ζ Pn [ µ] d ee iξλ e 2Pne 1 [ µ] d ωee iϕne 1

∑mq 0

[ X eiξ ] ne mPm [ µ] \(3.5.18)

ξ W arg1 X iq1d

iq\ ζ W i

2p A p 12ei ϕ

2 \ ϕ W argA\

ω W H2

p A p \ H W γ diσ d [ γ X iσ ] ee iξ Z

In (3.5.18)δ0n denotestheKroneckersymbolandPn [ µ] : W λ e nPn [ µ] thedampedLegendrepoly-

nomials(P0 1, Pe 1 0). In thenon–dissipative case(αb W 0) we have p λ p3W 1, µ m sX 1\ 1 ,andhence p Pn [ µ](p# 1. In the dissipative caseαb ^ 0 we have p λ p(^ 1, µ becomescomplex


and p Pn [ µ](p typically growswith n. In orderto evaluate n~ in a numericallystablefashionit is

thereforenecessaryto usethedampedpolynomialsPn [ µ] in (3.5.18).

5.3. The Asymptotic Behaviour of the Convolution Coefficients. Analogouslyto theconsiderationsin Subsection2.3 of Chapter1 onecanshow that the convolution coefficients(3.5.18)behaveasymptoticallyas n~ W X iH [ X 1] neinξ \ n ∞ \(3.5.19)

which may leadto subtractive cancellationin (3.5.17)(notethat ψmJ j ψmn 1

J in a reasonablediscretization).Thereforewe usethefollowing numericallymorestablefashionof theDTBCin theimplementation:

[ 1 diq ] ψn

J e 1 X 0~ ψnJ WYX [ 1 X iq ] ψne 1

J e 1d ne 1

∑mq 1

ψmJ s ne m~ \(3.5.20)

with thesummedcoefficientss n~ definedby s n~ : W n~ d eiξ ne 1~ , n ` 1. Thecoefficientss n~arecalculatedas

s n~ W 1 X iqd i

2[ γ X iσ ] δ1

nd ζ

Pn [ µ]UX λ e 2Pne 2 [ µ]2n X 1

Z(3.5.21)

The recurrenceformula for the summedcoefficients. In this subsectionwe shall givetwo different derivationsfor the recursionformula of the convolution coefficients s n~ . Thefirst oneis basedon the explicit representation(3.5.21)of s n~ by first calculatinga recursionformulafor Pnn 1 [ µ]X Pne 1 [ µ] . Thesecondderivationdoesnot requiretheexplicit form of thecoefficientss n~ but only thegrowthfunctionν1 [ z] from the ¤ –transformedDTBCs(3.5.10).

FIRST DERIVATION: We considerthe recursion formula (1.2.47)adaptedfor the scaledLegendrepolynomialsPnn 1 [ µ]UX Pne 1 [ µ] :(3.5.22)

Pnn 1 [ µ]X λ e 2Pne 1 [ µ]2n

d1

Wµλ e 1Pn [ µ]UX λ e 2Pne 2 [ µ]

nd

1X n X 2

nd

1λ e 2Pne 1 [ µ]UX λ e 2Pne 3 [ µ]

2n X 3\ n ` 2Z

From(3.5.21)weseetherecurrencerelation for thesummedconvolutioncoefficients:

s nn 1~ W 2n X 1nd

1µλ e 1s n~ X n X 2

nd

1λ e 2s ne 1~ \ n ` 2\(3.5.23)

whichcanbeusedaftercalculatings n~ , n W 0\ 1\ 2 by theformula(3.5.21).

SECOND DERIVATION: Now we presentan alternative derivation of the first convolutioncoefficientss n~ , n W 0\ 1\ 2andtherecurrencerelation(3.5.23).Theadvantageof thisalternativeapproachis thatwe shallonly needthegrowthfunctionν1 [ z] from the ¤ –transformedDTBCs(3.5.10).

In thissecondapproachweshallfirst deriveafirst orderODEfor thegrowth functionν1 [ z] .In fact,it is moreconvenientto consider

ν [ z] : W [ 1 diq ] z d

1 X iq ν [ z]W ∞

∑nq e 1

s nn 1~ ze n Z(3.5.24)


It solvesthequadraticequation

ν2 X 2 [ 1 diq ] z d

1 X iq X iR2

δb [ z X 1]X iκb [ z d1] ν d [ 1 d

iq ] z d1 X iq

2 W 0Zandthereforehastheexplicit form

ν1 ª 2 [ z]W z 1d

iq X i2[ γ d

iσ ] d1 X iq

d i2[ γ X iσ ]U« i

2 Az2 X 2Bz

dC Z(3.5.25)

Here,thesignhasto befixedsuchthat p ν1 [ z](p=_ 1 holds. This canbedonefor e.g.for z W ∞.

Multiplying νk W dνdz

by Az2 X Bzd

C thenyieldsaninhomogeneousfirst orderODEfor ν [ z] :A

Bz2 X 2z

d C

Bνk [ z]UX A

Bz X 1 ν [ z]W β [ z] : W β e 1z

d β0 \(3.5.26)

with

β e 1 WX A

B1 X iq

d i2[ γ X iσ ] d

1d

iq X i2[ γ d

iσ ] \β0 W 1 X iq

d i2[ γ X iσ ] d C

B1d

iq X i2[ γ d

iσ ] Z(3.5.27)

Its generalsolutionincludesν1 ª 2 [ z] asdefinedin (3.5.25). Recallingfrom (3.5.18)that Ac B Wλ c µ, C c B W 1c [ λµ] holdsandusingtheLaurentseries(3.5.24)of ν andνk in (3.5.26)immedi-atelyyieldsthedesiredrecursionfor thecoefficientss n~ :

X λµ

s 1~ X s 0~ W β e 1 \X 2

λµ

s 2~ d s 1~ d 1λµ

s 0~ W β0 \X[ n d

1] s nn 1~ d [ 2n X 1] µλ

s n~ X[ n X 2] 1λ2s ne 1~ W 0\ n ` 2\

(3.5.28)

which coincideswith (3.5.23).We remarkthat the startingcoefficientsof the recursion(cf. (3.5.18)) canbe determined

with [S3,Theorem39.1]:

s 0~ W 0~ W limz ∞

ν1 [ z]z

W 1d

iq X i2[ γ d

iσ ]X i2

A \s 1~ W lim

z ∞ν1 [ z]UX s 0~ z W 1 X iq

d i2[ γ X iσ ] d ζλ e 1µ W 1 X iq

d i2[ γ X iσ ] d i

2B

g AZ

Simplified DTBC. Usingasymptoticpropertiesof theLegendrepolynomials[S7]onefindss n~ W O [ ne 3 2 ] , n ∞ whichagreeswith thedecayof theconvolutionkernelin thedifferentialTBCs(3.2.14), (3.2.21).

Thisdecayof thes n~ motivatesconsideringalsoasimplifiedversionof theDTBC (3.5.20).The simplified DTBC canbe obtainedby cutting off the convolution coefficientsbeyond anindex M, takingonly the“recentpast”(i.e. M rangelevels)into account:

[ 1 diq ] ψn

J e 1 X 0~ ψnJ WhX[ 1 X iq ] ψne 1

J e 1d ne 1

∑mq ne M

ψmJ s ne m~ Z(3.5.29)


We remarkthat this numericallycheaperresultingschemedoesnot conserve the discreteL2–norm(cp. (3.5.2)) (in general,with theDTBC thediscreteL2–normdecreases),andhencethenumericallystabilityof thesimplifiedDTBC is notyetknown.

5.4. Discrete Treatment of the Density Jump. So far we did not considerthe (typical)densityjump at theseabottomin theDTBC (3.5.17).In thefollowing we review two possiblediscretizationsof the water–bottominterface. For the usualgrid zj , j m IN0 with Jh W zb thediscontinuityof ρ is locatedat thegrid pointzJ. In thiscaseit is astandardpractice[U2], [U20]to use(3.5.1)with

ρ j Wρw \ j _ J \2ρbρwρb n ρw

\ j W J \ρb \ j ^ J Z

(3.5.30)

As an alternative onemay usean offsetgrid, i.e. zj W¬[ jd 1

2 ] h, ψnj ψ [ zj \ rn ] , j W*X 1 [ 1] J,

wherethewater–bottominterfacewith thedensityjump lies betweenthegrid points j W J X 1andJ. For discretizingthematchingconditionsin this caseonewantsto find suitableapproxi-mationsfor ψ andρ at theinterfacezb, Ψ ψ [ zb ] andρef f W ρ [ zb] , suchthatbothsidesof thediscretizedsecondmatchingcondition(3.2.5b)

1ρw

ψnJ X Ψhc 2

W 1ρb

Ψ X ψnJ e 1

hc 2areequalto

1ρef f

ψnJ X ψn

J e 1

hZ(3.5.31)

Thisapproachresultsin aneffectivedensityρef f Wz[ ρwd ρb ] c 2 (basedonadifferentderivation

thiswasalsousedin [U7]). In numericaltestswefoundthattheoffsetgridwith theabovechoiceof ρef f producesslightly betterresultsthathave lessGibbs’ oscillationsat thediscontinuityofψz atzb. Thismaybeunderstoodby thefactthat(3.5.30)requiresahigherorderderivation(us-ing theevolutionequation)thanour derivation(3.5.31)(seealso[U7], [U19], [U20]). Becauseof thediscontinuityof ψz thehigherorderderivationyields(slightly) poorerresults.Thereforewechoosetheoffsetgrid for theimplementationof theDTBC. At thesurfaceweuseinsteadofψn

0 W 0 theoffsetBC ψn0 WhX ψne 1.

Finally it remainsto reformulatetheDTBC (3.5.17)suchthatthedensityjumpis takenintoaccount.We rewrite thediscretizationof theseconddepthderivativeat j W J from (3.5.1):

h2 ρJ D0h2

ρ e 1J D0

h2ψn

J W ∆2hψn

Jd

1 X ρb

ρef fψn

J X ψnJ e 1 Z(3.5.32)

Comparingther.h.s.of (3.5.32)to (3.5.6)weobservethatonly oneadditionaltermappears,andinsteadof (3.5.8)weget

ˆψJn 1 [ z]UX 1 X iRδb [ z X 1]X iκb [ z d

1]zd

1d

iq [ z X 1] ˆψJ [ z]W ρb

ρef f

ˆψJ [ z]UX ˆψJ e 1 [ z] Z(3.5.33)

Using ˆψJn 1 [ z]HW ν1 [ z] ˆψJ [ z] , whereν1 [ z] denotesthesolutionof (3.5.9),andconsideringthefact that ν1 [ z] d ν e 1

1 [ z] is equalto the term in the squaredbrackets in (3.5.33)we obtainthe¤ –transformedDTBC:

ˆψJ [ z]X ˆψJ e 1 [ z]W ρef f

ρb

ˆψJ [ z]UX ρef f

ρbν e 1

1 [ z] ˆψJ [ z]Z(3.5.34)


Hence,theDTBCincludingthedensityjumpreads

(3.5.35) [ 1 diq ] ρb

ρef fψn

J e 1d [ 1 d

iq ] 1 X ρb

ρef fX 0~ ψn

J

WYX[ 1 X iq ] ρb

ρef fψne 1

J e 1 X[ 1 X iq ] 1 X ρb

ρef fψne 1

Jd ne 1

∑mq 1

ψmJ s ne m~ \

with theconvolutioncoefficientss n~ givenby (3.5.21).

5.5. Nonuniform Depth Discretization. At the endof this Sectionwe now addressthequestionof nonuniformdepthdiscretizations. In the derivation of the DTBC we neededauniform z–discretizationfor the exterior problemon z ^ zb, i.e. j ` J X 1. For the interiorproblem,however, anonuniformdiscretization(evenadaptivein r) maybeused,andthiswouldnot changeour DTBC (3.5.35). For any given interior z–discretizationand a uniform gridspacinghb in the exterior domain,the DTBC will always yield, on the interior domain,thesamesolutionasthecorrespondingdiscretehalf–spacesolution.

Thisraisesanaturalquestion:givenaninterior(possiblynonuniform)z–discretization,whatis thebestuniformdiscretizationof theexteriordomain?To analyzethis questionwefirst con-siderthethreetypesof errorsthatarerelevanthere:Firstly, theerrorassociatedwith thegiveninterior discretizationdoesnot dependon thechoiceof hb. In orderto avoid strongreflectionsdueto thenonuniformgrid wewill assumethattheinteriorgrid spacingh j : W zj X zj e 1 “variesslowly with j” andcanbe representedash j W h [ zj ] with a “smooth” function h [ z] . To theauthor’s knowledge,the reflectionsin irregular grids have not yet beentheoreticallyanalyzedfor theSchrodingerequation,but very similar effectsappearin hyperbolicandparabolicequa-tions [F9], [F3]. In numericaltests,however, onecaneasilyverify thatdiscontinuitiesof h [ z]would introducespuriousnumericalreflectionsof an incidentwave (cp. [F9] andreferencestherein). Suchreflectionscanbe largely reducedby “smoothing”sucha discontinuityof h [ z](cf. Example4 of Section6).

Secondly, thediscreteBC atzb maycauseoutgoingwavesto bepartially reflectedbackintothecomputationaldomain,andthesereflectionsstronglydependon hb.

Finally, for thediscretizationerrorof the (uniformly discretized)exterior domainwe haveto distinguishbetweentravelingwavesandevanescentwaves.In thefirst casethediscretizationerror can be interpretedas a modificationto the dispersionrelation for the outgoingwaves(incomingwavesarenot presentin theexterior domain).But theaccuracy of their propagationspeedis irrelevant, as long as we are only interestedin the solution in the interior domain.Hence,the exterior discretizationerror canbe disregardedfor outgoingtraveling waves. Thediscretizationerrorof evanescentwaves,however, influencestheinterior solution.

Sinceour DTBC is fully equivalentto a discretehalf–spaceproblem,theabove discussionof thethreeerrortypescanbecompletelyreducedto theproblemof internalgrid reflectionsfortheSPEor theWAPE. In thecontinuouslimit (hb 0) of theexterior discretization,this alsoholdsfor thesemi–discreteapproachof [T23], [T24] for theSchrodingerequation.Followingthe above discussionwe cannow give the bestexterior discretizationin the ‘tr avelingwaveregime’: theuniformexteriorgrid spacinghb W h [ zb ] generatesacompletelyreflection–freeBCandtheuniformity of theexterior grid ensuresthat theoutgoingwaveswill never bereflectedback. Their inaccurateresolutionin the exterior domainonly causesinaccuratewave speeds,but this doesnotaffect theinterior solution.

6. NUMERICAL EXAMPLES 91

This behaviour is numericallyverified in the simulationsof Section5 in [T24], whereauniformly spacedgrid wascomparedto the semi–discreteapproachfor the exterior domain.There,a Schrodingerequationwith a constantpotential is considered,and hencethe initialGaussianwavepacket consistsonly of travelingwavemodesin theexteriordomain.

In the‘evanescentwaveregime’, however, thepictureis not thatsimple,andit is notknownyet whetherthereexistsa existsa unique‘bestexterior discretization’.Our simulationsof thefollowingSection6 indicatethatit mayindeedbeadvantageousto useaDTBC thatcorrespondsto a finer exterior discretization,aslong astheinterior andexterior grid spacingsaregraduallymatchedto eachother.

6. Numerical Examples

In the first two examplesof this Sectionwe shall considerthe SPEand the WAPE forcomparingthenumericalresultfrom usingournew discreteTBCto thesolutionusingeitherthediscretizedTBCof ThomsonandMayfield[T26] or anabsorbinglayer. Dueto its construction,our DTBC yieldsexactly (up to round–off errors)thenumericalhalf–spacesolutionrestrictedto the computationalinterval s 0\ zb . The simulationwith discretizedTBCs requiresthe samenumericaleffort. However, their solutionmay(oncoarsegrids)stronglydeviatefrom thehalf–spacesolution.

In eachexamplewe usedtheGaussianbeamfrom [U18] asinitial data.Below we presenttheso–calledtransmissionloss X 10log10 p p p 2, wheretheacousticpressurep is calculatedfrom(3.1.3).

Example 1. This is a well–known benchmarkproblemfrom the literature[U18], [T20],[T26]. In this example the oceanregion (0 _ z _ 240m) with the uniform density ρw W1Z 0gcme 3 is modeledby theSPE(3.1.6). It containsno attenuationanda large densityjump(ρb W 2Z 1gcme 3) at the water–bottominterface. Hence,this problemprovidesa test of thetreatmentof thedensityjump in theTBCsappliedalongzb W 240m.

Thesourceof f W 100Hz is locatedat a waterdepthzs W 30m andthereceiver depthis atzr W 90m. Thesoundspeedprofile in wateris givenby c [ z]W 1498

d p 120 X zp c 60mse 1, andthesoundspeedin thebottomis cb W 1505mse 1. For our calculationsup to a maximumrangeof 20km we useda referencesoundspeedc0 W 1500mse 1 anda uniform computationalgridwith depthstep∆z W 2m andrangestep∆r W 5m (thesamestepsizeswereusedin [T26]).

In Figure3.5thesolid line is thesolutionwith ournew discreteTBC (3.5.35)andthedottedline is obtainedwith the discretizedTBC (3.5.3). The discretizedTBC clearly introducesasystematicphase–shifterror, which is roughly proportionalto ∆z. The discretizedTBC alsoproducesartificial oscillations(cf. thezoomedregion),while ournew DTBC yieldsthesmoothsolutionwith thesamenumericaleffort.

Figure3.6comparestheresultsof ournew discreteTBC (solid line) to thesolutionobtainedwith an absorbinglayer of 240m thickness(dottedline) anda homogeneousDirichlet BC atzmax W 480m. Hencethecomputationtook abouttwice aslong asby usingthediscreteTBC.In our experimentswe obtaineda bettermatchto the ‘exact’ half–spacesolutionby usingtheexponentialabsorptionprofile

αb [ z]W 10 exp 4z X zb

zmax X zbX 1 dB c λb \ zb _ z _ zmax\(3.6.1)


0 2 4 6 8 10 12 14 16 18 20

30

40

50

60

70

80

90

100

Range r [km]

Tra

nsm

issi

on L

oss

−10

*log

_10

|p|^

2

Example 1

FIGURE 3.5. Transmissionlossat zr 90m for Example1: thesolutionwiththe new discreteTBC (—) coincideswith the half–spacesolution, while thesolution with the discretizedTBC ( ®®® ) introducesa phase–shiftandartificialoscillations.

ratherthana linear profile. We remarkthat the profile (3.6.1)andthicknessof the absorbinglayer weredesignedasto yield a closematchto the ‘correct’ solution. Without suchan ‘a–posterioridatafitting’, however, a calculationwith an absorbinglayer would usuallyyield asolutionwith a somewhat larger deviation thansuggestedby Figure3.6. With a thicker layeronecanof coursestill improve theresultsof Figure3.6,e.g.no moreartificial oscillationsarevisible whenusinga layerof 760m.

Figure3.7showsthesignificantdeviationsof thesolutionsusingeitherthediscretizedTBCor an absorbinglayer of 240m from the computedhalf–spacesolution,which coincideswiththesolutionusingour new DTBC.

Example 2. This exampleappearedas the NORDA test case3B in the PE WorkshopI[U15], [U18], [T20], [T26]. Theenvironmentfor this exampleconsistsof anisovelocitywatercolumn(c ¯ z° 1500ms± 1) overanisovelocityhalf–spacebottom(cb 1590ms± 1). Theden-sity changesat zb 100m from ρw 1² 0gcm± 3 in thewaterto ρb 1² 2gcm± 3 in thebottom.The sourceandthe receiver arelocatedat the samedepthnearthe bottom: zs zr 99² 5m.The sourcefrequency is f 250Hz. The attenuationin the wateris zero,andthe bottomat-tenuationis αb 0² 5dB ³ λb, whereλb cb ³ f denotesthewavelengthof soundin thebottom.


0 2 4 6 8 10 12 14 16 18 20

30

40

50

60

70

80

90

100

Range r [km]

Tra

nsm

issi

on L

oss

−10

*log

_10

|p|^

2

Example 1

FIGURE 3.6. Transmissionlossat zr 90m for Example1: thesolutionwithanabsorbinglayerof 240m( ®?®® ) is quitesatisfactoryin comparisonto the‘exact’solutioncomputedwith the discreteTBC (—). It is in phasebut shows someartificial oscillationsandoverestimatesthe transmissionlossat 6km, 7km, andin therange16–19km.

new discrete TBC

discretized TBC

absorbing layer

0 60 120 180 2400

0.01

0.02

0.03

0.04Example 1

depth [m]

|psi

|

FIGURE 3.7. Verticalcut of the3 solutionsat r 19km for Example1: ´ ψ ¯ zµ r 19km°(´Here,thesteepestangleof propagation(which is theequivalentray–angleof thehighestof the11propagatingmodes)is approximately20¶ (cf. [U15], [T26]). Sincethesourceis locatednear


thebottom,thehighermodesaresignificantlyexcited. Thereforethewide–anglecapabilityisimportanthereandwe usetheWAPE (3.1.8)(with thecoefficientsof Claerbout)to solve thisbenchmarkproblem.

The maximumrangeof interestis 10km andthe referencesoundspeedis chosenasc0 1500ms± 1. Thecalculationswerecarriedout usingthedepthstep∆z 0² 25m andthe rangestep∆r 2² 5m. Sincethe sourceis placedcloseto the bottom, the TBC wasapplied10mbelow theocean–bottominterface(thesamewasdonein [T26]).

new discrete TBCdiscretized TBC

5 6 7 8 9 10

60

70

80

90

100

Range r [km]

Tra

nsm

issi

on L

oss

−10

*log

_10

|p|^

2

Example 2

FIGURE 3.8. Transmissionlossatzr 99² 5m for Example2: thesolutionwiththenew discreteTBC coincideswith thehalf–spacesolution,while thesolutionwith the discretizedTBC still deviatessignificantlyfrom it for the chosendis-cretization.

The typical featureof this problemis the large destructive interferencenull at a rangeof7km. Figure3.8 comparesthe transmissionlossresultsfor thediscreteanddiscretizedTBCsin therangefrom 5 to 10km. In a secondcomparisonwe extendedthecomputationaldomainup to 200m. With the given bottomattenuationthis 100m layer is thick enoughto yield thereasonableapproximationshown in Figure3.9.

Figure 3.10 shows the deviation of the solutionswith the discretizedTBC and with theabsorbinglayerfrom thecomputedhalf–spacesolution,whichcoincideswith thesolutionusingour new discreteTBC.


new discrete TBC

absorbing layer

5 6 7 8 9 10

60

70

80

90

100

Example 2

Range r [km]

Tra

nsm

issi

on L

oss

−10

*log

_10

|p|^

2

FIGURE 3.9. Transmissionlossat zr 99² 5m for Example2: in comparisonto theexacthalf–spacesolution,the truncationof thecomputationaldomainat200m (thegivenbottomattenuationthenrepresentsanabsorbinglayerof 100m)introducesaslightphaseshift.

new discrete TBC

discretized TBC

absorbing layer

0 20 40 60 80 1000

0.04

0.08

0.12

0.16Example 2

depth [m]

|psi

|

FIGURE 3.10. Verticalcut of the3 solutionsat r 7km for Example2: ´ ψ ¯ zµ r 7km°#´Example 3. In this examplewe illustratethe theoreticalfindingsof Section3 on coupled

models.We usethephysicalparametersof thefirst two examplesbut differentmodelsfor thewaterandthebottomregion.


Westartwith consideringtheenvironmentof Example2 andcomparetheresultsof differentmodelcouplings.First we fix theWAPE of Claerbout(CWAPE; p0 1, p1 3

4, q1 14) in the

bottomandchooseadifferent(andin factbetter)rationalapproximation(GWAPE)for thewaterregionthatfulfils thecouplingcondition(3.3.9):p1 3q1. Thetwo remainingparametersp0, q1

arethendeterminedby minimizingtheapproximationerrorof ¯ 1 · λ ° 12 (in themaximumnorm)

over the interval ¸ 0² 0008µ 0² 103¹ , which containsthediscretespectrumof L: p0 1² 0000071,q1 0² 2501753.We comparethis approximationto thecaseof alsousingtheCWAPE in thewater. Furthermore,we show theresultswhenusingtheSPEin theseabottom(which clearlyviolates(3.3.9))andwhenusingtheSPEin bothregions.

GWAPE / CWAPE

CWAPE / CWAPE

CWAPE / SPE

SPE / SPE

7 8 9

60

70

80

90

100

Range r [km]

Tra

nsm

issi

on L

oss

−10

*log

_10

|p|^

2

Example 3

FIGURE 3.11. Transmissionloss at zr 99² 5m in several coupledmodels(waterandseabottom)for thesimulationof Example2.

Figure3.11displaysa comparisonof the transmissionlossfrom 6.5 to 9km for thesedif-ferentcouplings.It turnsout that thesolutionfor thecoupledGWAPE/CWAPE modelis verycloseto theoneusingtheCWAPE in bothmedia.While theCWAPE/SPEmodelviolatesthecoupling condition, it only deviatesfrom the above solutionsby a slight phase–shiftthat istypical for theSPEin this example(cp.alsothe“pure” SPEmodel).

Now we turn to the dissipation–freesituationof Example1 andfocusour attentionon aconservativediscretizationof coupledmodelsthatsatisfythecouplingconditionp1 ¯ z°³ q1 ¯ z° µ const, and hencepreserve the L2 ¯ IRº ; ¯ σρ ° ± 1dz° –norm (seeSection3). As a discrete


analogueof (3.2.9)weobtainin thedissipation–freecase

(3.6.2) hD»kJ ± 1

∑j ¼ 0

ψnj

2

σρ j · 2

p21k0 ρef f

Im ¯ p1 · q1 ° ψnº 12

J ± 1 ½ iq1k ± 10 D»k ψn

J ± 1 ¾¾ ¯ p1 · q1 ° D»hψnº 1

2J ± 1 ½ iq1k ± 1

0 D»kD»hψnJ ± 1 µ

with ρ j ρ ¯ zj ° andσ p21 ³¿¯ p1 · p0q1 ° . Analogously, a discreteversionof (3.2.10)canbe

shown for thebottomregion j À J:

(3.6.3) hD»k∞

∑j ¼ J

ψnj

2

σbρb 2

¯ pb1 ° 2k0 ρef f

Im ¯ pb1 · qb

1 ° ψnº 12

J ± 1 ½ iqb1k ± 1

0 D»k ψnJ ± 1 ¾

¾ ¯ pb1 · qb

1 ° D»hψnº 12

J ± 1 ½ iqb1k ± 1

0 D»kD»hψnJ ± 1 µ

with σb ¯ pb1 ° 2 ³¿¯ pb

1 · pb0qb

1 ° . For coupledmodelsσ usuallytakesdifferentvaluesin thewaterand bottom regions. It follows from (3.6.2), (3.6.3) that the weighteddiscreteL2–norm onj Á IN0 is preserved:

Âψn Â 2 h

J ± 1

∑j ¼ 0

ψnj

2

σρ j½ h

∞

∑j ¼ J

ψnj

2

σbρb const(3.6.4)

providedthatthecouplingcondition(3.3.9)is fulfilled.Figure3.12illustratesthatthediscreteL2–norm(3.6.4)is conservedaslongasthecoupling

condition(3.3.9) is satisfied. In all four simulationswe usedthe WAPE of Claerboutfor thewaterregion anddifferentmodelsin theseabottom: only thehybrid WAPE–modelwith con-stantp1 ¯ z°³ q1 ¯ z° µ 3 renderstheschemeconservative (only for this numericalillustrationwechoosethevaluesp0 0² 6,q1 0² 2). A couplingto theSPE(like in [T20]) or to aWAPEinthebottomwith pb

1 ³ qb1 µb Ã 3 all yieldsa non–conservativescheme.We point out thatthese

schemesarenot only non–conservative for the particularnorm (3.6.4)but alsofor any otherweightedL2–norm.

In thesimulationsfor Figure3.12,thesecondsumof (3.6.4)(for theexteriorof thecompu-tationaldomain)wasevaluatedvia (3.6.3).

Example 4. In this examplewe illustrateour discussionfrom Subsection5.5 on the ‘bestuniformexterior discretization’ for thecaseof evanescentwaves. We want to answerthe fol-lowing question:givena uniform interiordiscretization,cantheresultfrom aglobally uniformz–discretizationbe improvedby choosinga finer exterior z–discretization,or, equivalently, byusinga DTBC thatcorrespondsto suchafinerdiscretization?

As amodelproblemfor thistestweconsidertheSPE(3.1.6)onz Ä 0µ r Ä 0 with ahomoge-neousDirichlet BC atz 0, k0 2m± 1 andthe“potentialwell” V ¯ z° 0µ 0 Å z Å zb 100m,V ¯ z° Vb 0² 3µ z Ä zb. In thisexample,planewaveswith awavenumberk Å kcrit ÇÆ 1² 2m± 1

areevanescentin theexterior domainz Ä zb, andk Ä kcrit transmitsa traveling wave into theexterior. We chooseheretheGaussianbeamexp ¯ ikz · 0² 003m± 2 ¯ z · 50m° 2 ° with k 1m± 1

asaninitial condition.For this choiceof k ‘most’ of theFouriercomponentsof this wave cor-respondto evanescentmodesin thebottom. Hence,this wave will bepredominantlyreflectedbackinto theinteriordomain.


WAPE: p_1/q_1=3

SPE

WAPE: p_1/q_1=3.25

WAPE: p_1/q_1=2.75

0 2 4 6 8 10 12 14 16 18 200.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Range r [km]

(nor

mal

ized

) w

eigh

ted

L2−

norm

Example 3

FIGURE 3.12. CoupledWAPE–modelsconserve thediscreteL2–norm(3.6.4)only whensatisfyingthecouplingconditionp1 ³ q1 µ const(—).

Figure3.13comparestheeffectof choosingdifferent(uniformandnonuniform)z–discreti-zations.We show the resultsof this simulationat the ranger 200m whenthewave packethasbeenreflectedbackfrom thewater–bottominterface.Thesolid line wasobtainedwith theuniform z–discretizationh0 0² 05m, andit will serve asour ‘exact’ referencesolution. Thedashedline shows the solutionwith the uniform grid spacingh1 0² 25m. In the followingcomparisonswe will keepthis coarserinterior grid and will vary the uniform exterior grid.Following our discussionfrom Subsection5.5,we useda gradualtransitionbetweenthesetwogrid spacingsin thedepthinterval 100 · 110m (piecewiselineargrid spacingfunctionh ¯ z° ).

The dottedcurve of Figure 3.13 gives the resultswith the finer exterior z–discretizationh2 0² 1m. Closeto the seabottomit shows significantimprovementsover the uniform dis-cretizationwith h1. In the interval 0 Å z Å 60m both curvesalmostcoincideas the interiordiscretizationerroris dominantthere,andit impliesinaccuratewavespeedsthatarereflectedintheclearlyvisible phaseshift. Thedottedcurve still exhibits this phaseshift up to theseabot-tom at 100m, but for thedashedcurve theerror in the interval 80m Å z Å 100m is dominatedby the effect of the exterior discretization. It thusseemsthat the effect of the reducedexte-rior discretizationerror (dueto the finer exterior discretization)mayoutweigh(in the interiordomain!)theadditionalreflectionerrorsincurredby thenonuniformgrid.

TheL2 ¯ 0µ 100° –errors(w.r.t. thesolidcurve)of thesolutionswith theuniformh1–discretiza-tion andthenonuniformh1 ³ h2–discretizationare,respectively, 0² 0370and0² 0267. Usinganeven finer exterior discretizationdoesnot seemto improve the resultmuchfurther (L2–error


uniform dz=0.05m uniform dz=0.25m nonuniform dz=0.25−0.1m

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1Example 4

depth [m]

|psi

|

FIGURE 3.13. The ‘bestuniform exterior z–discretization’maybe finer thanthe interior discretization. Vertical cut of the threesolutionsat r 200m forExample4: thesolution( ®®?® ) calculatedon a nonuniformgrid (finer grid in theexteriordomainthanin theinterior) is moreaccuratein theinteriordomainthanthesolutionobtainedon a uniformly coarsegrid (– – –). Thereferencesolution(—) wascalculatedon auniformly finegrid.

0² 0266for theh1 ³ h0–discretization).A finer exterior discretizationwould, however, requireathicker region to adaptthetwo grids.

We thusconcludethatfinerexteriordiscretizationsmayindeedbeadvantageousin thecaseof evanescentwaves, and for large rangestheseare the importantmodesin the consideredapplicationsof underwateracoustics.

Conclusionsand Perspectives

Wehavederivedanew discretization(discreteTBC) of theanalyticTBC for theconvection–diffusionequationandSchrodinger–typepseudo–differentialevolution equationsin onespacedimension.It is of discreteconvolution form in t involving theboundarydatafrom all thepasttime levels.

TheconvolutioncoefficientssÈ nÉ canbecalculatedvia a simplethree–termrecurrencerela-tion andthey decaylikeO ¯ n± 3Ê 2 ° . Sinceour new DTBC hasthesameconvolutionstructureasexisting discretizations,it requiresthesamecomputationaleffort but improvestwo shortcom-ings: DTBCsaremore accurate(in fact,asaccurateasthediscretewhole–spaceproblem)andthey retainthestabilityof theunderlyingfinite differencescheme.

Theunderwateracousticscommunitytookagreatinterestin theresultsof Chapter3. Espe-cially the(continuous)TBC for anelasticbottomcaughttheir attentionandwill bediscretizedandimplementedin existingnumericalcodes.Our commentson themathematicaldifficulty ofcouplingdifferentmodelsfor thewaterandbottomregion arealsoof interest. In underwateracousticsthe computationsmust be very fast and thereforecoarsegrids arepreferredwhichmakestheusageof DTBCsadvantageous.

We point out that the superiorityof DTBCs over otherdiscretizationsof TBCs is not re-strictedto the presentedspecialtypesof partial differentialequationsor to our particularin-terior discretizationscheme(seee.g. [T1], [T2], [T3], [T4], [T5], [T10], [P2], [T11]). Thecrucial point of our derivation was to find the inverse Ë –transformationof the transformedDTBC explicitly. In moregeneralapplications,e.g.2D transientSchrodingerequation(corre-spondsto 3D–problemsin underwateracoustics),systemsof equationsor high–orderschemes,it might be necessaryto useour secondapproachof deriving the recurrencerelation for theconvolutioncoefficientsor to calculatetheconvolutioncoefficientsthroughanumericalinverseË –transformation(seeSection4 in Chapter1), but this doesnot changethe efficiency of thepresentedmethod.Moreover, in thegeneralcasetheDTBC will benonlocalin morethanonevariableandonehasto seekfor approximatelocal BCsthat leadto a stablescheme.As a gen-eralphilosophy, DTBCsshouldbeusedwheneverhighly accuratesolutionsareimportant.Ourstrategieshave alreadybeenadaptedby a Princetonworkgroupto systemsof wave equationsfor materialswith cracks.

Topics of future researchinclude DTBCs for higher order schemes, e.g. the Numerovscheme[U11], or moreefficient algorithmslike split–stepPade schemes[U9] in the appli-cationto underwateracoustics.Moreover we will usethe ideasof Section3 in Chapter1 andderiveaDTBC for theWAPE in thecasethatthestartingfield is notsupportedinsidethewaterregion (which is thecomputationaldomain). We intendto apply the ideasof Markowich andPoupaud[F5] in order to derive a finite differencediscretizationdirectly for the squarerootoperatorÆ 1 · L from Chapter3. Furthermorewe searchfor a conservative formulationof theelastic/fluidcouplingwhich is neededfor stablediscretizationsandasguidancefor deriving aDTBC for anelasticbottom.

101

102 CONCLUSIONSAND PERSPECTIVES

We will constructDTBCsfor situationsin which theexterior problemshave variablecoef-ficientsbut still areexplicitly solvable. For a linearCrank–Nicolson–typeschemeto solve theradialSchrodinger–Poissonsystempreservingboth theenergy andthemass[F6] we shall tryto designan appropriateboundaryconditionat the origin anda DTBC for r Ì ∞ [F7]. Alsowe will designDTBCsfor theSchrodingerequationwith a linearpotentialtermin theexteriordomain.This derivationis basedon theknowledgeof theasymptoticbehaviour of solutionstoa discreteAiry equation(cf. [E4], [E5]). This problemarisesin underwateracousticsin [T8]andin “parabolicequation”predictionsfor radarpropagationin thetroposphere[T13].

Finally to remedythe deficiency that our DTBC is nonlocal in the time variablewhichis noticeableespeciallyin long–timecalculationswe constructnew approximative transparentboundaryconditions[D1]. TheseBCsareanefficient convolution by anexponentialapproxi-mation:only onesimpleupdateis neededin eachtimestepto computethediscreteconvolution.We alsoprovecriteriafor thenumericalstabilityof theresultingIBVP.

Appendix

The Laplace Transformation

Herewe discussthe Laplacetransformationf f ¯ s° of a function f f ¯ t ° andpresentsomeelementaryresults.In this thesistheLaplacetransformationwasusedto explicitly solvetheexterior problemsin orderto derive thecontinuousTBCs.

DEFINITION L.1 (Laplacetransformation[S3]). If for a function f : IR» Ì IC thereexistsaσ Á IR, suchthattheintegral

ÍÏÎf Ð¿¯ s° : f ¯ s° : ∞

0f ¯ t ° e± st dt µ(L.1)

existsfor all s Á IC with Res Ä σ, then f ¯ s° is theLaplacetransformationof f .

Themostimportantpropertyof theLaplacetransformationis therelationshipbetweende-rivativeandtransformation:

THEOREM L.1 (DifferentiationTheorem[S3]). If the function f : IR» Ì IC is differentiableandtheLaplacetransformationof f Ñ exists,thenthelimit

f ¯ 0½ ° : limt Ò 0º f ¯ t °µ

existsandfor Res Ä σ theLaplacetransformationof f Ñ fulfilsÍÏÎf Ñ Ð¿¯ s° s f ¯ s°U· f ¯ 0½ °²(L.2)

Theinitial valuesenterinto thetransformationof thederivative.Theessentialrule for deriving thecontinuousTBCsis theLaplacetransformationof acon-

volution f Ó g of two functions f µ g : IR» Ì IC which is definedas

¯ f Ó g°B¯ t ° t

0f ¯ τ ° g ¯ t · τ ° dτ ²

THEOREM L.2 (Convolution Theorem[S3]). If theLaplacetransformation(L.1) existsforthefunctionsf µ g : IR» Ì IC andat leastoneLaplaceintegral is absolutelyconvergent thenwehave ÍÏÎ

f Ó gÐ¿¯ s° f ¯ s° g ¯ s°²(L.3)

It is alsonecessaryto havea link betweennormsin thephysicalandtransformedspace:

THEOREM L.3 (Plancherel’s Theorem[M5]) . If thefunction f : IR» Ì IC is continuousandsatisfiesan estimate

´ f ¯ t °(´ 2 Ô Ceσt µ t À 0µ103

104 APPENDIX

for somereal constantsC, σ, thentheLaplacetransformationof f (L.1) is ananalyticfunctionfor Res Ä σ and

∞

0e± 2ηt ´ f ¯ t °(´ 2 dt 1

2π

∞

± ∞f ¯ η ½ iξ ° 2

dξ µ η Ä σ ²(L.4)

holds.

The InverseLaplaceTransformation

Here we shortly give the basicfactsaboutthe inverseLaplacetransformationwhich areneededfor statingthecontinuousTBC in physicalvariables.The inverseLaplacetransforma-tion is givenasfollows:

THEOREM L.4 (InverseLaplacetransformation[S3]). Theinverseof theLaplacetransfor-mationis givenby theBromwich integral

Í ± 1 f ¯ t ° 12πi

γ º i∞

γ ± i∞est f ¯ s° dsµ(IL.1)

where γ is a vertical contourin thecomplex planechosensuch thatall singularitiesof f ¯ s° areto theleft of it.

As we haveseenin TheoremL.2 therule for inversetransformingaproductis simply

Í ± 1 f g ¯ t ° t

0f ¯ τ ° g ¯ t · τ ° dτ ²(IL.2)

Finally wepresentacollectionof usedinverseLaplace–transformationrulesfrom [S2].

Í ± 1 1s· 1

c ½ s 1 · e± ct µ s Ä maxÎ0µ· cÐ(IL.3)

Í ± 1 Æ s · γ · Æ s 12Æ π

¯ 1 · eγt ° t ± 32(IL.4)

Í ± 1 1

Æ s 1

Æ πt ± 1

2(IL.5)

Í ± 1 f ¯ s ½ σ ° f ¯ t ° e± σt(IL.6)

Í ± 1 s f ¯ s ½ σ ° ddt

f ¯ t ° e± σt µ if ψ ¯ 0° 0(IL.7)

Í ± 1 1

Æ sei Õ α Õ s 1

Æ πte

α4t(IL.8)

THE Ö –TRANSFORMATION 105

The Ë –Transformation

The main tool of this work is the Ë –transformationwhich is the discreteanalogueof theLaplace–transformation.The Ë –transformationcan be appliedto the solution of linear dif-ferenceequationsin order to reducethe solutionsof suchequationsinto thoseof algebraicequationsin the complex z–plane. In this theseswe usedit to explicitly solve the finite dif-ferenceschemesin theexterior domainin orderto constructthediscretetransparentboundaryconditions.The Ë –transformationis describedin moredetail in [S3], [S6], [S8]. It is definedin thefollowing way:

DEFINITION Z.1 ( Ë –transformation[S3]). Theformalconnectionbetweenasequenceandacomplex functiongivenby thecorrespondence

Ë Îfn Ð f ¯ z° : ∞

∑n¼ 0

fnz± n µ z Á ICµ ´ z3Ä Rf µ(Z.1)

is called Ë –transformation.Thefunction f ¯ z° is called Ë –transformationof thesequenceÎ

fn Ð ,n 0µ 1µ?²²² andRf À 0 denotestheradiusof convergence.

Thediscreteanalogueof theDifferentiationTheoremfor theLaplacetransformationis theshiftingtheorem:

THEOREM Z.1 (Shifting Theorem[S3]). If thesequenceÎ

fn Ð is exponentiallybounded,i.e.thereexistC Ä 0 andc0 such that

´ fn ´ Ô Cec0n µ n 0µ 1µ²²² µthenthe Ë –transformationf ¯ z° is givenbytheLaurentseries(Z.1)andfor theshiftedsequenceÎgn Ð with gn fnº 1 holds

Ë Îfnº 1 Ð zf ¯ z°U· zf0 ²(Z.2)

Theinitial valuesenterinto thetransformationof theshiftedsequence.As a usefulconse-quenceof theshifting theoremwehave:

Ë Îfnº 1 × fn Ð ¯ z × 1° f ¯ z°U· zf0 ²(Z.3)

Theconvolution fn Ó gn of two sequencesÎ

fn Ð ,Îgn Ð , n 0µ 1µ²?²² is definedby ∑n

k¼ 0 fkgn± k.For the Ë –transformationof a convolution of two sequenceswe formulatethefollowing theo-rem:

THEOREM Z.2 ( [S8]). If f ¯ z° Ë Îfn Ð existsfor ´ z3Ä Rf À 0 andg ¯ z° Ë Î

gn Ð for ´ z3ÄRg À 0, thentherealsoexists Ë Î

fn Ó gn Ð for ´ zAÄ max Rf µ Rg ° with

Ë Îfn Ó gn Ð f ¯ z° g ¯ z°²(Z.4)

Note that (Z.4) is nothing elsebut an expresssionfor the Cauchyproductof two powerseries.We furtherneedthe Ë –transformationof aproductof two sequences.

THEOREM Z.3 (Complex ConvolutionTheorem[S8]). If f ¯ z° Ë Îfn Ð existsfor ´ z4Ä Rf À

0 and g ¯ z° Ë Îgn Ð for ´ z#Ä Rg À 0, thenthere also exists Ë Î

fngn Ð for ´ zBÄ Rf Rg and thefollowing relationholds:

Ë Îfngn Ð 1

2πi Ø f ¯ ξ ° gzξ

ξ ± 1dξ ²(Z.5)

106 APPENDIX

Theintegration pathis thecircle Ù aroundtheorigin: ξ r eiϕ, Rf Å r Åt´ zÚ³ Rg, 0 Ô ϕ Ô 2π(if Rg 0 : Rf Å r Å ∞).

As aconsequenceoneobtainsimmediatelyPlancherel’s theoremfor the Ë –transformation:

THEOREM Z.4 (Plancherel’s Theorem[S3]). If f ¯ z° Ë Îfn Ð exists for ´ z(Ä Rf À 0 and

g ¯ z° Ë Îgn Ð for ´ z=Ä Rg À 0 with Rf Rg Å 1. Thenthere alsoexists Ë Î

fn gn Ð for ´ z=Ä Rf Rg

andthefollowing relationholds:∞

∑n¼ 0

fn gn Ë Îfn gn Ð¿¯ z 1° 1

2π

2π

0f ¯ r eiϕ ° g

eiϕ

rdϕ ²(Z.6)

Theintegration path is thecircle Ù definedby Rf Å r Å 1³ Rg (if Rg 0: Rf Å r Å ∞). Espe-cially, if Rf Å 1, Rg Å 1 thenr 1 canbechosento obtain:

∞

∑n¼ 0

fn gn 12π

2π

0f ¯ eiϕ ° g ¯ eiϕ ° dϕ ²(Z.7)

Notethat g ¯ z° is not the Ë –transformationof gn, but Ë Îgn Ð¿¯ z° g ¯ z° holds.

The Inverse Ë –Transformation

Now we presentthebasicrulesfor calculatingthe inverseË –transformationwhich arees-sentialfor formulatingthediscretetransparentboundaryconditions.

THEOREM Z.5 (InverseË –transformation[S3]). IfÎ

fn Ð is an exponentiallyboundedse-quenceand f ¯ z° the correspondingË –transformationthen the inverse Ë –transformationisgivenby

fn Ë ± 1 f ¯ z° 12πi Ø f ¯ z° zn± 1dzµ n 0µ 1µ²²?²4µ(IZ.1)

where Ù denotesa circlearoundtheorigin with sufficientlylarge radius.

Otherinversionformulascanbeobtainedby usingthefact that f ¯ z± 1° is a Taylor seriesorif f ¯ z° is a rationalfunction of z, analyticat ∞ (cf. Section1.3 in [S6]). The most importantformulais the inverse Ë –transformationof a product:

Ë ± 1 f ¯ z° g ¯ z° fn Ó gn n

∑k¼ 0

fk gn± k µ(IZ.2)

which is thediscreteanalogueof (IL.2).

THEOREM Z.6 (Initial ValueTheorem[S3]). If f ¯ z° Ë Îfn Ð existsthen

f0 limzÒ ∞

f ¯ z°²(IZ.3)

z cantendto ∞ on thereal axisor on an arbitrary path,since f ¯ z° is analyticat z ∞.

This theorem,whenrepeatedlyappliedto f ¯ z° , f ¯ z°U· f0, f ¯ z°U· f0 · f1z± 1, etc.,providesa methodfor theinversionof the Ë –transformation:

fn limzÒ ∞

zn f ¯ z°· n± 1

∑k¼ 0

fkz± k µ n 0µ 1µ 2µ²²² ²(IZ.4)

THE INVERSE Ö –TRANSFORMATION 107

Below we presentsomeinverseË –transformationrulesfrom [S3], [S6], [S8] thatwe usedbefore:

Ë ± 1 Î 1Ð δ0n µ(IZ.5)

Ë ± 1 1z δ1

n µ ´ zAÄ 1µ(IZ.6)

Ë ± 1 zz · a an µ ´ z3ÄI´ a ´AÄ 0µ(IZ.7)

Ë ± 1 1z · a a± 1 ¯ an · δ0

n °µ ´ zAÄ*´ a ´3Ä 0µ(IZ.8)

andfor 0 Å θ Å 1 holds:

Ë ± 1 12 ¯ θz ½ 1 · θ ° · 1

2 ¯ 1 · θ ° · 1 · θθ

n · δ0n µ ´ z3Ä 1 · θ

θµ(IZ.9)

Ë ± 1 z · 12 ¯ θz ½ 1 · θ ° 1

2θδ0

n ½ 12θ ¯ 1 · θ ° · 1 · θ

θ

n · δ0n µ ´ z3Ä 1 · θ

θ²(IZ.10)

Fromthegeneratingfunctionof theLegendrepolynomialsPn [S4] weseethat

Ë ± 1 z

z2 · 2µz½ 1 Pn ¯ µ°µ ´ z3Ä R max ´ z1 ´ÛµJ´ z2 ´ µ(IZ.11)

holdsandintroducingthescalingvariableλ weobtainfor Pn ¯ µ° : λ ± nPn ¯ µ°Ë ± 1 λz

¯ λz° 2 · 2µλz ½ 1 Pn ¯ µ°µ ´ z3Ä R max λ ± 1z1 µ λ ± 1z2 ²(IZ.12)

Herez1, z2 denotethezerosof thedenominatorin (IZ.11).

108 APPENDIX

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Curriculum Vitae

10. Nov. 1968 bornin Berlin/Zehlendorf

08/1975–06/1981Attendanceat theRiemeister–Grundschulein Berlin08/1981–06/1988Attendanceat theWerner-von-Siemens–Gymnasiumin Berlin

06/1988 A–levelsin mathematicsandphysics

10/1988–07/1995Studyof technomathematicsat theTechnicalUniversityin Berlin04/1991 Partials‘Vordiplom’ in technomathematics

09/1991–08/1995Tutorat theDepartmentof Mathematicsof theTechnical

Universityin Berlin08/1995 Masterof Science‘Diplom’ in technomathematics

09/1995-01/1999 Assistantat theDepartmentof Mathematicsof theTechnical

Universityin Berlin02/1999–03/1999TMR–scholarshipat theUniversityof Granada(Spain)

04/1999–09/1999Assistantat theDepartmentof Mathematicsof theTechnicalUniversityin Berlin

since10/1999 Assistantat theDepartmentof Mathematicsof theSaarland

University

Saarbrucken,Mai 2001

115

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