DISCRETE - asc.tuwien.ac.atarnold/papers/wape_old.pdf · DISCRETE TRANSP ARENT BOUND AR Y CONDITIONS F OR WIDE ANGLE P ARABOLIC EQUA TIONS IN UNDER W A TER COUSTICS ANTON ARNOLD AND

DISCRETE TRANSPARENT BOUNDARY

CONDITIONS FOR WIDE ANGLE

PARABOLIC EQUATIONS IN

UNDERWATER ACOUSTICS

ANTON ARNOLD�� AND MATTHIAS EHRHARDT �

�Fachbereich Mathematik

TU Berlin� MA ��

Stra�e des �� Juni ��

D�� Berlin� Germany

�Center for Applied Mathematics

Purdue University

West Lafayette� IN �� USA

E�mail� arnold�math�tu�berlin�de� ehrhardt�math�tu�berlin�de

Abstract� This paper is concerned with transparent boundary conditions�TBCs� for wide angle �parabolic� equations �WAPEs� in the application tounderwater acoustics �assuming cylindrical symmetry�� Existing discretiza�tions of these TBCs introduce slight numerical re�ections at this articialboundary and also render the overall CrankNicolson nite di�erence me�thod only conditionally stable� Here� a novel discrete TBC is derived fromthe fully discretized wholespace problem that is re�ectionfree and yieldsan unconditionally stable scheme� While we shall assume a uniform discreti�zation in range� the interior depth discretization �i�e� in the water column�may be nonuniform� and we shall discuss strategies for the best exteriordiscretization� �i�e� in the sea bottom�� The superiority of the new dis�crete TBC over existing discretizations is illustrated on several benchmarkproblems�

In the literature di�erent WAPEs �or WAPE and the standard �para�bolic� equation� have been coupled in the water and the sea bottom� Weanalyze under which conditions this yields a hybrid model that is conserva�tive for the acoustic eld�

Date� July �� revised March �� Mathematics Subject Classi�cation ��M�� Q�� S��Key words and phrases Underwater acoustics� wide angle parabolic equation� transparent

boundary conditions� nite di�erences� discrete transparent boundary conditions��

�� Introduction

This paper is concerned with a �nite di�erence discretization of wide an�

gle �parabolic� equations� These models appear as one�way approximationsto the Helmholtz equation in cylindrical coordinates with azimuthal symme�try� In particular we will discuss the discretization of transparent boundaryconditions�

In the past two decades �parabolic� equation �PE� models have been widelyused for wave propagation problems in various application areas e�g� seismo�logy �� optics and plasma physics �cf� the references in �� Here we willbe mainly interested in their application to underwater acoustics where PEshave been introduced by Tappert �� An account on the vast recent literatureis given in the survey article ��

In oceanography one wants to calculate the underwater acoustic pressurep�z� r� emerging from a time�harmonic point source located in the water at�zs� �� Here r � � denotes the radial range variable and � � z � zb thedepth variable� The water surface is at z � � and the sea bottom at z � zb�In our numerical tests of discrete transparent boundary conditions �in x�� wewill only deal with horizontal bottoms� However irregular bottom surfacesand sub�bottom layers can be included by simply extending the range of z�We denote the local sound speed by c�z� r� the density by ��z� r� and theattenuation by ��z� r� � �� n�z� r� � c��c�z� r� is the refractive index witha reference sound speed c� �usually the smallest sound speed in the model��Then the reference wave number is k� � ��f�c� where f denotes the �usuallylow� frequency of the emitted sound�

The pressure satis�es the Helmholtz equation

�

r

r

�rp

r

��

z

��

p

z

�� k�

�N�p � �� r � ��

with the complex refractive index

N�z� r� � n�z� r� � i��z� r��k� ��

In the far �eld approximation �k�r�� the �complex valued� outgoing acoustic�eld

��z� r� �pk�r p�z� r� e

�ik�r ��

satis�es the one�way Helmholtz equation�

�r � ik��p

�� L� �� r � � ��

Herep�� L is a pseudo�di�erential operator and L the Schr�odinger operator

L � �k�� z��

��z� � V �z� r� ��

with the complex valued �potential� V �z� r� � ��N��z� r��The evolution equation �� is much easier to solve numerically than the

elliptic Helmholtz equation �� Hence �� forms the basis for all standardlinear models in underwater acoustics �normal mode ray representation par�abolic equation� � �� Strictly speaking �� is only valid for horizontally

�

strati�ed oceans i�e� for range�independent parameters c � and �� In prac�tice however it is still used in situations with weak range dependence andbackscatter is neglected�

�Parabolic� approximations of �� consist in formally approximating thepseudo�di�erential operator

p�� L by rational functions of L which yields

a PDE that is easier to discretize than the pseudo�di�erential equation ��For a detailed description and motivation of this procedure we refer to �� The linear approximation of

p�� by ��

�gives the narrow

angle or standard �parabolic� equation �SPE� of Tappert ��

�r � � ik��L�� r � � ��

This Schr�odinger equation is a reasonable description of waves with a propa�gation direction within about �� of the horizontal� Rational approximationsof the form

��

� � f�� p� � p��

�� q��

with real p� p� q� yield the wide angle �parabolic� equations �WAPE�

�r � ik�

�p� � p�L

�� q�L� �

�� r � � ��

In the sequel we will repeatedly require the condition

f �� p�q� � p� � � ��

With the choice p� � � p� ��

� q� �

�

��Pad�e approximant of ��

�

� �one obtains the WAPE of Claerbout �� In �� Greene determines these

coe�cients by minimizing the approximation error of ��

� over suitable ��intervals� These WAPE models furnish a much better description of the wavepropagation up to angles of about �� Also higher order analogues of �� and split�step Pad�e algorithms �� have been successfully usedfor acoustic problems� While we will restrict ourselves here to the WAPE ��we remark that the construction of discrete transparent boundary conditions�see x�� could be generalized to higher order PEs and even �D�problems�

In this article we shall focus on boundary conditions �BC� for the WAPE�� At the water surface one usually employs a Dirichlet ��pressure release��BC� ��z � �� r� � �� At the sea bottom the wave propagation in water hasto be coupled to the wave propagation in the sediments of the bottom� Thebottom will be modeled as the homogeneous half�space region z � zb withconstant parameters cb �b and �b� Throughout most of this paper we willuse a �uid model for the bottom by assuming that �� also holds for z � zbpossibly with a di�erent rational approximation �� subject to the couplingcondition �� Only at the end of x� we will comment on the coupling ofscalar and elastic �parabolic� models�

In practical simulations one is only interested in the acoustic �eld ��z� r� inthe water i�e� for � � z � zb� While the physical problem is posed on the un�bounded z�interval �� one wishes to restrict the computational domain in

�

the z�direction by introducing an arti�cial boundary at or below the sea bot�tom� This arti�cial BC should of course change the model as little as possible�Until recently the standard strategy was to introduce rather thick absorbinglayers below the sea bottom and then to limit the z�range by again imposinga Dirichlet BC �� With a carefully designed absorption pro��le and layer thickness this strategy has been very successful� But without acomparison to the exact half�space solution it is hard to estimate how muchan absorbing layer modi�es the original problem� Also absorbing layers in�crease the computational costs for SPE� or WAPE�simulations typically by afactor around � �� However in simulations without attenuation ��falseabsorbing layer method� �� or over an elastic sea bottom �� muchthicker absorbing layers have been used to ensure accuracy and respectivelynumerical stability�

In �� and �� Papadakis derived impedance BCs or transparent boundaryconditions �TBC� for the SPE and the WAPE which completely solves theproblem of restricting the z�domain without changing the physical model�complementing the WAPE �� with a TBC at zb allows to recover � onthe �nite computational domain �� zb� � the exact half�space solution on� � z � �� As the SPE is a Schr�odinger equation similar strategies havebeen developed independently for quantum mechanical applications � � ��

Towards the end of this introduction we shall now turn to the main mo�tivation of this paper� While TBCs fully solve the problem of cutting o�the z�domain for the analytical equation their numerical discretization is farfrom trivial� Indeed all available discretizations are less accurate than thediscretized half�space problem and they render the overall numerical schemeonly conditionally stable � �� The object of this paper is to con�struct discrete transparent boundary conditions �DTBC� for a Crank�Nicolson�nite di�erence discretization of the WAPE such that the overall scheme isunconditionally stable and as accurate as the discretized half�space problem�

The paper is organized as follows� In x� we review the TBCs for the WAPEand discuss the coupling of the WAPE to the SPE and the elastic PE� Inx� discrete TBCs are derived and analyzed� their superiority over existingdiscretizations is illustrated in the numerical tests of x��

�� Transparent Boundary Conditions and Model Coupling

In this Section we shall �rst discuss the well�posedness of the evolutionproblem for the WAPE in the critical non�dissipative case i�e� for � � ��

�r � ik��f�L��

�� z � �� r � ��

subject to the BC �� r� � � and with the rational function f given in ��For simplicity of the analysis we only consider the range�independent situation�the functional analytic proof of this theorem is deferred to the Appendix�

Theorem �� Assume that the refractive index n�z�� the density ��z� � �� and��z� are bounded for z � �� Then� the WAPE has a unique solution for all

initial data in the weighted L��space L��IR�� dz� if and only if the pole of�

f�� at � � q��

is not an eigenvalue of the operator L with Dirichlet BCs at

z � ��

In applications of underwater acoustics the sound speed c�z� is typicallylarger in the sea bottom than in the water� Therefore V �z� forms a �potentialwell� in the water region � � z � zb which typically gives rise to bound statesof L that represent the propagating modes of �� and �� All of the corre�sponding eigenvalues satisfy � � �j � Vb � �� c�

��c�b � � if c� � minz�� c�z��

As q� is much smaller than � in all practical simulations ��in the WAPE of

Claerbout� also cf� �� usually lies in Vb�� the continuous spectrum ofL� Theorem � then guarantees the unique solvability of the evolution equation�� for any initial data� Let us compare the situation at hand �i�e� the WAPEon the original unbounded interval � and later also the WAPE with a TBC�to the WAPE restricted to the z�interval �� zmax� with a homogeneous RobinBC at zmax as a simple model for an absorbing layer� there L has a pureeigenvalue spectrum which inhibits the solvability of �� in several cases ofpractical relevance ��

Now we turn to the matching conditions and later the TBCs at the water�bottom interface �z � zb�� As the density is typically discontinuous there onerequires continuity of the pressure and the normal particle velocity�

��zb�� r� � ��zb�� r�� a�

�z�zb�� r�

�w�

�z�zb�� r�

�b� ��b�

where �w � ��zb�� r� is the water density just above the bottom and �b denotesthe constant density of the bottom�

With these matching conditions we shall now derive an estimate for the L��decay of solutions to the WAPE �� z � �� We assume � � ��z� and applythe operator �� q�L to ��

�� q�V � q�k�� z��

��z��r

� ik��p� � �� p� � q��V � �p� � q��k

�� z��

��z��

Multiplying �� by !�� integrating by parts on � � z � zb and taking thereal part gives

r

Z zb

�

j�j� ��dz

� ��p� � q��k�

�Z zb

�

Im V � j�j� ��dz � k��w Im

��z

!��z�zb�

� q�

�Z zb

�

�Re V �r j�j� � � Im V � Im �r

!��dz

�k��r

Z zb

�

j�zj� ��dz � �k��w Re

��zr

!��z�zb�

��

�

Analogously multiplying �� by !�r�� and taking the imaginary part we get

�p� � ��r

Z zb

�

j�j� ��dz

� ��q�k��

Z zb

�

Im V � j�rj� ��dz � �q�k��w Im

��zr

!�r

��z�zb�

� �p� � q��

�Z zb

�

�Re V �r j�j� � � Im V � Im �r

!��dz

�k��r

Z zb

�

j�zj� ��dz � �k��w Im

��z

!�r

��z�zb�

��

After an easy algebraic manipulation we obtain from ��

r

Z zb

�

j�j� �� dz

� ��C�

Z zb

�

�c�c

�� r�� dz � C�k��w Im

h r�z

r�i��

z�zb��

with

C� ��p� � q��

�

p� � p�q�� r � I �

iq�p� � q�

k��r

In the same way a similar equation can be derived for the bottom region z � zb�

r

Z �

zb

j�j� �� dz

� ��C��bc�cb

Z �

zb

�� r�� dz � C�k��b Im

h r�z

r�i��

z�zb� ��

Adding the two above equations with �� gives

rk�� r�k� � ��C�

Z �

�

�c�c

�� r�� dz ��

for the weighted L��norm

k�� r�k� �Z �

�

j��z� r�j� ��z� dz ��

In the dissipation�free case �� k�� r�k is conserved and for � � � andp�q� � p� � � it decays� The discrete analogue of this �energy��conservation�or �decay for � � �� will be the main ingredient for showing unconditionalstability of the �nite di�erence scheme in x��

Now we shall review the transparent bottom boundary condition for theSPE and sketch the derivation of the TBC for the WAPE� We assume thatthe initial data �I � ��z� �� which models a point source located at �zs� �� issupported in the interior domain � � z � zb� Also let the bottom region behomogenous let i�e� all physical parameters be constant for z � zb� The basicidea of the derivation is to explicitly solve the equation in the bottom regionwhich is the exterior of the computational domain �� zb�� The TBC for the

�

SPE �or Schr�odinger equation� was derived in � � �� for variousapplication �elds�

��zb� r� � ��k��

� e��i �b�w

Z r

�

�z�zb� r � � eib� ��

� d � ��

with b � k��N�b �� This BC is nonlocal in the range variable r and involves

a mildly singular convolution kernel� Equivalently it can be written as

�z�zb� r� � ��k��

� �

�

e��i eibr

�w�b

d

dr

Z r

�

��zb� � e�ib� �r � ��

�

� d � ��

and the r�h�s� can be expressed formally as a fractional �� derivative � � ��

�z�zb� r� � �p

�k�e��

�i eibr

�w�b��r

��zb� r� e

�ibr� ��

In �� this square root operator is approximated by rational functions whichleads to a hierarchy of highly absorbing �but not any more perfectly transpa�rent� BCs for the SPE� By introducing auxiliary boundary variables these BCscan be expressed through local�in�r operators� Hence this allows for a �local��level in r� discretization scheme � �� This scheme however introduces nu�merical re�ections at the arti�cial boundary whose amplitude depends on thechosen approximation order of the above square root operator�

In order to derive the TBC for the WAPE we consider �� in the bottomregion�

��b � q�k��z ��r � i

��b � �p��q��k��

�z�� z � zb� ��

with

�b � �� q��N�b �� b � k�

�p� � �� p��q��N�

b ��

After a Laplace transformation of �� in r we get�q�s� i�p��q��k�

�"�zz�z� s� � k�

��i�b � �bs� "��z� s� ��

Since its solution has to decay as z �� we obtain

"��z� s� � "��zb�� s� exp

�k� �

si�b � �bs

q�s� i�p��q��k� �z � zb�

�� z � zb� ��

and with the matching conditions �� this gives

"�z�zb�� s� � �k��w�b

�

si�b � �bs

q�s� i�p��q��k�"��zb�� s� ��

Here �p

denotes the branch of the square root with a nonnegative real part�An inverse Laplace transformation �� yields the TBC at the bottom for theWAPE �

��zb� r� � �i� �b�w

�z�zb� r�

� � ��b�w

Z r

�

�z�zb� r � � ei��ei��J�� iJ��

�d � ��

�

� ��

k��

rq��b� � � � p� � p�q�

� q�

k��b� � �

p� � q�q�

k��

where J� J� denote the Bessel functions of order � and � respectively� Thisis a slight generalization of the TBC derived in �� where p� was equal to ��Equivalently �� can be written as

�z�zb� r� � i��w�b

��zb� r�

� � ��w�b

Z r

�

��zb� r � � ei��ei��J�� iJ��

�d ��

Both TBCs are non�local in r� in range marching algorithms they thus requirestoring the bottom boundary data of all previous range levels�

We remark that the asymptotic behaviour �for r � �� of the convolutionkernel in the TBC �� is O

�r��

� which can be seen after an integration

by parts� Using the asymptotic behaviour of the Bessel functions �see ��one �nds that the convolution kernel of �� also decays like O

�r��

��

At the end of this Section we shall now brie�y comment on coupled modelsfor underwater acoustics as proposed in � �� In �� the WAPE for theocean �� z � zb� is coupled to the SPE for the sea bottom �z � zb�� In factthese models are coupled via a TBC corresponding to the SPE but this is equi�valent to the half�space problem� Here we want to point out a mathematicalambiguity of this coupling that may strongly in�uence the numerical stabilityof the discretization scheme� To this end we consider this model coupling inthe case of constant sound speed and density which is rather unrealistic butit illustrates the situation�

Let us �rst review the WAPE �� with the Schr�odinger operator L ��k��

��z � When discretizing �� one usually applies the operator � � q�L to

�� which gives the following PDE of �Sobolev type� �� q�L

��r � ik�

�p� � �� p� � q��L

��

Since the operators in the numerator and denominator of �� commute �evenfor non�constant c and �� this step is mathematically rigorous and �� iseasy to discretize �see x��

Disregarding for the moment the nonlocality of the involved pseudo�di�eren�tial operator one would formally want to write the evolution equation for thecoupled model �WAPE and SPE� as

�r � ik�A� ��

with

A� �

� � ��p� � p�k

��z

� � q�k�� z

� �

�� z � zb�

k��

��z�� z � zb

��a�

��b�

However the right hand side of �� is not well�de�ned due to the non�locality of the pseudo�di�erential operator in ��a�� Also its reformula�tion as in �� is not any longer justi�ed in the coupled case� Even in the

�

dissipation�free case it would result in a non�conservative evolution equationand hence in a non�conservative numerical scheme �nevertheless this strategyis used in �� This is illustrated in Example � of x�� Using more involvedpseudo�di�erential operators it is possible to �nd a correct and conservativeinterpretation of �� for mathematical details see Appendix B��However its discretization would be very di�cult�

From the above we conclude that it is not advisable to couple the WAPEand the SPE numerically� As an alternative we shall now analyze couplingsof WAPEs with di�erent parameters p� p� q� that can be reformulated as aPDE like in �� The coupled model

�r � ik�

�p��z�� p��z�L

�� q��z�L� �

��

is well�de�ned and can be transformed to �� if the numerator and deno�minator in �� commute� Under the condition

p��z��q��z� �� const ��

we can rewrite the pseudo�di�erential operator in �� as

p��z�� p��z�L

�� q��z�L�

�� z�� z�L

��

�z�� z�L� ��

with

�z� �� p��z�

�� z� � p�

��z��p��z�� p��z�q��z�

��

Here the numerator and denominator commute and hence �� can be writ�ten in the form of �� The resulting evolution equation is conservative inL��IR�� dz� and it allows for a conservative and unconditionally stablediscretization �see x� and Example � in x��

If the parameters p� p� q� are �xed in one medium condition �� stillleaves two free parameters to choose a di�erent rational approximation modelof ��

�

� for the second medium �cp� �� Hence one can in fact obtain abetter approximation in the second medium than with the originally intended�parabolic approximation��

Finally we add a small remark on the coupling of the SPE with an elastic

parabolic equation �EPE� for the sea bottom �� In � �� a TBCfor this coupling was derived� It reads for the Laplace transformed wave �eld�

"��zb� s� � � �b�w

�

k�N�s

��pMp�s�

��

�Ms�s� �N�

s

��

qMp�s�

�pMs�s�

�Ms�s� �N�

s

��"�z�zb� s��

with the notation

Mp�s� � ��N�

p ��i

k�s� Ms�s� � ��N�

s ��i

k�s ��

Here Np � np � i�p�k� and Ns � ns � i�s�k� denote the complex refractiveindices for the compressional and shear waves in the bottom �cp� �� In a

tedious calculation this BC can indeed be inverse Laplace transformed �using�� and it reads�

��zb� r� �

C

�Z r

�

�z�zb� r � � ei��g� � d � �i�

Z r

�

�zr�zb� r � � ei��

� d

��

with

C � � �b�w

�

k�� N�

s

r�

�e��i� � �

k��

�N�

p � �� k�

�

�N�

p �N�

s

��

g� � � �� ei��

�

� � ik��

��N�

p �N�

s � �N�

s ei��

�

� �

k��

�

�N�

p �N�

pN�

s � �

�N�

s �N�

p �N�

s

� �

�

� � O� �

�

�

�

While this inverse transformation was carried out numerically in � �� ouranalytical TBC may simplify the discretization of this coupled model� DTBCsfor the SPE�EPE coupling �in the spirit of x�� will be the topic of a subsequentpaper�

�� Discrete Transparent Boundary Conditions

In this Section we shall discuss how to discretize the TBCs �� in conjunction with a Crank�Nicolson �nite di�erence scheme for the SPEand the WAPE� Most of the time we shall only consider uniform grids in zand r� While a uniform range discretization is crucial for our construction ofdiscrete TBCs this construction is independent of the �possibly nonuniform�z�discretization on the interior domain�

For simplicity we �rst consider the uniform grid zj � jh rn � nk �h � #zk � #r� and the approximation �n

j ��zj� rn�� The discretized WAPE ��then reads��

�� q�Vn� �

�

j � q�k��jD

�h�

��j D�h�

��D�

k�nj

� ik��p� � �� p� � q��V

n� �

�

j � �p� � q��k��jD

�h�

��j D�h�

�� n

j � �n��j

��

��

with Vn� �

�

j �� V �zj� rn� �

�� and the usual di�erence operators

D�

k�nj �

�n��j � �n

j

k� D�

h�

�nj �

�nj� �

�

� �nj� �

�

h

It is well known that this scheme is second order in h and k and uncon�ditionally stable �� Proceeding similarly to the derivation of �� one can

�

show

D�

k

Xj�ZZ

��nj

��j

� �C�k��

Xj�ZZ

ImnV

n� �

�

j

o ��n� �

�

j �iq�

p� � q�k��D�

k�nj

��

�j�

��

with C� � ��p��q��p��p�q�� Hence the scheme �� preserves the discreteweighted L��norm in the dissipation�free case �V real�� This also holds whenusing a homogeneous Dirichlet BC at j � ��

In the literature three di�erent strategies have been proposed to discretizeTBCs mostly however just for the Schr�odinger equation� In �� Thomsonand May�eld used the following discretized TBC for the SPE�

�nJ � �n

J�� h

�Bk�

�

�nJ � B�

n��Xm��

��n�mJ � �n�m

J��

�"�m� ��

with

B � ��k��

� e��i �b�w

� B� � ei�bk sin�

�

�bk�

�

�bk

� "�m �eibmk

�qm� �

�

On the fully discrete level this BC is not perfectly transparent any more and itmay also yield an unstable numerical scheme� In analogy to the analytic TBC�� it requires the boundary data from the whole �past range� �� rn��

In the semi�discrete approach of Schmidt and Deu�hard �� a TBC is deri�ved for the semi�discretized �in r� SPE which also applies for nonuniform r�discretizations and range�dependent coe�cients in the exterior domain� ThisTBC yields an unconditionally stable method �in conjunction with an interior�nite element scheme� �� In �� this approach is also applied to uniformexterior z�discretizations and one then recovers � through a di�erent deri�vation � the discrete TBC from �� While the semi�discrete approach stillexhibits small residual re�ections at the arti�cial boundary the discrete TBCis re�ection�free �� at the end of this Section we shall return to this compa�rison when discussing the $best exterior discretization%�� In the recent article�� the methods of �� are extended to nonuniform r�discretizations andrange�dependent potentials�

In �� we constructed a discrete TBC for the fully discretized Schr�odingerequation and the resulting scheme elliminates any numerical re�ections� Thesame strategy was used in �� for advection di�usion equations and in �� forthe wave equation in frequency domain�

Here we shall generalize the latter approach �i�e� fully DTBC� to the WAPEand compare it numerically to the discretized TBC� To this end we use a

��

discretization of the TBC �� for the WAPE that is analogous to ��Z r

�

�z�zb� rn � � ei��ei��J�� iJ��

�d

�n��Xm��

Z rm��

rm

�z�zb� rn � � ei�� eJ�� i eJ�� d

�n��Xm��

�n�mJ � �n�m

J��

h

� eJ��rm��

�� i eJ��rm�

�

�� Z rm��

rm

ei�� d �

with the damped Bessel functions eJ�z� �� eizJ�z� z � IC� This yields thefollowing discretized TBC�

�nJ � �n

J�� ih

�

�w�b

�nJ �B�

n��Xm��

��n�mJ � �n�m

J��

� �m� ��

with

B� � i� ei��k sin�

�

��k�

�

��

� �m � ei�mk� eJ��rm�

�

�� i eJ��rm�

�

��

In far �eld simulations one has to evaluate J�z� for large complex z whennumerically calculating these convolution coe�cients �n� This however is arather delicate problem and many standard software routines are not able toevaluate J�z� for large complex z� This is due to the exponential growth ofthe Bessel functions for �xed � and jzj � � �see ��

J�z� �

��

�z

� �

� ncos

�z � �

�

��

�

�� ejIm zjO

�jzj��o � �� arg z � �

��

For this reason we used a subroutine of Amos �� to evaluate the damped Bessel

functions eJ�z� Im z � � �note that Im � � � for the standard parameterchoices in �� p� � p�q� � � and q� � ��

In �� May�eld showed for the attenuation�free case that the discretized

TBC for the SPE �� destroys the unconditional stability of the underly�ing Crank�Nicolson scheme and one can expect a similar behaviour for theWAPE� These existing discretizations also induce numerical re�ections at theboundary particularly when using coarse grids� Hence the existing discretizedTBC �� exhibits both stability problems and reduced accuracy whichmay require the usage of unnecessarily �ne grids�

Instead of using an ad�hoc discretization of the analytic TBCs like �� or�� we will construct discrete TBCs for the fully discretized half�space pro�blem as done in �� Our new strategy solves both problems of the discretizedTBC at no additional computational costs� With our DTBC the numericalsolution on the computational domain � � j � J exactly equals the dis�crete half�space solution �on j � IN�� restricted to the computational domain� � j � J � Therefore our overall scheme inherits the unconditional stabilityof the half�space solution that is implied by the discrete L��estimate ��

��

To derive the DTBC we will now mimic the derivation of the analytic TBCsfrom x� on a discrete level� For the initial data we assume ��

j � �� j � J � �and solve the discrete exterior problem in the bottom region i�e� the Crank�Nicolson �nite di�erence scheme �� for j � J ��

R�b � q#�

h

��n��

j � �nj � � i

�R�b �#�

h

��n��

j � �nj ��

with

�b � �� q��N�b �� R �

�k�p� � q�

h�

k� q �

k

�

q�p� � q�

k��

�b �k

�k��p� � �� p��q��N�

b ��

where #�

h�nj � �n

j��nj ��n

j�� and R is proportional to the parabolic meshratio� By using the Z�transform�

Zf�nj g � "�j�z� ��

�Xn��

�nj z

�n� z � IC� jzj � ��

�� is transformed to�z � � � iq�z � ��

�#�

h"�j�z� � �iR��b�z � �� i�b�z � ��

�"�j�z� ��

The solution of the resulting second order di�erence equation takes the form"�j�z� � �j� �z� j � J where ��z� solves

��

�� iR

�

�b�z � �� i�b�z � ��

z � � � iq�z � ��

��

For the decreasing mode �as j � �� we require j��z�j � �� We obtain theZ�transformed DTBC as

"�J��z� � ��

�z� "�J �z��

and in a tedious calculation this can be inverse transformed explicitly� TheDTBC for the SPE and the WAPE then reads�

�� iq��nJ�� n

J �n �nX

m��

�mJ �n�m� n � ��

with the convolution coe�cients �n �� iq�Z��f��

�z�g given by

�n �� iq �

i

�� i��e�i�

��n �

i

�H��nein�

� ��Qn�� e�i��Qn�� e�i

n��Xm��

��ei��n�mQm��

��

� � R �b� � � �R�b� � ��

rE

G� � �

F�pEG

� � �H�

jEj �

� � arg�� iq

� � iq� � � argE� � �

i

�jEj �� ei��

E � �� i�� q � i��

�� F � �� q� � ��

��

G � �� i�� q � i��

�� H � � � i� � �� i��e�i�

In �� n denotes the Kronecker symbol and Qn�� nPn�� the damped

Legendre polynomials �Q� � � Q�� In the non�dissipative case ��b � ��we have j�j � � � � �� and hence jPn��j � �� In the dissipative case�b � � we have j�j � � � becomes complex and jPn��j typically grows with n�In order to evaluate �n in a numerically stable fashion it is therefore necessaryto use the damped polynomials Qn�� in ��

The convolution coe�cients �� behave asymptotically as

�n � �iH��nein�� n��

which may lead to subtractive cancellation in �� note that �mJ � �m��

J ina reasonable discretization�� Therefore we use the following numerically morestable fashion of the DTBC in the implementation�

�� iq��nJ��

nJ � �� iq��n��

J�� n��Xm��

�mJ sn�m� ��

with sn �� n � ei��n�� n � �� The coe�cients sn are calculated as

sn �� iq�ei� �

i

�� i��

��n � �

Qn�� Qn��

�n� � ��

Alternatively they can be calculated directly with the recurrence formula

sn ��n� �

n��sn�� n� �

n��sn�� n � ��

once s� s� s� are computed from �� Using asymptotic properties of theLegendre polynomials �� one �nds sn � O�n�� n�� which agrees withthe decay of the convolution kernel in the di�erential TBCs ��

This decay of the sn motivates considering also a simpli�ed version of theDTBC �� with the convolution coe�cients being cut o� beyond an indexM � This means that only the �recent past� �i�e� M range levels� is taken intoaccount in the convolution in ��

�� iq��nJ��

nJ � �� iq��n��

J�� n��X

m�n�M

�mJ sn�m ��

This of course reduces the perfect accuracy of the DTBC �� but it isnumerically cheaper while still yielding reasonable results for moderate valuesof M � We remark that the resulting scheme does not conserve the discrete L��norm �cp� �� and hence the numerically stability of the simpli�ed DTBCis not yet known�

So far we did not consider the �typical� density jump at the sea bottom inthe DTBC �� In the following we review two possible discretizations ofthe water�bottom interface� For the usual grid zj j � IN� with Jh � zb thediscontinuity of � is located at the grid point zJ � In this case it is a standard

��

practice � �� to use �� with

�j �

� � ��w� j � J�� b�w�b��w

� j � J�

�b� j � J

��

As an alternative one may use an o�set grid i�e� zj � �j� �

��h �n

j �� zj� rn�j � ��J where the water�bottom interface with the density jump liesbetween the grid points j � J � � and J � For discretizing the matchingconditions in this case one wants to �nd suitable approximations for � and �at the interface zb & ��zb� and �eff � ��zb� such that both sides of thediscretized second matching condition ��b�

�

�w

�nJ � &

h��

�

�b

&� �nJ��

h��are equal to

�

�eff

�nJ � �n

J��

h ��

This approach results in an e�ective density �eff � ��w � �b�� based on adi�erent derivation this was also used in �� In numerical tests we found thatthe o�set grid with the above choice of �eff produces slightly better resultsthat have less Gibbs% oscillations at the discontinuity of �z at zb� This may beunderstood by the fact that �� requires a higher order derivation �using theevolution equation� than our derivation �� see also �� Becauseof the discontinuity of �z the higher order derivation yields �slightly� poorerresults� Therefore we choose the o�set grid for the implementation of theDTBC� At the surface we use instead of �n

� � � the o�set BC �n� � � �n

��Finally it remains to reformulate the DTBC �� such that the density

jump is taken into account� We rewrite the discretization of the second depthderivative at j � J from ��

h�h�J D�

h�

��J D�

h�

�nJ

�i� #�

h �nJ �

�� b

�eff

�� nJ � �n

J��

� ��

Comparing the r�h�s� of �� to �� we observe that only one additionalterm appears and instead of �� we get

" �J��z�� iR

�b�z � �� i�b�z � ��

z � � � iq�z � ��

�" �J�z� �

�b�eff

� " �J�z�� " �J��z��

��

Using " �J��z� � ��z�" �J�z� where ��z� denotes the solution of �� and

considering the fact that ��z� � ��

�z� is equal to the term in the squaredbrackets in �� we obtain the Z�transformed DTBC�

" �J�z�� " �J��z� ��eff�b

" �J�z��eff�b

��

�z� " �J�z� ��

��

Hence the DTBC including the density jump reads

�� iq��b�eff

�nJ��

�� iq�

�� b

�eff

�� nJ

� �� iq��b�eff

�n��J�� iq�

�� b

�eff

� �n��J �

n��Xm��

�mJ sn�m� ��

with the convolution coe�cients sn given by ��At the end of this Section we now address the question of nonuniform

depth discretizations� In the derivation of the DTBC we needed a uniformz�discretization for the exterior problem on z � zb i�e� j � J � �� For theinterior problem however a nonuniform discretization �even adaptive in r�may be used and this would not change our DTBC �� For any giveninterior z�discretization and a uniform grid spacing hb in the exterior domainthe DTBC will always yield on the interior domain the same solution as thecorresponding discrete half�space solution�

This raises a natural question� given an interior �possibly nonuniform� z�discretization what is the best uniform discretization of the exterior domain'To analyze this question we �rst consider the three types of errors that arerelevant here� Firstly the error associated with the given interior discretizationdoes not depend on the choice of hb� In order to avoid strong re�ections due tothe nonuniform grid we will assume that the interior grid spacing hj �� zj�zj��varies slowly with j� and can be represented as hj � h�zj� with a �smooth�function h�z�� To the authors% knowledge the re�ections in irregular gridshave not yet been theoretically analyzed for the Schr�odinger equation butvery similar e�ects appear in hyperbolic and parabolic equations � �� Innumerical tests however one can easily verify that discontinuities of h�z�would introduce spurious numerical re�ections of an incident wave �cp� � � andreferences therein�� Such re�ections can be largely reduced by �smoothing�such a discontinuity of h�z� �cf� Example � of x��

Secondly the discrete BC at zb may cause outgoing waves to be partiallyre�ected back into the computational domain and these re�ections stronglydepend on hb�

Finally for the discretization error of the �uniformly discretized� exteriordomain we have to distinguish between traveling waves and evanescent waves�In the �rst case the discretization error can be interpreted as a modi�cation tothe dispersion relation for the outgoing waves �incoming waves are not presentin the exterior domain�� But the accuracy of their propagation speed is irrele�vant as long as we are only interested in the solution in the interior domain�Hence the exterior discretization error can be disregarded for outgoing trave�ling waves� The discretization error of evanescent waves however in�uencesthe interior solution�

Since our DTBC is fully equivalent to a discrete half�space problem theabove discussion of the three error types can be completly reduced to theproblem of internal grid reections for the SPE or the WAPE� In the con�tinuous limit �hb � �� of the exterior discretization this also holds for thesemi�discrete approach of �� for the Schr�odinger equation� Following the

��

above discussion we can now give the best exterior discretization in the $tra�veling wave regime%� the uniform exterior grid spacing hb � h�zb� generates acompletely re�ection�free BC and the uniformity of the exterior grid ensuresthat the outgoing waves will never be re�ected back� Their inaccurate resolu�tion in the exterior domain only causes inaccurate wave speeds but this doesnot a�ect the interior solution�

This behaviour is numerically veri�ed in the simulations of x� in �� wherea uniformly spaced grid was compared to the semi�discrete approach for theexterior domain� There a Schr�odinger equation with a constant potentialis considered and hence the initial Gaussian wave packet consists only oftraveling wave modes in the exterior domain�

In the $evanescent wave regime% however the picture is not that simpleand it is not known yet whether there exists a exists a unique $best exteriordiscretization%� Our simulations of x� indicate that it may indeed be advan�tageous to use a DTBC that corresponds to a �ner exterior discretization aslong as the interior and exterior grid spacings are gradually matched to eachother�

�� Numerical Examples

In the �rst two examples of this Section we shall consider the SPE and theWAPE for comparing the numerical result from using our new discrete TBCto the solution using either the discretized TBC of Thomson and May�eld ��or an absorbing layer� Due to its construction our DTBC yields exactly �upto round�o� errors� the numerical half�space solution restricted to the com�putational interval �� zb�� The simulation with discretized TBCs requires thesame numerical e�ort� However their solution may �on coarse grids� stronglydeviate from the half�space solution�

In each example we used the Gaussian beam from �� as initial data� Belowwe present the transmission loss �� log�� jpj� where the acoustic pressure pis calculated from ��

Example �� This is a well�known benchmark problem from the literature�� In this example the ocean region �� z � ��m� with theuniform density �w � �� gcm�� is modeled by the SPE �� It contains noattenuation and a large density jump ��b � �� gcm�� at the water�bottominterface� Hence this problem provides a test of the treatment of the densityjump in the TBCs applied along zb � ��m�

The source of f � ��Hz is located at a water depth zs � ��m and thereceiver depth is at zr � ��m� The sound speed pro�le in water is givenby c�z� � �� j�� zj � �ms�� and the sound speed in the bottom iscb � ��ms�� For our calculations up to a maximum range of �� km weused a reference sound speed c� � ��ms�� and a uniform computationalgrid with depth step #z � �m and range step #r � �m �the same step sizeswere used in ��

In Figure � the solid line is the solution with our new discrete TBC ��and the dotted line is obtained with the discretized TBC �� The discreti�zed TBC clearly introduces a systematic phase�shift error which is roughly

��

proportional to #z� The discretized TBC also produces arti�cial oscillations�cf� the zoomed region� while our new DTBC yields the smooth solution withthe same numerical e�ort�

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|^

2Example 1

Figure �� Transmission loss at zr � ��m for Example ��the solution with the new discrete TBC �� coincides with thehalf�space solution while the solution with the discretized TBC�� introduces a phase�shift and arti�cial oscillations�

Figure � compares the results of our new discrete TBC �solid line� to thesolution obtained with an absorbing layer of ��m thickness �dotted line�and a homogeneous Dirichlet BC at zmax � ��m� Hence the computationtook about twice as long as by using the discrete TBC� In our experimentswe obtained a better match to the $exact% half�space solution by using theexponential absorption prole

�b�z� � ��hexp

n�

z � zbzmax � zb

o� �

idB ��b� zb � z � zmax� ��

rather than a linear pro�le� We remark that the pro�le �� and thicknessof the absorbing layer were designed as to yield a close match to the $correct%solution� Without such an $a�posteriori data �tting% however a calculationwith an absorbing layer would usually yield a solution with a somewhat largerdeviation than suggested by Figure �� With a thicker layer one can of coursestill improve the results of Figure � e�g� no more arti�cial oscillations arevisible when using a layer of � �m�

��

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 �� Transmission loss at zr � ��m for Example ��the solution with an absorbing layer of ��m �� is quite sa�tisfactory in comparison to the $exact% solution computed withthe discrete TBC �� It is in phase but shows some arti�cialoscillations and overestimates the transmission loss at km �kmand in the range � ��km�

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 �� Vertical cut of the � solutions at r � �� km forExample �� j��z� r � �� km�j

Figure � shows the signi�cant deviations of the solutions using either thediscretized TBC or an absorbing layer of ��m from the computed half�spacesolution which coincides with the solution using our new DTBC�

�

Example �� This example appeared as the NORDA test case �B in the PEWorkshop I � �� The environment for this example consists of anisovelocity water column �c�z� � ��ms�� over an isovelocity half�space bot�tom �cb � ��ms�� The density changes at zb � ��m from �w � �� gcm��

in the water to �b � �� gcm�� in the bottom� The source and the receiverare located at the same depth near the bottom� zs � zr � ��m� The sourcefrequency is f � ��Hz� The attenuation in the water is zero and the bottomattenuation is �b � �� dB ��b where �b � cb�f denotes the wavelength ofsound in the bottom� Here the steepest angle of propagation �which is theequivalent ray�angle of the highest of the �� propagating modes� is appro�ximately �� cf� � �� Since the source is located near the bottom thehigher modes are signi�cantly excited� Therefore the wide angle capability isimportant here and we use the WAPE �� with the coe�cients of Claerbout�to solve this benchmark problem�

The maximum range of interest is �� km and the reference sound speedis chosen as c� � ��ms�� The calculations were carried out using thedepth step #z � ��m and the range step #r � ��m� Since the source isplaced close to the bottom the TBC was applied ��m below the ocean�bottominterface �the same was done in ��

new discrete TBCdiscretized TBC

5 6 7 8 9 10

60

70

80

90

100

Range r [km]

Tra

nsm

issi

on L

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−10

*log

_10

|p|^

2

Example 2

Figure �� Transmission loss at zr � ��m for Example ��the solution with the new discrete TBC coincides with the half�space solution while the solution with the discretized TBC stilldeviates signi�cantly from it for the chosen discretization�

The typical feature of this problem is the large destructive interference nullat a range of � km� Figure � compares the transmission loss results for thediscrete and discretized TBCs in the range from � to �� km� In a second com�parison we extended the computational domain up to ��m� With the givenbottom attenuation this ��m layer is thick enough to yield the reasonableapproximation shown in Figure ��

�

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

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−10

*log

_10

|p|^

2

Figure �� Transmission loss at zr � ��m for Example ��in comparison to the exact half�space solution the truncationof the computational domain at ��m �the given bottom atte�nuation then represents an absorbing layer of ��m� introducesa slight phase shift�

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 �� Vertical cut of the � solutions at r � �km forExample �� j��z� r � �km�j

Figure shows the deviation of the solutions with the discretized TBCand with the absorbing layer from the computed half�space solution whichcoincides with the solution using our new discrete TBC�

��

Example �� In this example we illustrate the theoretical �ndings of x� oncoupled models� We use the physical parameters of the �rst two examples butdi�erent models for the water and the bottom region�

We start with considering the environment of Example � and compare theresults of di�erent model couplings� First we �x the WAPE of Claerbout�CWAPE� p� � � p� �

�

� q� �

�

�� in the bottom and choose a di�erent �and

in fact better� rational approximation �GWAPE� for the water region thatful�lls the coupling condition �� p� � � q�� The two remaining parameters

p� q� are then determined by minimizing the approximation error of ��

�

�in the maximum norm� over the interval �� which contains thediscrete spectrum of L� p� � �� q� � �� We compare thisapproximation to the case of also using the CWAPE in the water� Furthermorewe show the results when using the SPE in the sea bottom �which clearlyviolates �� and when using the SPE in both regions�

GWAPE / CWAPE

CWAPE / CWAPE

CWAPE / SPE

SPE / SPE

7 8 9

60

70

80

90

100

Range r [km]

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issi

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−10

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_10

|p|^

2

Example 3

Figure � Transmission loss at zr � ��m in several coupledmodels �water and sea bottom� for the simulation of Example ��

Figure � displays a comparison of the transmission loss from �� to � kmfor these di�erent couplings� It turns out that the solution for the coupledGWAPE(CWAPE model is very close to the one using the CWAPE in bothmedia� While the CWAPE(SPE model violates the coupling condition it onlydeviates from the above solutions by a slight phase�shift that is typical for theSPE in this example �cp� also the �pure� SPE model��

Now we turn to the dissipation�free situation of Example � and focus ourattention on a conservative discretization of coupled models that satisfy thecoupling condition p��z��q��z� � � � const and hence preserve the L��IR��

��

��dz��norm �see x�� As a discrete analogue of �� we obtain in thedissipation�free case

hD�

k

J��Xj��

�� nj

�� j

� � �

p��k��eff

Im

��p� � q��

n� �

�

J�� iq�k�� D�

k �nJ��

�

��p� � q��D

�

h �n� �

�

J�� iq�k��D�

kD�

h �nJ��

�i� ��

with �j � �� zj� and � � p��p��p�q�� Analogously a discrete version of ��

can be shown for the bottom region j � J �

hD�

k

�Xj�J

�� nj

��b�b

��

�pb��k

��eff

Im

��pb

�� qb

��

n� �

�

J�� iqb�k�� D�

k �nJ��

�

��pb

�� qb

��D�

h �n� �

�

J�� iqb�k��D�

kD�

h �nJ��

�i� ��

with �b � �pb��pb

��pb

�qb�� For coupled models � usually takes di�erent values

in the water and bottom regions� It follows from �� that the weighteddiscrete L��norm on j � IN� is preserved�

k �nk� � hJ��Xj��

�� nj

�� j

� h�Xj�J

�� nj

��b�b

� const ��

provided that the coupling condition �� is ful�lled�

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 � Coupled WAPE�models conserve the discrete L��norm �� only when satisfying the coupling condition p��q� �� const ��

��

Figure � illustrates that the discrete L��norm �� is conserved as longas the coupling condition �� is satis�ed� In all four simulations we usedthe WAPE of Claerbout for the water region and di�erent models in the seabottom� only the hybrid WAPE�model with constant p��z��q��z� � � � �renders the scheme conservative �only for this numerical illustration we choosethe values p� � � q� � �� A coupling to the SPE �like in �� or toa WAPE in the bottom with pb

��qb

�� b �� all yields a non�conservative

scheme� We point out that these schemes are not only non�conservative forthe particular norm �� but also for any other weighted L��norm�

In the simulations for Figure � the second sum of �� for the exterior ofthe computational domain� was evaluated via ��

Example �� In this example we illustrate our discussion from x� on the $bestuniform exterior discretization% for the case of evanescent waves� We want toanswer the following question� given a uniform interior discretization can theresult from a globally uniform z�discretization be improved by choosing a �nerexterior z�discretization or equivalently by using a DTBC that correspondsto such a �ner discretization'

As a model problem for this test we consider the SPE �� on z � �� r � �with a homogeneous Dirichlet BC at z � � k� � �m�� and the �potential well�V �z� � �� z � zb � ��m V �z� � Vb � �� z � zb� In this example planewaves with a wave number k � kcrit �

p��m�� are evanescent in the exterior

domain z � zb and k � kcrit transmits a traveling wave into the exterior� Wechoose here the Gaussian beam exp�ikz��m��z��m�� with k � �m��

as an initial condition� For this choice of k $most% of the Fourier componentsof this wave correspond to evanescent modes in the bottom� Hence this wavewill be predominantly re�ected back into the interior domain�

Figure � compares the e�ect of choosing di�erent �uniform and nonuniform�z�discretizations� We show the results of this simulation at the range r � ��mwhen the wave packet has been re�ected back from the water�bottom interface�The solid line was obtained with the uniform z�discretization h� � ��mand it will serve as our $exact% reference solution� The dashed line showsthe solution with the uniform grid spacing h� � ��m� In the followingcomparisons we will keep this coarser interior grid and will vary the uniformexterior grid� Following our discussion from x� we used a gradual transitionbetween these two grid spacings in the depth interval �� m �piecewiselinear grid spacing function h�z��

The dotted curve of Figure � gives the results with the �ner exterior z�discretization h� � ��m� Close to the sea bottom it shows signi�cant impro�vements over the uniform discretization with h�� In the interval � � z � �mboth curves almost coincide as the interior discretization error is dominantthere and it implies inaccurate wave speeds that are re�ected in the clearlyvisible phase shift� The dotted curve still exhibits this phase shift up tothe sea bottom at ��m but for the dashed curve the error in the interval��m � z � ��m is dominated by the e�ect of the exterior discretization� Itthus seems that the e�ect of the reduced exterior discretization error �due to

��

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 �� The $best uniform exterior z�discretization% may be�ner than the interior discretization� Vertical cut of the three so�lutions at r � ��m for Example �� the solution �� calculatedon a nonuniform grid ��ner grid in the exterior domain than inthe interior� is more accurate in the interior domain than the so�lution obtained on a uniformly coarse grid �� The referencesolution �� was calculated on a uniformly �ne grid�

the �ner exterior discretization� may outweigh �in the interior domain)� theadditional re�ection errors incurred by the nonuniform grid�

The L�� errors �w�r�t� the solid curve� of the solutions with the uni�form h��discretization and the nonuniform h� � h��discretization are respec�tively �� and �� Using an even �ner exterior discretization doesnot seem to improve the result much further �L��error �� for the h� � h��discretization�� A �ner exterior discretization would however require a thickerregion to adapt the two grids�

We thus conclude that �ner exterior discretizations may indeed be advan�tageous in the case of evanescent waves and for large ranges these are theimportant modes in the considered applications of underwater acoustics�

��

�� Conclusions

We have derived a new discretization �discrete TBC � of the TBC for theWAPE of acoustics� It is of discrete convolution form involving the boundarydata from the whole �past range�� The convolution coe�cients sn are calcula�ted via a simple three�term recurrence relation and they decay like O�n��Since our new DTBC has the same convolution structure as existing discretiza�tions it requires the same computational e�ort but improves two shortcomings�DTBCs are more accurate �in fact as accurate as the discrete half�space pro�blem� and they yield an unconditionally stable scheme�

We point out that the superiority of DTBCs over other discretizations ofTBCs is not restricted to the WAPE or to our particular interior discretizationscheme �see e�g� � �� The crucial point of our derivation was to �ndthe inverse Z�transformation of �� explicitly� In more general applications�e�g� higher order Pad�e approximations or �D�problems� it might be necessaryto derive the convolution coe�cients in �� through a numerical inverse Z�transformation �� but this does not change the e�ciency and stability of thepresented method� As a general philosophy DTBCs should be used wheneverhighly accurate solutions are important�

Appendix A� Proof of Theorem ��well posedness of the WAPE�

In Theorem � we assumed that V � �� L��IR�� Then the Schr�odingeroperator

L � �k�� z��

��z� � V �z� �A��

with a homogeneous Dirichlet BC at z � � is self�adjoint in L��IR�� dz�with the dense domain

D�L� � H�

� �IR�� j��z � H��IR��

� �A��

We now consider the operator f�L� � p��p�L��q�L

de�ned as

f�L� �

Z �

��

f�� dP�� A��

with dP� denoting the projection valued spectral measure of the operator L�cf� �� According to �� Th� XII�� the domain of f�L� is dense in

L��IR�� dz� if and only if � � q��

the pole of f�� is not an eigenvalue of L�In this case f�L� is self�adjoint and by Stone%s Theorem �� ik�f�L� generatesa unitary C��group on L��IR�� dz� which yields the unique solution to��

If � coincides with an eigenvalue �j of L then �� still admits a uniquemild solution for all initial data in the orthogonal complement of �j the uniqueeigenfunction corresponding to �j� Theorem � generalizes the well�posedness

analysis for the WAPE on �nite intervals given in �� There however � caneasily lie in the �pure eigenvalue� spectrum of L what then restricts the classof admissible initial conditions�

��

Appendix B� WAPE SPE coupling

Here we discuss the mathematically sound formulation of the coupledWAPE�SPE model for the simple model case of constant c and �� We �rst consi�der the pseudo�di�erential operator f�L� appearing in the WAPE �� withL � �k��

��z � Due to the BC at z � � it can be expressed in terms of Fourier�

sine transforms as�f�L��

��z� �

�

�

Z �

�

Z �

�

*��y� sin��y� sin��z� dy d�� B��

with the symbol

*�� p� � p�k

��

�� q�k��

�B��

In the coupled WAPE�SPE model one would formally want to write theevolution equation as

�r � ik�A� �B��

with

A� �

� � ��p� � p�k

��z

� � q�k�� z

� �

�� z � zb�

k��

��z�� z � zb

�B��a�

�B��b�

However as the pseudo�di�erential operator in �B��a� is non�local actingon L��IR�� it cannot be simply restricted to the interval � � z � zb� It istherefore appropriate to de�ne the coupled evolution equation on the symbollevel of the two involved operators �cf� �� Without attenuation both theSPE and the WAPE conserve the L��norm and the discrete analogue of thisconservation is the main ingredient for showing unconditional stability of the�nite di�erence scheme in x�� Therefore we postulate that the coupled modelalso has to conserve the L��norm� This can be achieved if the operator A onthe right hand side of �B�� is interpreted as the Weyl operator �see ��

A��z� ��

�

Z �

�

Z �

�

a

�y � z

��

��y� sin��y� sin��z� dy dz �B��

to the symbol

a�z� ��

*�� z � zb�

�k��

�� z � zb

�B� �

As a�z� �� is real one readily veri�es that the evolution equation �B�� B��conserves the L��norm�

Due to the pole of the symbol *�� it would be quite di�cult to appropriatelydiscretize �B�� B�� and it is beyond our scope here� We remark that �nitedi�erence schemes of pseudo�di�erential equations with smooth symbols haverecently been studied in ��

��

ACKNOWLEDGEMENT

The �rst author acknowledges partial support by the DFG �Grant No� MA� �(�� and the NSF under Grant No� DMS�� and the second aut�hor was funded by the DFG under Grant No� MA � �(��

The �rst author acknowledges fruitful discussions with J� Douglas Jr� andthe second author interactions with F�B� Jensen� We are grateful to the twoanonymous reviewers for many suggestions to improve and clarify this work�

��

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