transparent boundary conditions for the wave equation—a kirchhoff point of view

Research Article

Received 12 April 2011 Published online in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.1524MOS subject classification: 35C15; 35L05; 65M99

Transparent boundary conditions for the waveequation—a Kirchhoff point of view‡

George C. Hsiao*† and Richard J. Weinacht

Communicated by Y. S. Xu

We reduce an exterior initial boundary value problem for the wave equation in three space dimensions to an initial bound-ary value problem on a bounded computational domain bounded by an artificial boundary as well as the original bound-ary. From a Kirchhoff representation formula for the solution of wave equation in an exterior domain, we derive on theartificial boundary (exact) transparent boundary conditions that are nonlocal in space and time. These lead to local approx-imate transparent boundary conditions of the first and second orders. It is shown that these approximate transparentconditions are satisfied exactly for a related spherically symmetric problem. Copyright © 2012 John Wiley & Sons, Ltd.

Keywords: wave equation; exterior initial boundary value problem; transparent boundary conditions; Kirchhoff representationformula

1. Introduction

The method of artificial boundaries is a device to obtain numerical solutions of partial differential equation problems in unboundeddomains. For equations of elliptic type (nonlinear as well as linear), this approach has led to the coupling procedure that combinesboundary elements and finite elements and for which there is a huge literature (see, for example, [7, 14, 15]). It is recognized nowa-days as a very efficient method. The basic idea for the coupling procedure in the elliptic case is the reduction of the partial differentialequation in the exterior domain to suitable boundary integral equations on the artificial boundary. The transmission conditions yieldnonlocal boundary conditions (in the spatial variables).

It is tempting to apply the coupling procedure to time-dependent problems, in particular, for wave propagation on unboundeddomains. But the situation in this case is much more complicated than in the elliptic case because the boundary conditions derived fromthe relevant boundary integral equations are nonlocal in time as well as in space. One way to approach the problem is to approximatethe (exact) transparent boundary conditions by accurate approximate boundary conditions that are local in time so that at various dis-crete time levels, existing numerical schemes for elliptic problems can be readily applied. We do not pursue this approach further here.

There has been a considerable amount of work concerning transparent boundary conditions for the wave equation ([1, 2, 4, 5], toname a few; see the many citations in the review [20]). Looking through the existing literature, most of the transparent conditions arederived by using series solutions or asymptotic developments for either the symbols or the Laplace transform of the solutions of thewave equation. In this paper, we derive (exact) transparent conditions on the artificial boundary directly from the Kirchhoff representa-tion formula for the solution of the wave equation in the domain exterior to the artificial boundary. Ting and Miksis [19] and Givoli andCohen [9] also used Kirchhoff’s formula but introduced an intermediate surface on which to compute numerical boundary values.

In the next section, we begin with the formulation of an exterior initial boundary value problem and reduce it to an initial boundaryvalue problem on a bounded domain bounded partially by an arbitrary smooth surface on which a nonlocal boundary condition is pro-vided by Kirchhoff’s formula. In Section 3, we choose the artificial boundary to be a sphere and explore (exact) transparent boundaryconditions and local approximations of first and second orders.

Remark 1The term transparent boundary condition on an artificial boundary for an exterior problem is generally accepted to mean a relation thatis satisfied exactly by the solution of the exterior problem in the absence of the artificial boundary. For emphasis, in this paper, we usethe admittedly redundant term (exact) transparent boundary condition to make a contrast with approximations thereto. In particular,

Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA*Correspondence to: George C. Hsiao, Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA.†E-mail: [email protected]‡Dedicated to Robert P. Gilbert collaborator, colleague and friend.

Copyright © 2012 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2011

G. C. HSIAO AND R. J. WEINACHT

we use the term local approximate transparent boundary condition (LATBC) to refer to a local boundary condition that approximates an(exact) transparent boundary condition. In the relevant literature, the terms nonreflecting boundary conditions and absorbing boundaryconditions are also used, sometimes synonymously with our meanings (see, e.g., [8]).

2. Exterior initial boundary value problem

Let � be a bounded domain in R3 with smooth boundary @�. A general exterior initial boundary value problem (EIBVP) for the waveoperator consists of the nonhomogeneous wave equation

utt � c2�xuD f .x, t/ in �c � .0,1/, (1)

together with the initial conditions

u.x, 0/D u0.x/, ut.x, 0/D v0.x/, and x 2�c (2)

and the Dirichlet boundary condition

u.x, t/D g.x, t/ on @�� .0,1/. (3)

Here, �c D R3 n N� denotes the exterior domain, c > 0 is the wave speed, and f , u0, v0, and g are given sufficiently smooth func-tions satisfying compatibility conditions to ensure the existence of classical solutions with a certain degree of smoothness. Only fordefiniteness we have considered a Dirichlet boundary condition.

For numerical computation, it is usual to replace the above problem by one in a bounded computational domain by introducingan artificial smooth surface � that completely surrounds � and by restricting t to Œ0, T� for some chosen positive T . It is known thatthe solution of EIBVP for the domain exterior to � has a representation via Kirchhoff’s formula (usually given [18, 22] for homogeneousinitial conditions, but see [12]). For simplicity, in this paper, we confine ourselves to f D u0 D v0 D 0. Then, the solution u of EIBVPadmits the representation

u.x, t/D

Z�

@

@n�E.x, �/ u

�� , t�

jx � �j

c

�ds�

�

Z�

E.x, �/

@

@n�

jx � �j

c

!Pu

�� , t�

jx � �j

c

�ds� �

Z�

E.x, �/@u

@n�

�� , t�

jx � �j

c

�ds� (4)

for .x, t/ in .ext�/� .0,1/where E.x, �/ is the fundamental solution for the negative of the Laplacian

E.x, �/ :D1

4�

1

jx � �j,

and Pu is the derivative with respect to the time variable so that in equation (4),

Pu

�� , t�

jx � �j

c

�:D

@

@zu.x, z/j

zD�� ,t� jx��jc

�.

In equation (4) and below, it is understood that the integrands vanish when the retarded argument t � jx � �j=c is negative; withappropriate compatibility conditions there is no jump in u or its derivatives when the retarded argument becomes negative.

For fixed t, as x approaches � from outside, we obtain the boundary integral equation of the form

u.x, t/D1

2u.x, t/C KŒu�.x, t/� QVŒPu�.x, t/� V

�@

@nu

�.x, t/ (5)

for .x, t/ in � � .0,1/, where Œ � indicates a retarded argument so that in equation (5), Œv� :D v.� , t� .jx � �j/=c/. The boundary integraloperators V , QV , and K are defined, respectively, by

KŒ��.x, t/D

Z�

@

@n�E.x, �/ �

�� , t�

jx � �j

c

�ds� , (6)

QVŒ P��.x, t/D

Z�

E.x, �/@

@n�

�jx � �j

c

�P�

�� , t�

jx � �j

c

�ds� , (7)

VŒ�.x, t/D

Z�

E.x, �/

�� , t�

jx � �j

c

�ds� . (8)

Then, the boundary integral equation (5) may be written more briefly as

1

2uD KŒu�� QVŒPu�� V

�@u

@n

�on � � .0,1/, (9)



which can be considered as a nonlocal boundary condition for the solution in the annular domain bounded by @� and � . It isworth mentioning that in contrast to the elliptic case, the nonlocal boundary condition is nonlocal both in space and time. Thismakes the implementation of equation (9) in numerical computation more complicated. In the present context, the explicit use ofthe Dirichlet-to-Neumann map [8] further complicates the numerical computation, and we do not discuss it further.

3. Spherical artificial boundaries

We now consider the special case when � is a sphere of radius R with R chosen so that the corresponding ball BR :D fx 2 R3 : jxj < Rgcontains�; the origin of coordinates is chosen to be in�. We rewrite equation (9) in the form

u.x, t/D OKŒu�.x, t/� QVŒPu�.x, t/� V

�@u

@n

�.x, t/, (10)

where in equation (10) and in what follows the notation OKŒu� indicates subtraction of u at .x, t/ in �R � .0, T/:

OKŒu�.x, t/ :D

Z�R

@

@n�E.x, �/

nu�� , t�

r

c

�� u.x, t/

ods�

with rD jx � �j. The boundary operators QV and V are defined in equations (7) and (8) but with � replaced by �R.Equation (10) is an (exact) transparent boundary condition on �R � .0, T/, that is, it is satisfied (exactly) by the solution of the EIBVP. If

g has a compact support, then for any given positive T , one can choose R sufficiently large so that u.x, t/� 0 for x on �R and 0� t � T ,thanks to Huygens’ principle. Thus, u D 0 on �R � .0, T/ is an approximate transparent boundary condition. And, indeed, this bound-ary condition has been sometimes used for numerical purposes. But it is known that one can do better, especially since the work ofEngquist and Majda [4, 5] .

3.1. First-order local approximate transparent boundary conditions

At first glance, it is tempting to get an LATBC by merely disregarding the nonlocal terms, that is, replacing the argument .� , t � .r=c//by .x, t/. This results in the boundary condition

u.x, t/C R@u

@n.x, t/C

R

2cPu.x, t/D 0 (11)

because Z�R

E.x, �/ds� D R,

Z�R

E.x, �/@r

@n�ds� D

R

2.

Like the boundary condition u D 0, the boundary condition (11) is a bit heavy-handed especially if we seek an approximate localboundary condition that is a good approximation to an (exact) transparent boundary condition—good not merely for large R. Moreover,the boundary condition (11) ignores the fact that in the integralZ

�R

@

@n�E.x, �/u

�� , t�

r

c

�ds� ,

there are contributions to the first derivatives of u as we now show.For fixed x, we may rotate the coordinate system so that x is at the north pole. Then, an integration by parts with respect to the polar

angle ' yields from equation (11)

u.x, t/D� Vu.x, t/� 2 QV.x, t/

C

Z�R

E.�x, �/ ru�� , t �

r

c

�� .�/ds�

C1

2

�u.x, t/� u

��x, t�

2R

c

�, (12)

where is a unit tangent vector,

.�/D cos.�/ cos.'/OiC sin.�/ cos.'/Oj� sin.'/Ok.

Again, equation (12) is an (exact) transparent boundary condition, but it lacks symmetry, which can be attained by first adding andsubtracting in equation (10)

@u

@n.x, t/ in the integrand for V ,

Pu.x, t/ in the integrand for QV ,

u.� , t/ in the integrand for OK



and then performing the above integration by parts. This results in the (exact) transparent boundary condition

LuD Nu, (13)

where L is the local term,

Lu.x, t/ :D u.x, t/C R@u

@n.x, t/C

R

cPu.x, t/,

and N is nonlocal in space and time,

Nu.x, t/ :D

Z�R

@

@n�E.x, �/ fu.� , t/� u.x, t/gds�

�

Z�R

E.x, �/

(@u

@n�u�� , t�

r

c

��@u

@nxu.x, t/

)ds�

�2

c

Z�R

E.x, �/@r

@n�

nPu�� , t �

r

c

�� Pu.x, t/

ods�

C

Z�R

E.�x, �/nru

�� , t �

r

c

��ru.� , t/

o� .�/ds�

C1

2

�u.�x, t/� u

��x, t �

2R

c

�,

which has a nice symmetry of differences of functions.For an approximate boundary condition that is local in time but still nonlocal in space, one merely chooses all time variables equal

to t, yielding

L.u/D

Z�R

@

@n�E.x, �/ fu.� , t/� u.x, t/gds�

�

Z�R

E.x, �/@r

@n�fPu.� , t/� Pu.x, t/gds�

�

Z�R

E.x, �/

(@u

@n�u.� , t/�

@u

@nxu.x, t/

)ds� ,

which is still fairly cumbersome for purposes of computation.If we summarily disregard all nonlocal terms in equation (13), we arrive at the LATBC

u.x, t/C R@u

@n.x, t/C

R

cPu.x, t/D 0, (14)

which could be regarded as the first iterate in the iteration scheme

L.umC1/D N.um/,

with u0 D 0. Moreover, equation (14) enjoys the feature of being an (exact) transparent boundary condition in the case where � is aball and the data are spherically symmetric, that is, for the EIBVP when

�D fx : jxj< ag

and

g.x, t/D Qg.t/ on jxj D a

with solution

Qu.x, t/D

(ajxjQg�

t � .jxj � a/=c�

, t > .jxj � a/=c,

0, otherwise.

with the compatibility condition Qg.0/D 0 and higher-order compatibility conditions for a smooth classical solution. We will refer to thisparticular exterior problem as the spherically symmetric case.



3.2. Second-order local approximate transparent boundary conditions

Remark 21. It is inviting to get second-order LATBCs by exploiting the fact that terms in Nu consists of departures of values from a local point

and to expand in a Taylor series (or use a Pade approximation). For example, in the second term in Nu,

@u

@n�

�� , t�

� � xj

c

��@u

@nx.x, t/D .� � x/ � r

@u

@n.x, t/

�j� � xj

c

@Pu

@n.x, t/CO.j� � xj2/

so that upon integration, the term

�

Z�R

E.x, �/

(@u

@n�

��t�j� � xj

c

��@u

@n.x, t/

)ds�

equals explicitly the given second-order derivatives

�r@u

@n.x, t/ �

Z�R

.� � x/E.x, �/ds� C1

c

@Pu

@n.x, t/

Z�R

j� � xjE.x, �/ds�

D2

3R2 @

2u

@n2.x, t/C

R2

c

@Pu

@n.x, t/

plus Z�R

E.x, �/O.j� � xj2/ds� .

But the last term is O.1/ in j� � xj, that is, of the same order as the previous terms. Thus, the terms omitted are of the same orderas those included, and this casts doubt on the value of the expansion.

One gets the same disappointing result if one expands in the angular variables in a coordinate representation of �R. With arotation so that x is at the north pole of �R, the second term in Nu equals

R

4�p

2

Z 2�

0

Z �

0

sin'

.1� cos'/1=2

(@u

@n�

R cos � sin', R sin � sin', R cos', t �

p2R.1� cos'/1=2

c

!�@u

@n.0, 0, R, t/

)d'd� ,

from which an expansion about ' D 0 gives an integrand

' sin'

.1� cos'/1=2

(Rr

@u

@n.x, t/ � .�/�

p2R

c

@Pu

@n.x, t/

sin'

.1� cos'/1=2

)CO.'2/

But again upon integration, the terms omitted are of the same order as the explicitly given second-order derivative terms. Thus,this expansion is of doubtful value.

2. Below, we use the following heuristic. From the view on �R, the region� for large R does not differ much from a point, or better, asmall ball. Moreover, if for each t one uses the (trivially spherically symmetric) mean value of the Dirichlet datum g over @� as Qg forthe spherically symmetric case, one can hope that for large R, the spherically symmetric Qu will not differ greatly on �R�.0, T/ fromthe solution u of the original EIBVP (at least when g has compact support). This hope is encouraged by the first-order boundarycondition (11).

For an LATBC of second order, we can for fixed x differentiate equation (14) with respect to t to have

PuC R@Pu

@nC

R

cRuD 0 (15)

The left-hand side of equation (14) when differentiated in the normal direction, however, need not yield 0. But in the sphericallysymmetric case, the equation

Qu.�, t/C �@Qu

@�.�, t/C

�

cPQu.�, t/D 0

is an identity in � and t, and hence, differentiation with respect to � yields

2@Qu

@�C �

@2 Qu

@�2C

1

cPQuC

�

c

@PQu

@�D 0,



which corresponds in the general (not spherically-symmetric) case to the boundary condition

2@u

@nC R

@2u

@n2C

1

cPuC

R

c

@Pu

@nD 0, (16)

which is another LATBC and which is expected to a good approximation for large R.Adding R=c times equation (15) to R times equation (16) yields another second-order LATBC

R2 @2u

@n2C 2

R2

c

@Pu

@nC

R2

c2RuC 2R

@u

@nC 2

R

cPuD 0. (17)

Finally, if we add 2 times equation (14) to equation (17), we obtain

R2 @2u

@n2C 2

R2

c

@Pu

@nC

R2

c2RuC 4R

@u

@nC 4

R

cPuC 2uD 0,

which is the second-order LATBC given by Bayliss and Turkel [2] and which is claimed to be a good approximation for large R. Clearly,all of the LATBCs given above are satisfied in the spherically symmetric case.

We have not completed numerical computations that compare the relative merits of the various second-order LATBCs.

Remark 3For smooth classical solutions u of the wave equation (1) (with f � 0), the functions Pu and u,j (the partial derivative with respect to thejth Cartesian space coordinate) also satisfy the wave equation, and thus, Kirchhoff’s formula is valid if u is replaced by Pu or u,j . Therefore,we have exactly

LPuD N Pu and Lu,j D Nu,j

and approximately

LPuD 0 (18)

and

Lu,j D 0. (19)

The boundary condition (18) is precisely equation (15) and is satisfied (exactly) in the spherically symmetric case, whereas condition(19) is not satisfied in the spherically symmetric case and does not transform into equation (16). If we substitute equation (18) intoequation (14), we obtain as a second-order LATBC

R

cPuC

R2

c

@Pu

@nC

R2

c2RuD 0, (20)

which is satisfied (exactly) in the spherically symmetric case. This is to be expected because equation (20) is R=c multiplied by the tderivative of equation (18). Thus, we have equation (20) as another second-order LATBC.

It is interesting to note the following point concerning equation (19), which has not been used thus far. Multiplying equation (19) byxj=R (and summing on j) yields

@u

@nC R

xj

R

xk

Ru,jk C

R

c

@Pu

@nD 0,

and then substituting both equations (18) and (19) into equation (14) results in the second-order LATBC

R

cPuC R2 @

2u

@n2C 2

R2

c

@Pu

@nC

R2

c2RuD 0 (21)

because

xj

R

xk

Ru,jk D

@2u

@n2�

1

R

@u

@n. (22)

Using equation (22), we see that equation (19) is equivalent to

R@2u

@n2C

R2

c

@Pu

@nD 0,

so that if we use both equations (18) and (19), then the LATBCs (20) and (21) are equivalent, but condition (20) holds without the use ofequation (19).



4. Concluding remarks

1. Clearly the procedure given above can be used to generate higher-order LATBCs.2. We note that we have obtained (exact) transparent boundary conditions and LATBCs for exterior problems in linear elastodynam-

ics by using the corresponding Kirchhoff formula (Love’s theorem) [6, 21]. An analogous (exact) transparent boundary conditionwas obtained in [9] but again as in [19] with an additional artificial boundary.

References1. Barry A, Bielak J, MacCamy RC. On the absorbing boundary conditions for wave propagation. Journal of Computational Physics 1988; 79:449–468.2. Bayliss A, Turkel E. Radiation boundary conditions for wave-like equations. Communications on Pure and Applied Mathematics 1980; 33:707–725.3. Engquist B, Halpern L. Far field boundary conditions for computation over long time. Applied Numerical Mathematics 1988; 4:21–45.4. Engquist B, Majda A. Absorbing boundary conditions for the numerical simulation of waves. Mathematics of Computation 1977; 31:629–651.5. Engquist B, Majda A. Radiation boundary conditions for acoustic and elastic wave calculations. Communications on Pure and Applied Mathematics

1979; 32:313–357.6. Cemal EA, Suhubi ES. Elastodynamics, Linear Theory 2. Academic Press: New York, 1975.7. Gatica GN, Hsiao GC. Boundary-field Equation Methods for a Class of Nonlinear Problems, Pitman Research Notes in Mathematics Series 331. Longman:

New York, 1995.8. Givoli D. Numerical Methods for Problems in Infinite Domains. Elsevier: Amsterdam, 1992.9. Givoli D, Cohen D. Nonreflecting boundary conditions based on Kirchhoff-type formulae. Journal of Computational Physics 1995; 117:102–113.

10. Hagstrom T. Radiation boundary conditions for the numerical simulation of waves. Acta Numerica 1999; 8:47–106.11. Hagstrom T, Hariharan SI, MacCamy RC. On the accurate long-time solution of wave equation in exterior domains: asymptotic expansions and

corrected boundary conditions. Mathematics of Computation 1994; 63:507–539.12. Hsiao GC, Weinacht RJ. A Representation Formula for the Wave Equation Revisited. Technical Report, Department of Mathematical Sciences,

University of Delaware, May, 2009.13. Ionescu D, Igel H. Transparent boundary conditions for wave propagation on unbounded domains. International Conference on Computational

Science—ICCS, 2003; 500–506.14. Johanson C, Nedelec JC. On the coupling of boundary integral and finite element methods. Mathematics Competition 1980; 35:1063–1079.15. MacCamy RC, Marin SP. A finite element method for exterior interface problems. International Journal of Mathematics and Mathematical Sciences

1980; 3:311–350.16. Sofronov IL. Conditions for complete transparency on the sphere for the three-dimensional wave equation. Russian Academy of Sciences Doklady

Mathematics 1993; 46:397–401.17. Sofronov IL. Artificial boundary conditions of absolute transparency for two- and three-dimensional external time-dependent scattering problems.

European Journal of Applied Mathematics 1998; 9:561–588.18. Stakgold I. Boundary Value Problems of Mathematical Physics, Classics in Applied Mathematics 29. SIAM Publications: Philadephia, 2000.19. Ting L, Miksis MJ. Exact boundary conditions for scattering problems. Journal of the Acoustical Society of America 1986; 80(6):1825–1827.20. Tsynkov SV. Numerical solution of problems on unbounded domains. A review. Applied Numerical Mathematics 1998; 27:465–532.21. Wheeler LT, Sternberg E. Some theorem in classical elastodynamics. Archive for Rational Mechanics and Analysis 1968; 31:51–90.22. Kirchhoff formula. Encyclopedia of Mathematics, Springer Online Reference Works, Springerlink, http://eom.springer.


transparent boundary conditions for the wave equation—a kirchhoff point of view

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