New Results on Absorbing Layers and
Radiation Boundary Conditions
Thomas Hagstrom?
Department of Mathematis and Statistis
The University of New Mexio
Albuquerque, NM 87131
and
Institute for Computational Mehanis in Propulsion
Ohio Aerospae Institute and NASA Glenn Researh Center
Cleveland, OH 44142
hagstrommath.unm.edu
1 Introdution
Perhaps the dening feature of waves is the fat that they propagate long
distanes relative to their harateristi dimension, the wavelength. This allows us to use them to probe the world around us - optially, aoustially,
and now at a wide range of wavelengths in a variety of media. For numerial simulations, it is preisely this essential harateristi - the radiation of
waves to the far eld - that leads to the greatest diÆulties. One may view
this fundamental diÆulty as rooted in the existene of (at least) two widely
separated spatial sales. The rst are the small sales assoiated with the
wavelengths and the satterer, and the seond is the long distane between
the satterer and the observers.
The tehniques disussed in this review are designed to make the aurate and eÆient solution of typial wave propagation problems possible by
restriting the omputation to the small sale only. We do this, of ourse, by
introduing an artiial boundary, , and either imposing absorbing boundary onditions on it or surrounding it with an absorbing layer. Typial ongurations are shown in Figure 1 below.
Our fous here will be on time-domain problems for the standard equations of wave theory - the salar wave equation, rst order hyperboli systems,
and the Shrodinger equation. Although frequeny-domain alulations still
dominate muh applied work, we believe that time-domain simulations will
beome inreasingly important to eÆiently study broadband problems and
nonlinear satterers and soures. Moreover, in the frequeny domain aurate
boundary onditions and integral equation methods are well-established, and
?
Supported in part by NSF Grant DMS-9971772 and NASA Contrat NAG32692. Any opinions, ndings, and onlusions or reommendations expressed in
this paper are those of the author and do not neessarily reet the views of NSF
or NASA.
2
Thomas Hagstrom
An Exterior Domain
A Waveguide
Typial domain ongurations. is the omputational domain,
artiial boundary, is the tail.
Fig. 1.
is the
the osts assoiated with them are fundamentally easier to ontrol. (See, e.g.
[33, 38, 85℄.)
In 1999, the author wrote a lengthy review of the state-of-the-art in radiation boundary onditions for time-dependent problems [47℄. Even then it was
lear that for many important problems a number of satisfatory methods
were available. Sine then, new developments have provided the pratitioner
with a wider range of tools and raise the hope of solving the still-open problems listed in [47℄. Within the narrower sope of the urrent work, we will
review that state-of-the-art along with later ontributions. Our primary goal
is to explain in detail the methods that work and to illustrate their performane. We will also revisit some unsolved ases and speulate on their
potential resolution.
Prior to the early nineties, the usual approah to trunating the domain
was to x some low order absorbing boundary ondition. If the auray provided was insuÆient, improvements ould be made by inreasing the domain
size. The latter approah is both inonvenient to arry out automatially and,
for three-dimensional problems in partiular, rather expensive. Thus, in pratie, there was no way to ahieve onvergene to a presribed tolerane. The
fundamental breakthroughs whih hanged the situation were:
New Results on Absorbing Layers and Radiation Boundary Conditions
3
i. The realization that the boundary ondition hierarhies proposed a deade
earlier [78, 26, 27, 16, 61, 60℄ ould, in fat, be stably and onveniently
implemented using auxiliary funtions dened only on the boundary [18,
49, 35℄ ;
ii. Proofs and/or numerial demonstrations of the rapid onvergene with
inreasing order of the solutions produed by these hierarhies [45, 46,
49, 104℄;
iii. Low storage and fast methods to diretly impose nonloal onditions [89,
40, 41, 5, 6, 28, 86, 80℄;
iv. Parallel development of the perfetly mathed layer (PML), an absorbing
layer with a reetionless interfae [14, 17, 94, 20, 96, 82℄.
1
As mentioned earlier, the new methods provide, in many irumstanes,
a satisfatory solution to our problem. That is, rst, they enable us to meet
any presribed error tolerane at a ost no greater than the ost of solving
the interior problem. And, seond, they allow us to ompute on a domain
whih sales with the size of the satterer, independent of solution time or
error tolerane. Our hope in this paper is to explain how and why they work,
fousing on the ase we understand best: the salar wave equation and its
relatives.
We will illustrate many of the tehniques with numerial experiments.
Often these were exeuted with the author's researh odes. The latter an
be downloaded from www.math.unm.edu/~hagstrom/downloads. The reader
is autioned, however, that these are researh odes whih are not onsistently
doumented, and that they ome with no warranty whatsoever. I would also
like to aknowledge the ollaboration of a number of others in the work
desribed below: Dr. Brad Alpert of NIST-Boulder, Dr. John Goodrih of
NASA Glenn, Prof. Leslie Greengard of the Courant Institute, Prof. S.I.
Hariharan of the University of Akron, Prof. Tim Warburton of UNM, Dr.
Liyang Xu of MZA, and Igor Nazarov, urrently a dotoral student at UNM.
2 Boundary Conditions
To develop useful approximate boundary onditions, it is fruitful to onsider
rst the onstrution of exat boundary onditions. Plainly speaking, the
ondition we wish to impose is that the solution and any neessary derivatives
at the boundary be the trae of an element of the set of outgoing solutions preisely solutions of the homogeneous problem in whih are initially zero.
One an generally derive onrete expressions for the boundary ondition by
performing a Laplae transformation in time with dual variable s. Taking
<s suÆiently large, the transformed problem typially has an exponential
dihotomy. The exat boundary ondition is then given by an appropriate
1
Lindman, in the earliest of these works [78℄, atually suggests the auxiliary variable method.
4
Thomas Hagstrom
projetion of the solution onto the exponentially deaying subspae. Although
this ookbook onstrution may seem diÆult, we see that sometimes we an
arry it through.
2.1 The Wave Equation with a Cylindrial Tail
We begin with the simplest ase, namely when the equation in the tail, , is
the salar wave equation. Suppose we are in the rst ase shown in Figure 1,
so that is the ylinder (x; y ) 2 (0; 1) . We suppose the equation in the
tail is given by:
1 u u
=
+ L(y; =y )u;
(1)
2
t
2
2
x
2
2
with some homogeneous boundary onditions on . We expliitly assume
that u is initially zero in . We suppose L in onert with the boundary onditions is some negative, self-adjoint, linear ellipti operator with assoiated
eigenvalues-eigenvetors:
L
j=
j j ; j = 1; : : : ; 1;
2
Z
2
j dy = 1:
(2)
(3)
Expanding u in a Fourier series in the j and performing a Laplae transformation in t we obtain the equation:
s
u^j
=
+ j u^j ; x > 0:
x
2
2
2
2
2
(4)
Clearly, the ausal solution must vanish for x large and t small. Choosing
<s > 0 we see that (4) has only one bounded solution, whih must solve the
original problem:
u^j (x; s) = e
2 s2 +2j )1=2 x u
^j (0; s);
(
(5)
where the branh is hosen so that:
<(
2
s + j )
2
=
2 1 2
> 0;
<s > 0:
(6)
This implies the following exat boundary ondition at x = 0, whih we write
in two forms:
j
u^j
+ su^j +
x
s + ( s + j )
2
1
1
2
2
^j = 0;
= u
2 1 2
u^j
+ u^j = 0:
( s + j ) =
2
2
2
1 2
Inverting these we obtain, respetively:
x
(7)
(8)
New Results on Absorbing Layers and Radiation Boundary Conditions
u 1 u
+
+F
x t
F
Wj (t) 1
(Kj (t) (F u(0; ; ))) = 0;
1
u
(0; ; )
F x
5
(9)
+ u = 0:
(10)
(The rst form is the one we have typially used, while the seond has been
used in [80℄.) Here, F is the Fourier series with respet to the eigenfuntions of
L. The onvolution kernels, Kj and Wj , an, in fat, be expliitly represented
in terms of Bessel funtions, but in general we work with their transforms
diretly.
Both (9) and (10) are nonloal in spae and time. However, this nonloal
operator fators into the omposition of purely spatial and purely temporal
operators. In many ases, for example when L = r , F and F an be
applied using the FFT. Also, the temporal onvolution an be treated by
the algorithm in [54℄. Then the ost of applying the exat nonloal ondition
is aeptable. However, these algorithms require full history storage at the
boundary, whih is prohibitive exept when the solution times are relatively
short. We remark (see also [80℄) that the kernels do dene standard Volterra
integral operators. In ontrast, the retarded potential operator arising in
diret intergal equation formulations of the wave equation is nonstandard.
The improved operators are a diret onsequene of using the Dirihlet-toNeumann or Neumann-to-Dirihlet maps.
The primary diÆulty, then, is to remove the temporal and, possibly, the
spatial nonloality. The main observation is that onvolution with exponential
funtions an performed without storing the history, but instead by solving
a dierential equation. If
2
Z t
1
(t z) v (z )dz;
(11)
d
+ = v; (0) = 0:
dt
(12)
(t) =
e
0
then
For the transformed variables we have:
^(s) =
v^:
s+
(13)
The idea, then, is to onstrut onvergent sequenes of exponential approximations, Aqj j , to the kernels, Kj in (9) or Wj in (10), at least on nite time
intervals. Working in transform spae, this is equivalent to onstruting rational approximations to their transforms in right half planes. To make this
preise we reall the simple estimate based on Parseval's relation:
k(Kj
Aqjj ) vkL2
;T )
(0
Cet
sup jK^ j (s)
<s
A^qjj (s)j kvkL2
;T ) :
(0
(14)
6
Thomas Hagstrom
We onstrut A^qj j to be a rational funtion of degree (qj
admits a partial fration deomposition:
A^qjj =
and
Aqjj
qj
X
k=1
v =
jk
;
s + jk
qj
X
k=1
jk (t);
djk
+ jk jk = jk v; jk (0) = 0:
dt
1; qj ). It then
(15)
(16)
(17)
Thus we have approximated the temporal onvolution by a system of
dierential equations for auxiliary funtions dened on the boundary only.
The work per time step and storage required is proportional to the number
of eigenfuntions, j , the number of auxiliary funtions, qj , and the ost of
omputing and inverting the Fourier series.
We note that the boundary ondition is also loalizable in spae if the j dependene of the rational approximations an be desribed by polynomials
in the eigenvalues, j . We then have:
2
A^qjj =
N (s; j )
:
D(s; j )
2
2
(18)
Formally (9) beomes:
u u
+
+ w = 0;
(19)
x t
D
;L w = N
; L u:
(20)
t
t
Of ourse we rewrite the equation for w as a rst order system using additional
auxiliary funtions to avoid high order derivatives.
Later on we will enounter some variations of these approximations. First
of all, we will often nd it onvenient to write Aqj j as a ontinued fration
rather than by a partial fration expansion. The ontinued fration representation seems to allow more diretly an adaptive determination of qj , though
we have not yet implemented suh a sheme. Seond, it is possible to ompute
dierent exponential approximations to the kernels on dierent time intervals - a tehnique whih is more general than the one we've outlined. Suh a
method is proposed in [80℄ and will be disussed in more detail below.
Loal Approximations
Already in [26℄ it was noted that Pade approximation produes boundary onditions whih lead to well-posed problems.
There the approximation was entered at normal inidene, whih an also
be thought of as an expansion of Kj valid for short time. We will follow the
derivation due to Xu [104℄ whih employs ontinued frations.
New Results on Absorbing Layers and Radiation Boundary Conditions
7
To onstrut the approximations we note the relation:
j
:
s
^ j (s)
2 + K
2
K^ j (s) =
(21)
Then we dene A^qj reursively:
A^qj =
j
s
2 + A^jq
2
1
:
(22)
In [104℄ a number of hoies for A^j are onsidered. For example, the hoie
A^j = jj j leads to spatially nonloal onditions whih are exat at steadystate, generalizing to high order the onditions of the type proposed in [25,
31℄. Loal onditions follow from the hoie:
0
0
A^j = 0;
(23)
0
and may be written as the ontinued fration (terminated after q terms with
q independent of j ):
j
2
A^qj =
2 s +
2j
2j
s
2 + s
2 +:::
:
(24)
To apply the approximate operator rst dene
w = u; wk = Aq
0
+1
k
wk
1
:
(25)
Here we have dropped the j dependene, as the multipliations by j may be
replaed by appliations of L. Now the boundary ondition approximating
(9) is written in terms of:
w = Aq u:
(26)
2
1
We also have that
wq = 0:
(27)
u 1 u
+
+ w = 0;
x t
(28)
wq
+1
=A
0
We thus obtain the form:
1
2 wk
t
= Lwk
1
wk ; k = 1; : : : q:
+1
(29)
As a nal step we reformulate the reursion so that only seond order equations are solved. The details, along with rst order reformulations suitable
for inorporation into standard time-marhing shemes, are found in [53℄.
Note that we are using the same symbols, wk , to denote in general dierent
auxiliary funtions, exept for w . Also, we have assumed that q = 2P .
1
8
Thomas Hagstrom
1 w
2
t
2
1 u 3
= L + Lw
2 x 4
1
2
1 wk
=
1
Lw ;
4
1
(30)
2
1
1
1
Lwk + Lwk
Lw ; k = 2; : : : P:
(31)
t
4
2
4 k
Again we emphasize the ease of applying this boundary ondition to arbitrary order; it simply requires inreasing the parameter P . We also note
that the size of the nal term, wP , provides some measure of the error in
terminating the fration. If this measure ould be made more preise, an
adaptive implementation ould be developed. The upoming analysis and experiments will larify the potential utility of an adaptive implementation. In
partiular we will see that the worst ase analysis indiates that the neessary
order ould be quite large, while numerial experiments show that it is often
possible to use a relatively low order.
To study the onvergene of the method using (14) we must estimate:
2
2
1
2
+1
sup <s= s
Aqj (s) :
j
2
=
s2
+ 2 + j
2
1 2
(32)
Note that the branh points at s = ij make it diÆult to derive good
estimates when = 0. Thus we aept > 0. Estimates, derived in detail in
[45, 104℄, follow reursively from the equation:
Rq
where
+1
=
(z + (z + 1) =
2
1 2
Rq
+ j Rq )(z + (z + 1)
2
2
j
s
2
Rq = Aqj (s)
= );
1 2
; z=
:
s + s2 + 2 1=2
jj j
2
j
(33)
(34)
Using (33) we nd that the error is given by:
jRq j C
1+
jj j
q
(2 +1)
:
(35)
As we must hoose = O(T ) to obtain a meaningful estimate, we nd that
to ahieve a relative error tolerane of for the j th harmoni we require:
1
1
q = O(jj jT ln ):
(36)
It is relatively straightforward to turn this into an estimate depending on
various Sobolev norms of u on . It tells us that for a xed problem and xed
time that the method is spetrally onvergent with inreasing q . However, for
jj jT large many terms may be needed.
New Results on Absorbing Layers and Radiation Boundary Conditions
9
We note that to omplete the onvergene argument we must derive stability estimates. This is arried through in detail in [46, 104℄. A ompliating
feature is the fat that the exat boundary onditions themselves do not satisfy the uniform Kreiss ondition [73℄. However, we an show that any growth
of the stability onstant with q is at worst linear, so that spetral onvergene
is retained. In our numerial experiments here and in [48℄ we have used very
large values of q > 100 without observing any loss of stability.
To illustrate the stability and onvergene of the loal onditions we
present the results of a simple numerial experiment. We solve the wave equation with = 1 in the two-dimensional domain ( 2; 2) (0; 1) for 0 t 50
with homogeneous Neumann boundary onditions on y = 0; 1. The initial
data is generated by a Gaussian foring at negative times whih is shut o
before t = 0. We use the boundary onditions (28), (30), (31), an eighth order
two-step method in time and eighth order dierening in spae. The spatial
mesh was 264 66 and the number of time steps was 20000. Evaluating to
high auray an integral formula for the exat solution, we are able to generate preise error data, whih is shown in Figure 2. (The solution is O(1) so
that the absolute errors shown are omparable to relative errors.)
Waveguide Problem − Pade Approximation
−1
10
−2
10
P=5
−3
10
P=10
−4
10
P=20
−5
error
10
P=40
−6
10
−7
10
P=60
−8
10
−9
10
−10
10
0
5
Fig. 2.
10
15
20
25
time
30
35
40
45
50
Errors as a funtion of time and boundary ondition order.
The results learly show that the Pade approximation an be used to
ahieve exellent auray at modest ost. For example, P = 20 suÆes for
a tolerane of 10 up through t = 50. The results are also onsistent with
the error analysis. Note that we are plotting the total error due both to the
3
10
Thomas Hagstrom
boundary approximations and the disretization, whih limits the gains in
auray whih an be attained by inreasing P alone. The stability of the
onditions for P large are also onrmed.
We nally note that a number of other loal approximations are possible.
In [56℄ various alternatives are disussed, and a reasonable ase is made that
they should perform better. To implement them to high order, we note from
[60, 61, 56℄ that they are formally equivalent to:
q
Y
k=1
1 u = 0; 0 < k :
+
k t x
(37)
Diretly, (37) an be diÆult to implement. However, Givoli and Neta [35℄
and Guddati and Tassoulas [44℄ have reently shown how these an be reformulated using auxiliary funtions. The preise result in [35℄ is that (37) is
equivalent to:
u 1 u
+
+ w = 0;
(38)
1
k
+
1
k
+1
wk
=
t
x
1
2
t
1 k t
1
1
2
2
2
L wk
1
wk ; k = 1; : : : q:
+1
(39)
In [44℄, on the other hand, a ontinued fration interpolant of the wave speed
funtion is diretly omputed, leading to a distint but equivalent form, again
without high order derivatives.
Clearly, (28)-(29) orrespond to the speial ase k = . The automati
optimization of these parameters as well as their appliation to more diÆult
problems are subjets with great potential. (Note that Higdon has applied
his boundary onditions to a variety of problems, but not to high order [63,
64, 65℄.)
Spatially Nonloal Approximations
An unpleasant result of the preeding analysis is the poor behavior of of the approximations for jj jT large.
As shown in [46℄, this sort of long time behavior is to be expeted from
any homogeneous spatially loal approximation. It is reasonable to ask how
muh better we an do if we allow spatially nonloal onditions and onstrut
norm-minimizing rational approximations.
Results along these lines have been obtained in [6℄. Theoretially it is
shown that a tolerane of an be met with:
1
(40)
q = O ln + ln jj jT ;
whih is learly superior to (36). Near-optimal approximations were also onstruted numerially using a nonlinear least squares proedure. (The resulting
approximations are also aurate in the maximum norm as required by the
error estimate.) For example, rational approximations are onstruted in [6℄
satisfying the toleranes in Table 1.
New Results on Absorbing Layers and Radiation Boundary Conditions
Table 1.
11
jj jT q
10 4 104 21
10 6 104 31
Number of poles required for various toleranes and times.
The rational funtion A^qj is expressed in partial fration form:
q
X
k
;
k jj j
(41)
u 1 u X
= 0;
+
+
x t k k
(42)
A^qj = j
2
k=1
s
leading to the boundary ondition
q
=1
^jk
k jj j^jk = k j u^j :
(43)
t
Note that the oeÆients j and j are independent of and j and have
2
been omputed one and for all. In partiular the poles lie stritly in the
left half omplex plane. (Their tabulated values an be downloaded from the
address mentioned above.)
We have repeated the numerial experiment desribed above for the wave
equation in a waveguide using (42),(43) instead of (28),(30),(31). An FFT,
modied for the nonuniform mesh, is used to ompute the diret and inverse
Fourier series. Note that we use an eighth order quadrature rule to ompute
the transforms, so that if we use too many Fourier modes the auray is
degraded. The results, shown in Figure 3, onrm the theoretial preditions.
The errors when we use 16 modes are an order of magnitude smaller than
the toleranes used to dene the approximations themselves.
An alternative approah to onstruting eÆient nonloal approximations
has been proposed by Lubih and Shadle [80℄. First, they reformulate the
exat boundary ondition using the Neumann-to-Dirihlet map (10). Seond,
they simplify the approximation problem somewhat by onstruting loal exponential approximations to the onvolution kernels, Wj (t), on time intervals:
Il = B l t; (2B l
1
1)t ;
(44)
where t is the time step employed in the disretization and B is an integer.
(They reommend B = 10.) The positive result of this simpliation is that
they an onstrut eetive approximations with predetermined pole loations, simply approximating the inverse Laplae transform by a quadrature
rule on the so-alled Talbot ontour. Thus they an onstrut the approximations themselves far more rapidly than possible with the method used
12
Thomas Hagstrom
Waveguide Problem − 21−pole Uniform Approximation
−4
10
m=8
m=32
−5
10
−6
error
10
m=16
−7
10
−8
10
−9
10
−10
10
0
5
10
15
20
25
time
30
35
40
45
50
40
45
50
Waveguide Problem − 31−pole Uniform Approximation
−4
10
m=8
−5
10
m=32
−6
error
10
−7
10
m=16
−8
10
−9
10
−10
10
0
5
10
15
20
25
time
30
35
Errors for the 21-pole and 31-pole least squares approximations. The parameter m is the number of Fourier modes used to evaluate the boundary ondition.
Fig. 3.
New Results on Absorbing Layers and Radiation Boundary Conditions
13
in [5, 6℄. This allows them to eÆiently evaluate exat boundary onditions
for the disretized problem, whih an be more aurate when the solution
is marginally resolved. On the negative side, the use of dierent approximations to the onvolution kernel on dierent time intervals leads to a more
omplex implementation, and they require a little more work and storage
than the method of [5, 6℄. Preisely they require O(ln 1= ln jj jT ) auxiliary
funtions.
2.2 The Wave Equation in an Exterior Domain
We now onsider the wave equation in the seond standard onguration for
our lass of problems, namely an exterior domain. Preisely, we suppose the
tail, , is suh that Rn is bounded and that within the governing
equation is:
1 u
= r u; x 2 :
(45)
2
t
2
2
2
We note that within the omputational domain we may have inhomogeneities,
nonlinearities, or any other perturbations.
Performing our usual Laplae transformation in time and supposing <s
suÆiently large (typially <s > 0) we are led to the problem of desribing
the trae on of all bounded solutions of the Helmholtz equation:
r u^
2
s
u^ = 0:
2
2
(46)
A useful format for desribing this trae is the Dirihlet-to-Neumann (or
Neumann-to-Dirihlet) map, whih we express as:
u^
= D(s)^u; x 2 :
n
(47)
(Preisely, we may dene D if is suÆiently smooth by solving the exterior
Dirihlet problem with data u^ and omputing the trae of the normal derivative. Here =n denotes the derivative into , that is out of the omputational
domain.)
To derive onrete expressions for D it is neessary to restrit ourselves
to simple boundaries, for example boundaries assoiated with oordinate systems in whih the Helmholtz equation is separable. Even then there an be
diÆulties in transforming bak to the time domain if the eigenfuntions of
D are s-dependent. Thus the ideal ase, from the point of view of analysis, is
when is hosen to be a sphere, and we will treat this ase in great detail.
However, the sphere is a wasteful hoie for highly elongated or nononvex
satterers. We will disuss some tehniques whih an be applied on highaspet ratio or even nononvex artiial boundaries. However, the problem
of generalizing some of the more eÆient and exible methods remains open.
14
Thomas Hagstrom
Assuming now that is a sphere of radius R and expanding the solution
of (46) in spherial harmonis:
u^ =
1 X
l
X
ulm (r)Ylm (; );
l=0 m= l
we nd that ulm satises:
ulm 2 ulm
+
r
r r
2
2
l(l + 1)
s
ulm = 0:
+
r
2
2
2
(48)
(49)
The bounded solution of (49) is given by the modied spherial Bessel funtion:
k (rs=)
ulm (r; s) = l
u (R; s);
(50)
kl (Rs=) lm
kl (z ) =
e
2z
z
l
X
(l + k )!
(2z ) k :
k!(l k)!
k=0
(51)
To onstrut the Dirihlet-to-Neumann map we must ompute the logarithmi derivative of kl whih from (51) is given by:
skl0 (Rs=)
s
=
kl (Rs=)
1
R
1
Pl
R
1
(2l k)!
k
k=0 k!(l k 1)! (2Rs=)
:
(2l k)!
k
k=0 k!(l k)! (2Rs=)
Pl
(52)
Summing over the harmonis and inverting the Laplae transform we nally
nd:
1
u 1 u 1
(53)
+
+ u + H (Sl (Hu)) = 0;
r
t
R
R
1
2
where H denotes the spherial harmoni transform and the Laplae transform
of Sl is given by the last term in (52).
The primary observation to be made is that S^l is a rational funtion, albeit
of degree (l 1; l). Thus Sl an be written as a sum of exponential funtions
and, for eah xed harmoni, (53) an be loalized in time. This fat was
independently notied by Sofronov [89, 90℄ and Grote and Keller [40, 41℄,
who used it to onstrut temporally loal boundary onditions whih are
exat on funtions with nite harmoni ontent.
A seond onsequene of (50)-(51) is found by summing the series over l:
u^ =
1
X
k=0
e
(2rs=)k
rs=
+1
1
X
l
(l + k )! X
u (R; s)
:
Ylm (; ) lm
k
!(
l
k
)!
kl (Rs=)
l k
m l
=
(54)
=
Dening:
f^k (s=; ; ) =
(2s=)k
1
X
+1
l
u (R; s)
(l + k )! X
;
Ylm (; ) lm
k
!(
l
k
)!
kl (Rs=)
m l
l k
=
=
(55)
New Results on Absorbing Layers and Radiation Boundary Conditions
15
and inverting the transform we produe the progressive wave or multipole
expansion of u:
1 f (t r)
X
k
u=
; r R:
(56)
k
k=0
r
+1
Note again that if we assume nite harmoni ontent, ulm = 0 for l > M ,
then the series terminates with k = M .
From (56) we immediately derive the well-known Bayliss-Turkel onditions [16℄:
1 2q + 1
+
+
r t
R
1 1
u = 0;
+
+
r t R
(57)
whih we note are loal and exat in the same sense as the onditions of
[89, 90, 40, 41℄. In the next setion we will reformulate them using auxiliary
funtions to enable their high-order implementation.
Loal Boundary Conditions Inspired by the reformulation of the BaylissTurkel onditions in [15℄, Hagstrom and Hariharan [49℄ gave the rst reformulation of (57) in terms of auxiliary funtions on satisfying seond
order hyperboli equations. Soon thereafter Huan and Thompson [70, 100℄
demonstrated the equivalene between the auxiliary funtions of [49℄ and the
residuals of the Bayliss-Turkel ondition and also developed nite element
implementations. Following them we set:
wk
+1
=
1 2k + 1
+
+
r t
R
Obviously,
1 1
u:
+
+
r t R
1 2k + 1
wk = w k ;
+
+
r t
R
(59)
+1
and (57) is equivalent to:
1
1
u = w ; wq
+
+
r t R
1
+1
(58)
= 0;
(60)
ombined with (59). The only problem with this formulation is the presene
of the radial derivative in (59), whih would fore us to dene the auxiliary
funtions in the interior. To eliminate it we note the identity derived in [70℄,
whih an easily be proven by indution:
r
1
t
+
1
r
wk =
1
r
2
r
2
+ k (k
1) wk ;
1
(61)
where r denotes the Laplae-Beltrami operator on the sphere:
2
r
2
w
1 w
1 sin +
:
w=
sin sin 2
2
2
(62)
16
Thomas Hagstrom
k
Using (61) to eliminate w
r from (59) we obtain our desired system:
1
1
k
w =
+
t R k 2R
2
r
2
+ k (k
1) wk
1
1
+ wk :
2
+1
(63)
Equations (63) together with (60) are our nal reformulation of (57). We
again remark that they are easily implemented to any order by introduing
additional auxiliary funtions, that is by inreasing q , and retain the property
of being exat on funtions desribed by harmonis of index up through q . We
also note their obvious similarity to the planar boundary onditions based on
the Pade approximation, (28)-(29). (The dierene in salings orresponds to
a dierent saling of the auxiliary funtions.)
We have not yet arried through a omplete onvergene proof for these
boundary onditions for xed R and inreasing q . We note that if we write:
u=u q +Æ q ;
( )
(64)
( )
where u q is the projetion of u onto the span of the harmonis of index
up through q , then the onsisteny error is simply the result of applying the
boundary ondition to Æ q . Moreover, for smooth solutions kÆ q kL2 deays
to zero faster than any power of q . Thus it is reasonable to onjeture that
the proposed onditions are spetrally onvergent uniformly in time.
An analagous sequene of boundary onditions an be onstruted on
irular boundaries in two spae dimensions (and ylindrial boundaries in
three). However, unlike the three-dimensional ase, we have no expetation
of time-uniform onvergene. Indeed, the results of [50℄ essentially prelude
it. In [53℄ we show how to rewrite these onditions so that only rst order
derivatives of the auxiliary funtions appear, implement the onditions in a
disontinuous Galerkin spetral element ode for Maxwell's equations [59℄,
and arry out numerial experiments. In Figure 4 we plot the error over
time for a two-dimensional simulation of a TE-pulse. Clearly the results are
analagous to those obtained in the plane ase, noting that for the waveguide
experiments the solution was better resolved. (Compare with Figure 2.)
In addition to the appliations to Maxwell's equations in [53℄, some other
generalizations of this method have reently been ompleted. In [51℄ boundary
onditions analagous to (60),(63) are onstruted for the onvetive wave
equation,
( )
( )
( )
+M
t
x
2
u = r u;
2
(65)
where 0 < M < 1 is the Mah number. These will be appliable to problems
governed by the linearized subsoni Euler equations in . Generalizations to
the wave equation in a layered half spae, motivated by problems in soilstruture interation, have also been proposed [105℄.
A defet of the preeding analysis from the point of view of ertain appliations is the fat that the underlying expansion (56) only holds exterior
New Results on Absorbing Layers and Radiation Boundary Conditions
1
10
0
||Ez - exact Ez||∝ /||exact Ez||∝
10
17
M=0
M=1
M=4
M=8
M=12
M=16
M=20
10-1
10-2
10
-3
10
-4
10-5
10
-6
5
10
time
Errors as a funtion of order, M , and time. Reformulated ylindrial BaylissTurkel ondition. DG Maxwell solver.
Fig. 4.
to a sphere ontaining the problem's inhomogeneities. This ould be quite
wasteful of omputational volume for problems with very high aspet ratio
satterers. (This defet is shared by many of the nonloal formulations we
will disuss below.) This suggests that it ould be useful to nd an alternative expansion with a onvergene domain exterior to a high-aspet ratio
boundary, whih would serve as . For the Helmholtz equation, Holford [67℄
has onstruted suh an expansion with an oblate or prolate spheroid.
Translated to the time domain, the results in [67℄ suggest that (56) holds
exterior to a spheroid with the r oordinate replaed by its analogues in the
relevant spheroidal oordinate system. On this basis we believe that a sequene of boundary onditions generalizing (60),(63) ould be onstruted
whih might prove eÆient for a large lass of problems. We also note that
related methods based on spheroidal innite elements have been onsidered
[8, 9, 10℄.
A seond approah to eÆiently bounding a high-aspet ratio satterer
is to use a retangular box, applying onditions suh as (28), (30),(31) or
(38),(39) on eah fae. There are two issues whih must be onsidered in
this ontext. First, it is neessary to provide some boundary onditions at
the edges whih relate the auxiliary funtions on adjaent faes. Seond, it is
neessary to develop some argument that the approximations will onverge
with inreasing q .
18
Thomas Hagstrom
In [102℄, Vaus makes substantial progress on the rst issue. Considering
the boundary onditions (37), he proves the existene of a unique smooth
solution in the retangular domain assuming suÆiently regular data. He also
desribes an algebrai proedure for deriving orner ompatibility onditions.
Vaus' onstrution involves taking high order spae-time derivatives of both
the equations and the boundary onditions and nding linear ombinations
whih an be integrated to yield nontrivial onstraints. It will take some
additional eort to translate his results into usable ompatibility relations for
our preferred auxiliary variable formulations. We note that earlier Collino [18℄
derived orner ompatibility relations for an auxiliary variable formulation of
the boundary onditions based on Pade approximations. He used a sequene
of exat solutions of the wave equation to determine the relations. As it is not
obvious how to generalize either onstrution to more omplex situations, a
new and somewhat more straightforward approah would be very useful.
Vaus' uniqueness theorem lends some redene to the belief that the
method will onverge. In partiular, onvergene on the planes ontaining
eah fae follows from the results above, and the orner ompatibility onditions should lead to auxiliary funtions whih agree with the restritions of
the planar auxiliary funtions. However, this argument is far short of a proof.
Thus a denitive analysis of onvergene for retangular would be an important advane. Note that it is plausible that the error estimates will be
better than for the plane. The nonuniformity in time is due to the possibility
of late-time glaning inidene. However, waves whih impinge on one part
of the boundary at glaning angles impinge on other parts nearly normally,
and thus are very well-absorbed.
Nonloal Conditions
Nonloal boundary onditions for exterior problems
have been onstruted from at least three distint formulations of exat
boundary onditions. The most straightforward are based on the evaluation
of (53). Diretly, Grote and Keller [40, 41℄ ompute some (typially small)
number of spherial harmoni expansion oeÆients and solve a system of
ordinary dierential equations equivalent to the order l system dened by S^l .
For l large, however, this method (and the ompanion loal methods deribed
earlier) require a large number of auxiliary variables. This number an be signiantly redued if least squares approximations are used instead. Note in
this ase we are approximating the degree (l 1; l) rational funtion, S^l , by a
rational funtion of lower degree, (ql 1; ql ). A fundamental theoretial result
of [5℄ is that for an absolute error tolerane we an take:
ql = O(ln l + ln 1=):
(66)
Note that, unlike the ase of a plane boundary, this estimate is uniform in
time. That is, we an approximate S^l on the imaginary axis.
In either ase, the nonloal term in (53) is replaed by an expression of
the form:
New Results on Absorbing Layers and Radiation Boundary Conditions
1
R
2
P X
l
X
l=0 m= l
Ylm (; )
ql
X
k=1
lmk ;
19
(67)
with the auxiliary funtions satisfying ordinary dierential equations. For
example, using the representation from [5, 6℄:
1 lmk
t
+
1
R
lk lmk = lk u~lm ;
(68)
where u~lm (t) is a oeÆient in the spherial harmoni expansion of u on .
The nonlinear least squares proedure has also been employed to ompute
the pole loations and strengths for near-optimal approximations. That is, we
have omputed ql and the omplex numbers lk , lk for the spherial version
of (68). (They an be obtained from the website mentioned earlier.) In Table
2 we list the number of poles, and thus the number of auxiliary funtions,
needed for eah harmoni with = 10 for both the sphere and ylinder
kernels. We note that the numbers are very small - no more than 21-poles
per harmoni are needed to guarantee exellent time-uniform auray for
harmonis up through index 1024. By way of omparison, the methods of
[89, 90, 40, 41, 49℄ would all require 1024 to be exat for suh modes. Thus
for a diÆult problem with high harmoni ontent at the boundary it learly
pays to use the least squares approximations. The primary ost assoiated
with the appliation of the boundary ondition is the omputation of the diret and inverse spherial harmoni transformations. Mohlenkamp [81℄, Suda
and Takami [92℄, and Healy and oworkers [57℄ have devised fast algorithms
for this purpose. Using them, the formal operation ount for applying the
boundary ondition is of muh lower order than the ount assoiated with
the interior solve, though for moderate size problems asymptotially more
omplex methods an be faster. Tables of poles and amplitudes an be downloaded from the web site mentioned above.
The ost of the spherial harmoni transform aside, the eÆieny of the
least squares approximations in the spherial ase is unmathed. However,
just as the sequenes of loal boundary onditions, their urrent formulation
requires a spherial (ylindrial) artiial boundary, whih is an expensive
hoie for a high-aspet ratio satterer. Possible solutions, again just as in
the loal ase, are to extend the onstrution to spheroids or boxes. A problem here is that exat onditions on these boundaries are fundamentally more
omplex. In partiular, they an not be diagonalized by a xed spatial basis.
We feel that the development of eÆient representations of boundary onditions on high-aspet ratio surfaes is an important problem whih should be
studied further.
The other two nonloal formulations both allow for the exible hoie
of , but are otherwise more ostly. The rst, suggested in [101℄, is based
on a formulation of exat boundary onditions using retarded potentials. It
involves enlosing the satterer by two surfaes, I and O , as indiated in
Figure 5.
8
20
Thomas Hagstrom
S^l
ql
l
ql
0 44
1 15
2 9
3{ 8 7
0{ 7 l
9{ 10 8
8{ 10 8
11{ 14 9 11{ 14 9
15{ 20 10 15{ 19 10
21{ 28 11 20{ 28 11
29{ 41 12 29{ 40 12
42{ 58 13 41{ 57 13
59{ 84 14 58{ 83 14
85{ 123 15 84{ 123 15
124{ 183 16 124{ 183 16
184{ 275 17 184{ 275 17
276{ 418 18 276{ 418 18
419{ 638 19 419{ 637 19
639{ 971 20 638{ 971 20
972{1024 21 972{1024 21
Table 2. Number of poles needed to approximate the exat boundary ondition
kernels to an auray of 10 8 . From [5℄.
l
C^l
I
Fig. 5.
O
A domain ongurations with two boundaries.
New Results on Absorbing Layers and Radiation Boundary Conditions
21
Then we an evaluate u on O using past values of u and its derivatives
on I . Preisely:
u(x; t) =
1
4
Z
u
I
1
n r
1 u
1 r u
r n
r n t
dy;
(69)
where x 2 O , =n is the normal derivative, r = jx y j and the time
argument of u in the integral is t r=. We note that although the formula
is nonloal in time, the required history extends over only a nite interval
determined by the maximum travel time between points on the boundary.
The rst implemention of (69) appears in [34℄. However, it employs a diret numerial evaluation of the integral, whih implies a ost per time step
proportional to the square of the number of boundary points. This is more
ostly by one order than the interior solve, and is thus not ompetitive with
other methods. Reently, a fast algorithm for evaluating retarded potentials
has been devised [28℄. Using this algorithm, the formal operation ount assoiated with (69) beomes omparable to the ounts for the methods desribed
above, though in pratie it is still quite expensive. However, the method is
ompetitive in ertain speial ases, for example for highly nononvex satterers.
The nal method, due to Ryaben'kii and oworkers [86, 87℄, is based
diretly on the strong Huygens' priniple (or the existene of launae) whih
itself follows from (69). As a rst step, onsider again Figure 5 and introdue
a smooth uto funtion, (x), satisfying = 1 at and beyond O and = 0
inside I . Introduing the auxiliary funtion:
v = u;
(70)
we see that v satises a fored wave equation in all spae:
1 v
2
t
2
2
= r v + f; x 2 R ;
2
3
(71)
where f is supported between O and I . Sine u = v on O , an exat
Dirihlet boundary ondition an be imposed if a solution of (71) an be
omputed. To solve (71) we note that if f is supported only in the time
interval (tj ; tj ), the strong Huygens' priniple would imply that v 6= 0
on O only on some larger time interval, (tj ; Tj ). Moreover, on this time
interval, v restrited to O is idential to the solution of a periodi problem
in spae with suÆiently large period.
The algorithm proposed in [87℄ is based on these observations. By a
smooth partition of unity in time, f is written as the sum of funtions, fj ,
whih are supported on time intervals of xed duration. Then v on O is written as the sum of funtions vj whih solve spatially periodi wave equations
with foring fj . The periodi spatial domains must be hosen fairly large larger than the domains required by a spherial artiial boundary. However,
+1
+1
22
Thomas Hagstrom
the equation for the funtions vj is simple and an be very eÆiently solved
using a Fourier spetral method. Thus the resulting algorithm is ompetitive.
We nally mention an intriguing result of Warhall [103℄. He proves that
if u solves the wave equation, with all inhomogeneities and initial data supported inside a onvex domain, , bounded by , the solution at times
t tp 0 is ompletely determined by the data within \ B (x; (t tp ))
at time tp . This seems to imply that an exat boundary ondition with no
history dependene or auxiliary variables exists. However, to date this exat
operator has not been found.
2.3 First Order Hyperboli Systems
We now return to the ase of a ylindrial tail, = (0; 1) . However we
assume that is retangular and that the equation in is the rst order,
onstant oeÆient hyperboli system:
u X u
u
Bj
+ Cu;
= +
t
x
yj
j
(72)
where the diagonal matrix is partitioned into inoming and outgoing piees.
(We are assuming that x = 0 is nonharateristi, but that assumption an
be relaxed.)
= O O
+
; > O; < O:
+
(73)
We also suppose that appropriate boundary onditions are imposed on .
After a Fourier-Laplae transformation, we derive the system of equations in
x:
1
0
X
u
ikj Bj C A :
= sI
(74)
1
x
j
For any xed transverse mode, this system learly possesses an exponential
dihotomy for <s suÆiently large. Moreover, the dimension of the subspae
of growing solutions is that of . An exat boundary ondition is then given
by:
P (s; k)u = 0;
(75)
where P is a projetor into the subspae of growing solutions. Noting further
that in the large s limit this equation redues to u = 0 we reah our nal
form:
u = F (R(t; k) (F u)) ;
(76)
where R is now a matrix.
We note rst that the projetion operator is not unique and not every
hoie leads to a well-posed problem. For example, the obvious hoie of left
eigenvetors to dene P leads to an ill-posed problem for the linearized, subsoni Euler equations [32, 48℄. Seond, we have restrited the ross-setion
+
+
+
+
+
+
1
New Results on Absorbing Layers and Radiation Boundary Conditions
23
to a retangle so that we ould redue the problem to a linear algebra problem using Fourier series. More generally the projetion onto outgoing waves
(deaying solutions after Laplae transformation) is more diÆult. Finally,
even if we an redue the problem to linear algebra, we may be unable to
nd analyti representations of the projetors.
Despite these diÆulties, there are a number of important examples where
we an make progress. Not surprisingly, in eah of these ases the basi propagating modes either satisfy the wave equation or the simple transport equation. Below we will disuss two suh ases in some detail, Maxwell's equations
and the linearized Euler equations. Other systems whih an be disussed inlude the linearized shallow water equations, whih are losely related to
the Euler equations, and the equations of linear elastiity, whih are typially formulated as a seond order system. For a treatment of the latter see
[83, 62, 63, 39, 43℄.
Maxwell's Equations Maxwell's equations in free spae using appropriate
units are given by:
1 E
r B = 0;
t
1 B
+ r E = 0:
t
(77)
(78)
Systemati derivations of exat boundary onditions an be found in [42, 47℄.
Here we'll take a shortut, noting that eah of the Cartesian omponents
of the eld vetors satises the salar wave equation. Thus, we an use the
boundary onditions disussed above to eliminate the normal derivatives of
any variable. We simply use the normal harateristi analysis to predit
whih variables we need to speify. Note that for Maxwell's equations there
are, at any boundary, two inoming, two outgoing and two harateristi
variables. Thus we will impose two boundary onditions. See [53℄ for more
details and numerial experiments with loal onditions.
In our waveguide geometry, where x is the normal oordinate and (y; z )
the tangential oordinates, the inoming normal harateristi equations are:
1
t
(Ey
x
Bz ) +
Ex
y
Bx
= 0;
z
(79)
E
B
(Ez + By ) + x + x = 0:
(80)
t x
z
y
Using (9) to eliminate the x-derivatives yields the exat boundary ondition:
2
+F
1
2
+F
1
t
1
t
Æ (Kj ) Æ F
Æ (Kj ) Æ F
(Ey
Bz ) +
(Ez + By ) +
Ex
y
Bx
= 0;
z
(81)
Ex Bx
+
= 0:
z
y
(82)
24
Thomas Hagstrom
Finally, we an replae the exat nonloal term by either the loal sequenes
or the nonloal least squares approximations.
A similar proedure an be arried out on a sphere. Eah Cartesian omponent of the elds satises (53). One simply writes the equations for the inoming variables at eah point and replaes r derivatives using (53) and whatever loal or nonloal approximation to the nonloal term, H (Sl (Hu)),
is hosen. As desribed, this method would require the denition of auxiliary
funtions for all six omponents. However, the more omplex derivations in
[42, 47℄ show that only two sets are required.
1
Linearized Euler Equations
As a seond example, onsider the ompressible Euler equations linaerized about a uniform, subsoni ow:
D
+ r u = 0;
Dt
Du 1
+ rp = 0;
Dt Dp
+ r u = 0:
Dt
(83)
(84)
(85)
Here the material derivative is dened by:
D
Dt
t + U r;
(86)
and U is the uniform ow eld about whih we've linearized. Note that units
have been hosen so that the sound speed is one. Thus:
3
X
j =1
Uj = M < 1:
2
2
(87)
We also assume U > 0.
The linearized Euler equations support waves of diering types. Combining (84) and (85) we nd that the pressure, p, satises the onvetive wave
equation:
1
Dp
= r p:
(88)
Dt
On the other hand, the entropy, S = p , and the vortiity, ! = r u
2
2
2
satisfy the transport equation:
D!
DS
= 0;
= 0:
Dt
Dt
(89)
Thus the boundary ondition problem for the pressure is similar to the
boundary ondition problem for the wave equation. Indeed, we have shown
in [48, 51℄ that the loal boundary ondition sequenes disussed above for
New Results on Absorbing Layers and Radiation Boundary Conditions
25
both waveguide and exterior geometries an be generalized to the onvetive
wave equation. In addition, exat boundary onditions for (89) are extremely
simple; namely:
S = 0; ! = 0; if U n < 0;
(90)
where U is the outward normal - that is if the ow is inoming. The primary
diÆulty is to ombine these relationships to produe a well-posed problem
at inow.
In [48℄ the full range of well-posed formulations of exat boundary onditions is displayed. (See also [91, 84℄.) At outow, where there is a single
inoming aousti mode, the boundary ondition is essentially uniquely dened:
u
u
D
= 0;
(91)
+
(p u ) + Kp + U
tan
Dt
1
1
x
2
x
2
3
3
At inow there are many possibilities. The simplest realization, whih is used
in the numerial experiments, is:
p = 0;
(92)
1
1 U
D
(p + u ) + K(p + u ) +
Dt
2
2
D u
u
p
+
+U
Dt
x
x
u
p
D u
+
+U
Dt
x
x
1
tan
1
1
tan
2
1
1
2
tan
2
1
3
1
3
3
u
u
+
x
x
2
3
2
3
= 0;
(93)
= 0;
(94)
= 0:
(95)
Here, K denotes a nonloal operator very losely related to the operator
appearing in (9). Preisely:
Kw = F
1
(1
q
U ) k K ( 1 U kt) (F w) ;
1
2
2
2
1
(96)
where K (j t) is the kernel we had before. In partiular, both the loal and
nonloal approximations we have disussed an be used here. In [48℄, for
example, we use the Pade sequene for a problem involving a periodi array
of pressure pulses. The results are similar to those reported in Figure 2.
As mentioned above, it is also possible to generalize the loal boundary
ondition sequene we have disussed from the wave equation to the onvetive wave equation [51℄. However, we have yet to omplete the onstrution
of a well-posed inow ondition. We note that for the exterior problem a
smooth artiial boundary must ontain a point where U n = 0. Thus the
swith between inow and outow onditions may require some additional
ompatibility onditions for the auxiliary variables. Given the importane of
ompressible ow problems, the resolution of this issue is ertainly important.
26
Thomas Hagstrom
Variable CoeÆients and General Systems
We've now ompleted the
list of hyperboli problems whih we an satisfatorally solve. Although it
ontains many of the more important equations arising in appliations, it
ertainly doesn't ontain them all. Partiularly glaring is our inability to
treat variable oeÆient problems and general hyperboli systems.
We note that in their original work [27℄, Engquist and Majda do treat
the variable oeÆient ase from the perspetive of the reetion of singularities. More reently, pratial algorithms for onstruting the Engquist-Majda
onditions to high order have been proposed [7℄. However, the underlying expansions are only onvergent up to smooth errors, whih may not be small.
Thus, unless a stronger form of onvergene an be established for some ases,
it is not obvious that high order onstrutions will be useful. (We note that
a sort of onvergene has been established in the highly osillatory limit in
[55℄ for the lowest order Engquist-Majda onditions.)
A seond approah to these issues is the reformulation of Higdon's boundary onditions in [35, 36℄. Already in their original version appliations to
problems in stratied media have been undertaken [64℄. For problems with
traveling waves only, the input to the method is simply a range of wave
speeds whih an, in priniple, be estimated from the oeÆients. However,
the reformulations urrently require a seond order system and need to be
generalized to the rst order ase. Moreover, modal solutions to variable oefient problems may be growing or deaying and it is unlear if the onditions
need to be reformulated in that ase. In any event, we believe the method has
great promise, but it is lear that muh more experimentation and analysis
is needed.
2.4 The Shrodinger Equation
Waves, partiularly dispersive waves, an also be desribed by higher order
equations. Work on these systems is less well-developed, though low order
onditions of Higdon type have ertainly been used. In reent years, however,
there have been advanes in the treatment of the most important equation
from this lass, the Shrodinger equation. In partiular, two reent dissertations, by Jiang at NYU [72℄ and Shadle in Tubingen [88℄ deal with the
onstrution, analysis and testing of highly aurate nonloal onditions. (See
also [80℄.)
We begin again with the ase of a ylindrial tail:
i
u u
+ L(y; =y )u:
=
t x
2
2
(97)
Expanding in a Fourier series in the ross-setion based on the eigenfuntions
of L and performing a Laplae transform in time we obtain a formula for an
exat boundary ondition:
u^j
+ ( is + j ) = u^j = 0:
(98)
x
2 1 2
New Results on Absorbing Layers and Radiation Boundary Conditions
27
We immediately note an important dierene between (98) and its wave
equation analogue. To derive boundary onditions for the wave equation,
we simply removed the large s behavior from the nonloal term until the
remaining operator was equivalent to onvolution with a bounded funtion.
As for large s the symbol was linear, the total operator was the sum of
a dierential operator and the onvolution. Here, in ontrast, the symbol
behaves like (is) = for s large. To desingularize this somewhat, we an use
the Neumann-to-Dirihlet map as in [88, 80℄,
1 2
F
Vj u
F x
+ e i 4 u = 0;
(99)
(s + ij )
;
( is + j ) = = e i 4
(s + ij ) =
(100)
1
or, as in [72℄, write:
2
2 1 2
2 1 2
leading to:
u
+e i4F
x
1
Vj F
u
t
iLu
= 0:
(101)
In either ase, the boundary ondition is expressed in terms of onvolution
with:
2
e ij t
(102)
Vj (t) = L (s + ij ) = = p :
1
2
1 2
t
Unlike the kernels we dealt with earlier, the singularity at t = 0 preludes
the uniform approximation of Vj on (0; 1). Thus both authors split the
onvolution into two piees, a loal piee on (0; Æ ) to be handled by diret
disretization, and a global piee where Vj is replaed by exponential or
pieewise exponential funtions. Jiang's global approximation is based on
approximating the integral formula:
1
p
t
=
2
Z
1
e
z2 t dz;
(103)
0
and leads to the estimate:
ql = O (ln 1= (ln T=Æ + ln ln 1=)) :
(104)
Lubih and Shadle, on the other hand, use the same algorithm they suggest
for kernels arising from the wave equation. They require about the same
number of poles, though the implementation is more omplex. They do treat
the disretized problem, however.
Similarly, exat boundary onditions an be formulated on spherial and
ylindrial boundaries. Transforms of the resulting kernels are still expressed
28
Thomas Hagstrom
by logarithmi derivatives of modied Bessel funtions, but now the arguments are proportional to s = . Theoretially, Jiang shows that the approximation theorems of [5℄ an be applied in this ase, preisely to:
1 2
L
p
1
p
!
isK0 ( is)
p :
(s s )K ( is)
(105)
(A multipliation and division by s s has been performed to regularize
the kernel.) Choosing Æ as above he shows that for harmonis of index l the
number of poles required is:
ql = O ((ln 1= + ln 1=Æ) (ln l + ln 1= + ln 1=Æ)) :
(106)
Atual approximations are also omputed in [72℄ by a least squares proedure. Generally, more poles are required than for the simpler ase of the
wave equation. Nonetheless, these nonloal approximations do provide arbitrary auray at relatively small ost. Moreover, high-order loal alternatives
do not as yet exist.
As an example of the diÆulties in onstruting high-order loal boundary onditions for the Shrodinger equation, we point to the reent paper of
Allonso-Mallo and Reguera [4℄, where they study the stability of Higdon-type
boundary onditions proposed for the Shrodinger equation in [29, 24℄. They
nd mild instabilities whih worsen with inreasing order for semidisretizations of the problem. Their analysis ends at rather low order by our standards
- a rational approximation of degree (3; 2). It leads to doubt about the utility
of very high order onditions, though this still needs to be tested.
3 Absorbing Layers
The alternative to domain trunation by aurate boundary onditions is
the use of an absorbing or sponge layer. The idea is simple - extend the
omputational domain from the boundary of the physial domain, I , to
a seond boundary, O , and hange the equations in the new buer zone
so that waves deay. It is obviously not diÆult to write down dissipative
wave equations whose solutions deay rapidly as they propagate. However,
impinging waves generally reet o the interfae or transition zone and thus
return to . As a result, early methods of this type (e.g. [71℄) required a
gradual inrease in the absorption parameters, leading to fairly thik layers.
All of this hanged with Berenger's introdution of the perfetly mathed
layer (PML) for Maxwell's equations [14℄.
The new property of the PML is a reetionless interfae with the physial domain. In the original paper, the origins of this property are somewhat obsured. However, soon thereafter it was noted that the PML was
New Results on Absorbing Layers and Radiation Boundary Conditions
29
in fat equivalent to a ontinuation into the omplex plane of the real, normal oordinate, for example x for our ylinder domains [17℄. This observation has led to a number of important generalizations to urvilinear oordinates [94, 20, 96, 82℄, whih we'll disuss below, and to more omplex media
[95, 97, 98℄. We will present a nonstandard derivation of the PML whih
generalizes the oordinate transformation approah. A result of this generalization has been the extension of the PML to the linearized Euler equations.
Interestingly, in this framework the reetionless interfae property is easily
satised, but the dissipativity ondition is not.
3.1 Perfetly Mathed Layers for Hyperboli Systems
Consider our usual ylindrial tail and suppose we are solving a rst order
hyperboli system whih we allow to have varying oeÆients in the ross
setion;
u
u
u X
Bj (y)
+ C (y )u = 0:
(107)
+ A(y ) +
t
x
yj
j
Just as before, the physial domain is loated in x < 0, the interfae, ,
between the physial domain and the absorbing region is loated at x = 0,
and the absorbing region itself lies between x = 0 and x = L.
Performing a Laplae transformation in time we nd modal solutions of
(107):
u^ = ex ;
(108)
where
0
sI
+ A +
X
j
1
Bj (y)
+ C (y )A = 0:
yj
(109)
We label inoming and outgoing solutions by looking at < for <s suÆiently
large. However, in the limit <s ! 0, we typially have < ! 0. The idea
behind the layer onstrution is to modify the equation so that modes are
damped as they propagate, meaning that < is bounded away from zero for
<s 0. We also want to assure that there is no reetion at the interfae
between the physial and absorbing regions. The most diret way to do this
is to design the problem in the layer so that its modal solutions for any s
have the same eigenfuntions, , as (109).
Thus our starting point for onstruting the layer is the formal modal
solution:
Rx
1
u^ = ex (R ) 0 z dz :
(110)
+
^
( )
In the examples we have arried through so far, R is a rst order dierential
operator in time and the transverse variables with Laplae transform R^ ,
R=
X j
+ ;
+
t
yj
j
(111)
30
Thomas Hagstrom
is some number, and 0 is the absorption parameter. We note that
more omplex hoies for R and are ertainly possible, for example R and
ould be dened by higher degree rational funtions of s with operator or
matrix oeÆients. This may prove to be neessary to extend the method to
other problems. We also note that if = 0 and R^ is represented by a funtion
of s (and tangential wave numbers for onstant oeÆient problems), we an
view the transformation as the result of extending x into the omplex plane,
as in [17℄ and its desendents.
It is now a simple matter to write down a system of equations satised by
u in the layer. In partiular, when we substitute (110) into the new system
we want (109) to be the result. This yields:
0
sI
+ (I
1
X
Bj
+ C A u^ = 0:
+ +
(R + ) )A
x
y
j
j
1
(112)
Introduing auxiliary funtions w we nally obtain:
X
u
u
u
Bj (y)
+ C (y )u + w = 0:
+ A(y )
+ u +
t
x
yj
j
(113)
u
Rw + w + A(y)
+ u = 0:
(114)
x
We emphasize that the interfae, x = 0, is nonreeting for any hoie of
R, and . The diÆulty is to hoose them so that all waves are damped.
Below we will examine some speial onstant oeÆient ases where this an
be ahieved, albeit by inspetion rather than by some systemati proess.
A pratial diÆulty in implementing the PML is the hoie of the absorption prole, . This is purely a numerial issue, as theoretially we an
hoose it to be onstant and arbitrarily large. In our experiments we make it
linear, and as our methods are high order we treat the interfae as an internal boundary and use harateristi mathing aross it. However, we typially
need to experiment to nd near optimal slopes for the linear prole. The only
serious analysis of this question whih we are aware of is [19℄, where numerially optimal proles for a spei disretization in the frequeny domain are
found.
Maxwell's Equations
We return to (77)-(78), whih we reall was the rst
system for whih a PML was onstruted [14℄. Reordering the unknowns into
the 6-vetor u = (Ex ; Bx ; Ey ; By ; Ez ; Bz )T we note that the matrix A is given
in blok form by:
0
1
O O O
01
A = O O T A; T =
:
10
O T 0
(115)
New Results on Absorbing Layers and Radiation Boundary Conditions
31
From (114) and the struture of A we onlude that the rst two omponents
of w are zero. Thus we will take w to be a 4-vetor appearing in the last
four equations of (113) and (114). As the system has onstant oeÆients we
may additionally perform a Fourier transformation in y and z . This leads to
expliit expressions for :
= 0; s
+ kx + k y
2
2
1=2
2
2
;
(116)
whih, of ourse, are by now quite familiar. Letting R denote the symbol of
R our problem is to guarantee:
0
s2 + k 2 + k 2
x
y
B 2
<
1
1=2
C
A > 0:
R
(117)
A simple hoie whih by an easy alulation an be shown to satisfy (117)
is:
s
R = + ; 0; = 0;
(118)
leading to the equations:
0
1 E
t
E
rB+B
wy
(
t
1
t
1
t
)
wzE)
C
A
= 0;
(119)
(
0
1 B
1
0
1
0
B
+rE+B
wy
(
)
wzB)
C
A=
0;
(120)
(
++ w E +
(
++ w
)
B) + (
Bz x
By = 0;
x
Ez x
Ey = 0:
x
(121)
(122)
Although the equations themselves look quite dierent, a study of the
eigenvalues reveals the formal equivalene between solutions of (119)-(122)
and solutions in the layer onstruted in [13℄. In partiular, the hoie of
> 0 orresponds to the omplex frequeny shift disussed in [13℄.
Linearized Euler Equations We now onsider the linearized Euler equa-
tions, (83)-(85), again assuming a subsoni ow with U 6= 0. Then A is
nonsingular so that we require the full omplement of auxiliary variables.
Again we perform a Fourier-Laplae transformation and nd that is given
by:
1
32
Thomas Hagstrom
1
=
s~ U s~ s~ + (1 U )(ky + kz )
;
;
(123)
=
U
1 U
s~ = s + iky U + ikz U :
(124)
We note the two distint forms for , orresponding to the two types of waves.
We rst hoose R in analogy with the ase of Maxwell's equations,
2
1
2
1
2
2
2
2
1
1
2
3
R = s~;
(125)
is suggested. Then =R is given by:
1
U
1
;
1
U
1
U
2
1
1=2
s~ + (1 U )(ky + kz )
(1 U )~s
2
2
1
2
2
1
2
:
(126)
Clearly when the square root is zero we an't have the orret sign on both
terms. Thus we hoose:
U
:
(127)
=
1 U
Now all terms now have the orret sign. We note that, as in (118), we ould
add a positive lower order term, , to R. However, it would then be neessary
to make dependent on s~, leading to additional auxiliary variables.
We have arried out preliminary numerial experiments with the new
layer for a problem dened by a periodi array of pressure pulses in the twodimensional uniform ow U = 0:3, U = 0:4. The numerial method is eighth
order in spae and time. Relative errors as a funtion of time and average
absorption parameter are listed in Table 3. Here the interior mesh is 258 64
while we have 34 64 points in eah layer. The absorption prole is linear,
and the layer is terminated by harateristi end onditions. Clearly, the
auray is aeptable. Moreover, the insensitivity of the results to hanges
in indiates that the primary error soure is the disretization itself. We
note that the same problem has been solved in [48℄ using the Pade sequene
of loal boundary onditions, but there the solutions were better resolved so
that the dominant error was due to the boundary ondition. We plan to redo
these omputations so that diret omparisons an be made.
1
2
1
1
Table 3.
2
t = 50 = 75 = 100
1 1:6( 3) 1:6( 3) 1:6( 3)
2 1:4( 3) 1:4( 3) 1:4( 3)
4 1:1( 3) 1:1( 3) 1:1( 3)
8 1:3( 3) 1:4( 3) 1:4( 3)
16 2:4( 3) 2:5( 3) 2:9( 3)
32 3:4( 3) 2:4( 3) 2:4( 3)
Relative errors - periodi array of pressure pulses, Euler PML L = 1=2
New Results on Absorbing Layers and Radiation Boundary Conditions
33
There have been other earlier attempts to generalize the PML to the Euler
equations, beginning with Hu [68℄. However, these all had stability problems
[37, 58, 93℄. Reent xes in [2, 69℄ are restrited to ows aligned with the layer
oordinate, while others [79℄ sarie the perfet mathing property. Thus
the general formulation developed here seems to have provided a onvenient
framework for onstruting a true PML. We note that similar PMLs for
advetive problems have been onstruted in [69, 11, 23℄.
Returning to the variable oeÆient formulation, we have also begun experiments with the PML for linearizations about jet ow proles [52℄. A
onfounding fator in this ase is the instability of the ow prole itself. Now
the parameter is hosen numerially to suppress instabilities in the layer.
So far, we have been unable to onstrut an absolutely stable layer, but we
have been able to make the growth rate small enough that aurate results
an be obtained over long time intervals.
Exterior Problems
There are a variety of ways that the PML an be
applied to problems in exterior domains. In partiular, retangles or onvex
regions of arbitrary aspet ratio an be used. This is, in our view, the primary
advantage of the PML over high-order absorbing boundary onditions.
The standard approah is to use a retangular box. Then, besides unidiretional layers attahed to eah fae, there must be orner layers attahed
to the layers themselves. In these we use multiple sets of auxiliary variables.
In a three dimensional orner, for example, we use three sets of auxiliary
funtions. Thus our general formulation would be:
X
u X
w j = 0;
Aj
+ j u + Cu +
+
t
x
j
j
j
( )
Rj w j + j A j
+ j u + j w j = 0:
xj
( )
( )
(128)
(129)
(Here we assume onstant oeÆients and rename the oeÆient matries.)
An alternative to a retangular box is a urved boundary. Using the oordinate mapping tehnique, the PML has been extended to spheres and
ylinders [94, 20, 82℄. The idea is simply to apply the omplex hange of variables to the radial oordinate. For example, onsider Maxwell's equations for
a TE mode in two spae dimensions using polar oordinates. Following [20℄
we make the hange of variables:
r ! r 1 + ; = r
s
1
Z r
()d;
(130)
= 0;
(131)
R
the equations for the Laplae transforms beome:
s^
E
r
s
1 + 1
1 B^z
r 34
Thomas Hagstrom
B^z
s^ E + 1 + = 0;
s
r
1^ E^ E 1 + + 1 + r
s
r
s
1
s^ Bz + 1 + s
1
1
(132)
1
1 E^r
r = 0: (133)
Returning to the time domain we see that only a single auxiliary variable is
needed:
1
1 Bz
+ Er
= 0;
(134)
t
r 1
B
+ E + z = 0;
t
r
E
1
1 Er
1
+ Bz + + E
+ w = 0;
t
r
r
r 1
E
+ w + ( ) = 0:
t
r
(135)
(136)
(137)
For satterers whih are well-t by a sphere, this method has the advantage that it avoids the expensive orner regions where multiple sets of
auxiliary funtions are needed. Even more general urvilinear oordinates
are onsidered in [96, 99℄. Here the only restrition is that the loal radii of
urvature of the oordinate system remain positive.
Stability and Convergene
To prove the onvergene of the solutions
obtained using the PML to the restrition to of the unbounded domain
solution one must establish the onsisteny of the method, as we have done
in some ases for the boundary ondition sequenes, and its stability. In the
frequeny domain this has been arried out [74, 75, 66℄, but we know of no
analagous results in the time domain. Stability, however, has been the fous
of a great deal of analysis, whih we will disuss below.
As rst analyzed in [1℄, Berenger's original formulation is not strongly
well-posed, suggesting the possibility of instability and ill-posedness under
perturbation. Later formulations, suh as the ones presented in [82℄, are
strongly well-posed. Strong well-posedness for (113)-(114) is easily analyzed.
Freezing oeÆients and performing a Fourier expansion in all spatial variables leads to the symbol of the PML equations:
P^ =
P
ik A + j ikj Bj
ik A
1
1
P
O
:
j ikj j
(138)
Clearly, P^ is diagonalizable
assuming the strong hyperboliity
P of the original
P
problem whenever j ikj j is not an eigenvalue of ik A + j ikj Bj . If it is,
the assoiated eigenvetors must be null vetors of A or we must have k = 0.
These onditions an be heked for our two examples.
Beyond strong well-posedness, whih we see is independent of the sign
of the damping term, , we require that the layer be asymptotially stable;
1
1
New Results on Absorbing Layers and Radiation Boundary Conditions
35
that is we require that no growing modes exist. This analysis is more diÆult.
However, for the Maxwell PML Beahe and Joly have reently developed an
energy method whih establishes this stronger form of stability [12, 13℄. Interestingly the method does not apply when is variable, whih is standard
in pratie. Therefore, the analysis of the asymptoti stability of the layer
equations is an important open problem. One might hope that a new tehnique for treating this issue would lead to new insights into the onstrution
of stable layers for variable oeÆient problems.
3.2 Other Absorbing Layer Tehniques
Although the PML has dominated the attention of the omputational ommunity in reent years, there are important problems to whih it has not
been applied. In many of these problems, older, ad ho methods are still in
use. A ase of partiular interest arises in the attempt to simulate the aeroaoustis of shear ows suh as jets. Due to ow instabilitites, both ne sale
turbulene and large vortial strutures are present at the outow boundary.
These ertainly invalidate the analysis used to derive the PML and radiation
boundary ondition for the linearized Euler equations. Moreover, in pratie
the auray is also degraded, as shown in [21℄. These authors suggest a different method, whih seems to have beome the method of hoie at present
for diret simulations of aeroaousti phenomena [30, 76℄.
The basi desription of the method of [21℄ is as follows; apply a smooth
grid strething in the absorbing region, whih has the eet of making the
propagating waves of shorter wavelength relative to the grid, and use some
form of artiial visosity or ltering to damp them. The omputational experiments of [21℄ indiate that the method an be eetive so long as the
buer zone is suÆiently wide.
In omparison with the PML, even more parameters need to be hosen without muh theoretial guidane: grid strething proles, buer zone
lengths, and low-pass lters. As the method is learly useful, we believe it
would be of interest to develop some theory. An interesting move in this diretion is the work in [22℄, whih asts suh proedures as a omposition of
grid mapping and ltering using the language of supergrid models. Although
we emphasize that no hard error estimates diretly follow, the formulation
may be a good starting point.
3.3 PML for the Shrodinger Equation
Finally, we onsider the onstrution of a PML for the Shrodinger equation.
In [3, 77℄ they have been onstruted using the omplex oordinate strething
tehnique in more ompliated settings than we will onsider here, inluding
systems desribing exitons and one-way wave equations.
After Fourier-Laplae transformation, solutions of (97) are of exponential
form with:
36
Thomas Hagstrom
=
is + j
2
1=2
:
(139)
As before, we seek to replae this in the layer by:
+ R
Z x
(z )dz:
(140)
0
We note that we must now also require ontinuity of u
x , so we will impose
the additional ondition:
(0) = 0:
(141)
Also, as the system is isotropi we don't require 6= 0.
Unlike the exponents whih we dealt with in the previous ases, the argument of now varies only between 0 and when <s > 0. Thus it an
made to have a nonvanishing real part by a simple rotation. That is we an
hoose R to be any omplex number whose argument lies between 0 and .
We thus obtain:
2
2
i
1
u
u
1
+ Lu; 0 < < :
=
i
i
t 1 + e x 1 + e x
2
(142)
Equation (142) has a lear interpretation. We simply make the system
paraboli in the layer. However, note that inreasing the absorption parameter orresponds to dereasing the diusion oeÆient, thus spawning a
boundary layer at the interfae. We also emphasize that, unlike the PML for
hyperboli equations, no additional variables are needed in the layer.
We have performed some simple one-dimensional numerial experiments
with the Shrodinger PML. Choosing = we use seond order dierenes
and a seond order BDF method in time. The layers ontained twenty-ve
points, the absorption prole was quadrati, and the termination was with
Dirihlet onditions. The solution, whih we ompute up to T = 50, was given
initially by (32= ) = exp (ix 16x ), = 1; 10. The errors as a funtion of
and t are plotted in Figure 6. The results are quite good, though the reader
should be autioned that this is a one-dimensional experiment. They do show
a diÆult-to-predit dependene on .
4
1 4
2
4 Conlusions and Open Problems
In summary, for a small but important list of problems domain trunation
methods apable of delivering arbitrary auray at aeptable ost are now
available to the omputational sientist. These inlude the equations of aoustis, eletromagnetis and elastodynamis in homogeneous media and waveguide or exterior geometries. Certainly there is room for improvement in the
eÆieny and the mathematial analysis of these suessful methods, but we
believe they are already well-grounded, reliable tools.
New Results on Absorbing Layers and Radiation Boundary Conditions
37
eta=1 nL=25
−2
10
−3
10
sigma = 6400/3
−4
relative error
sigma = 800/3
sigma = 3200/3
10
sigma = 1600/3
−5
10
−6
10
−7
10
−8
10
−9
10
−2
−1
10
0
10
1
10
time
2
10
10
eta=10 nL=25
0
10
sigma = 800/3
−1
10
−2
10
sigma = 1600/3
−3
relative errors
10
sigma = 6400/3
−4
10
−5
10
sigma = 3200/3
−6
10
−7
10
−8
10
−3
10
−2
10
−1
0
10
10
1
10
time
Fig. 6.
Errors for the Shrodinger PML, 25-point layer, = 1; 10.
2
10
38
Thomas Hagstrom
As we've presented a number of dierent tehniques, it is natural to try to
ompare them. At present, it would be premature to denitively laim that
one or the other is best. For a xed domain onguration, it is the author's
experiene that high-order boundary ondition methods are somewhat more
eÆient and easy to use than the PML. This is due to the fat that the PML
depends on the hoie of the absorption prole and on adding points in a
volume distribution to ahieve onvergene, whereas the boundary ondition
methods only require the addition of auxiliary funtions. On the other hand,
the PML an be used in arbitrary onvex domains, and thus is urrently more
eÆient for high-aspet ratio satterers. Comparing boundary onditions,
the nonloal approximations are the lear winners from the point of view
of omplexity analysis. However, the loal sequenes are easier to implement,
and often require many fewer auxiliary funtions than the error analyses
suggest. Thus they an be ompetitive if a good adaptive strategy an be
found.
We have identied a number of important open problems whose resolution
is likely to have the nal say on the future importane of the various aurate
tehniques. These inlude:
i. Extension of the high-order loal and/or nonloal boundary onditions to
high-aspet ratio boundaries suh as boxes or spheroids.
ii. Sharp error analysis for the time-domain PML, partiularly in high-aspet
ratio domains.
iii. Convergene analysis of Higdon-type boundary onditions for problems
with variable oeÆients.
iv. Improved understanding of the stability of PMLs leading to their extension to a wider range of problems inluding those with variable oeÆients.
v. Adaptive order determination for loal boundary ondition sequenes.
vi. Optimization of absorption parameters in the PML.
vii. Mathematial analysis of alternative absorbing layer methods.
We believe that progress on these issues is possible, and that it will lead to
substantial improvements in our ability to simulate waves.
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