one needs a finite region 1n which to solve these problems. ... Applications in Science and Engineering, NASA Langley Research Center,. Hampton, VA 23665.
NASA Contr.lctor Report 177944
lCASE REPORT NO. 8533
NASACR177944
19850027363
leASE Absorbing Boundary Cond1tions for Exter10r Problems
S. I Hariharan
Contract No. NAS117070 July 1985
INSTITUTE FOR CO~WUTER APPLICATIONS IN SCIENCE AND ENGINEERING NASA Langley Research Center, Hampton, Virginia 23665 Operated by the Univers1ties Space Research Association
NI\5I\
Natlonal Aeronautics and Space Administration
Langley Research Center Hampton, Virginia 23665
1111111111111 IIII 11111 11111 11111 11111 IIII 1111
NF00703 •
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1 Report No
NASA CR177944 lCASE Report No. 8533
3
2 Government Accession No
Recipient's Catalog No
5 Repon Date
4 Title and Subtitle
Absorbing Boundary Conditions for Exterior Problems
Julv 1985 6 Performing Organization Code
7 Author(s)
8 Performing Organization Report No
8533
S. I Hariharan
10 Work Unit No 9
Performing Organization Name and Address
Institute for Computer Applicat10ns in Science and Engineer1ng Mall Stop 132C, NASA Langley Research Center Hampton, VA 23665
11
Contract or Grant No
NASl17070 13 Type of Repon and Period Covered
12 Sponsoring Agency Name and Address
National Aeronaut1cs and Space Administrat10n Washington, D.C. 20546
r.nnrr",rtnr Rnort 14 Sponsoring Agency Code
505318301 AdditTonaT support: NSl" Grant NAljl)U Langley Technical Monitor: J. C. South, Jr.; Final Report To appear in Numerical Methods for Partial Differential Equations, (5. I. Hariharan and T.H. Moulden, eds.), Pitmas Press, 1986.
15 Supplementary Notes
16 Abstract
In th1s paper we consider ellipt1c and hyperbolic problems in unbounded These problems, when one wants to solve them numerically, have the regions. difficulty of prescr1bing boundary conditions at infinity. Computationally, one The corresponding needs a finite region in which to solve these problems. conditions at infinity imposed on the finite distance boundaries should dictate the boundary condition at infinity and be accurate with respect to the interior numerical scheme. Such boundary conditions are commonly referred to as absorbing boundary conditions. This paper presents a survey and covers our own treatment on these boundary cond1tions for wavelike equat10ns.
17 Key Words (Suggested by Author(s))
18 Distribution Statement
64  Numer1cal Analysis exter10r problems, absoring boundary cond1tions, integral equations, finite elements, spectral approximations, and f1nite differences. Unclassified  Unlimited 19 Security Oasslf (of thiS report)
Unclass1f1ed
20 Security Classlf (of thiS page)
Unclass1f1ed
21
No of Pages
34
22
~Ice
A03
For sale by the National Technical Information SerVice, Springfield, Virginia 22161 NASALangley, 1985
ABSORBING BOUUNARY CONDITIONS FOR EXTERIOR PROBLKKS
S. I Har1haran Univers1ty of Tennessee Space Institute Tullahoma, Tennessee
Abstract
In this paper we cons1der elliptic and hyperbolic problems in unbounded regions.
These problems, when one wants to solve them numerically, have the
difficulty of prescribing boundary conditions at infinity.
Computationally,
one needs a finite region 1n which to solve these problems.
The corresponding
conditions
at
infinity
1mposed
on
the
finite
distance
boundaries
should
d1ctate the boundary condition at inf1nity and be accurate with respect to the interior numerical scheme.
Such boundary conditions are commonly referred to
as absorbing boundary cond1tions.
This paper presents a survey and covers our
own treatment on these boundary conditions for wavelike equations.
Research was supported in part by the National Aeronaut1cs and Space Administration under NASA Grant NAGl527 and in part by NASA Contract No. NASl17070 while the author was 1n residence at the Inst1tute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23665.
i
ABSORBING BOUNDARY CONDITIONS FOR EXTERIOR PROBLEMS by
S.
r.
Hartharan *
Many fOrlnuleLLlons elristng from physical Ildtllre YIeld problems Ifl IlnbOllndt'd regions MdLht'matical formulatIons of such probh~rns ~ Ield governing partlCll dilferentlel.l equations in or near a given domain in such a fashIon that.: i) the equations may Iw linear but \\ith nonconstant coelficients, or ii) the equation., may be nonlinear, but at leLrge dl.,tances essentially behave linearly and with comtdnt coelficients. ThIS note presents a survey of the treatment of ~Ilch problems, when the desired C;;ollltlonc;; are governed by elliptic or hyperbolIc partiell differential equation:, Thf'o.;e problem:, are called exterior problems and commonly artse In the field., of aerodynanllcc;;. mett'Orology. electromagnetIc c;;cattering, and atmospherIC acoustical \"a\(' propagation. The rIldUl difficulties with these probl(~ms are the boundar), condItIOn., that need to be preSCribed at large distances from the regIOn of intc'r('..,t. CSllaliy only an a...,ymptotic behaVIor is known Such conditions may 1)(' "ulficient to check the weliposedness of the problem however. If one wants to compute the solutions of these problems numerically, inlirute distances need to be truncated to linite distances. The boundary condItion" imposed on the.,e finite distanct' boundartes should dictate the behaVIor eLt infinity dnd be c\ccurate \\ it h the illterior numencal scheme. furthermOrE'. the shorter tht' dic;;tann's. the mure efficipnt the solutions in term~ of (,0111putatioll ttnl(, rpcl'lin'd. ThIS con"iderat.ion is the (~ssential need tn ",{'\eral pl'Obl(,1I15. c\t'(>t>I1
L) say with bounddry cOlldl
(I :!) ( l.:~ ) u, u
l
contlnuous on lnterfaces,
(1 I)
where ill and ill are boundaQ operators and 9 are given are to be cho",('11 according to the physics of t tlf' prohlem. For example. If thert> I" a \Hill
amplitude wave traveling from solution may be written as
00
Incident on the slab, then for x
< 0 the
(1 1) where R(k) is the rellcction coefficient and measures the part of the wave which h, reflected from the boundary x = 0 Elimtnating R(k) from (1.5) yields
u'(O)
+ ~ku(O)
= 2lk
(1 6)
=
which h, ('(1I\(,d cut inflow condition. Thus in view of (1.2) 8 I d/d.r + ik clnu y = '!.tk. Now we can do a similar calculation to obtain a boundar) comittion at U For x > L all the Wclves transmit and do not reflect back as there are no other bounuaries for x > L. [n this region the solution may I,l' written as u(x) = T(k)e'knnx (1. 7) where T(k) is titt' tran1'>rtllssion coefficient which measures the transmitted part of the wave. f:liminating T(k) in (l.i) ~e obtain
Ii (L')  lkn.o 11.( L') =
o.
( 1.8)
Again comparing with (1.:l) we see that the operator 8 2 is
81
=dxd  lknO.
(1 9)
The boundary condition (1 8) IS the desired absorbing boundary condition for this problem, which is exact. We note that thi., condition could have been appllt'd ('''{actly at J = "and this. as \\(' will .,ct'. clop., not d.tways holu in high('r' dtnlfmsions. TllffiCient does not exact Iy \\ork In thIS ~itudtion This :,how~ the difficult), ill higher dimensions. However. If we settle for loss of accuracy, in particular elullInating ao{O),we have
(26) Thus we see that !
r2(llr  tklL} = O(l/r} 6
\
+  r +  r2 + "J
..
, ( .) )
or simply rL~: r ~ (tL r in two dirnem:ilons.

zktL) = 0 which is Sommerfeld's radiation comlttion
Taking a c1o.,cr look at the right hand .,ide of (2.6), in particular, the first exprC'ssion, we sce that it is nothing more than u/2r. Thus we have
(2.8) Comparing (2 i) cmd (2 8) shows the error introduced in the boundary U)(ldition at large distanccs is reduced by a factor of L/r Tllls was first studlcd in corlllt'ction wit.h nUlIlcrical implementation for tune dependent problern~ by Bayli~s and Turkel[31 and also by Engquist and \lajda[11 For time harmOniC' casps with time dependence of the form e  ,kt, both of their boundary conditions reduce to the form (2.8). Bayliss, Gunzburger ,and Turkel[21 generalized these conditions. They defined the boundary operator on the left side of (2.8) by
Bt
=
:r 
ik + 1/2r.
(2.9)
Wht'n (2.8) is explicitly written it has the form (2.10) They oh'H~rved that now the second coefficient al(O) can also be eliminated If one does that by setting v = BtU it is readily venfied thdt
av.   zkv + :) v = 0 ( 1/ r9 /) 2 ar
r
or
lh IL = ( iJ ar
tk
. + 1 ) + :;) (()   tA.·
2r
ar
2r
It
= 0 (1.)/2) 1/ r .
(2 (1)
This process can be repeated to generalize
(2 12)
where
8m =
IT (aar, + (2j r ~) Tn
2

l.k
)
•
(2 13)
J=I
Thus in crdcr to IInplement this approach the problem can be considered in a truncilL~d fhavior. To apply the above process let us consider the following problem: ~1L
= ()
in
n+
(2.19)
'PtTk('PJ)'
(:121) We can now c\
(r,e)
Figure 7
a
~ra.ppillg
of physical d(lmain to computational domam
is introduc('d,and this is why we call this form the infinite order radirl.tion condition. Thc actudl implementation in a simplified form is as follows: Suppose the boundary r is polar representable as r = R( 0). Wt> use Cheby~hev polynomial approximation in the radial direction and FourIer (if'composition ill the angular direction. F'or this we need to make a coordinatp transform as depicted in figure i. In particular the transformation is
'P=o r =
r(05,O).
( 1.16)
So that
r(  L, 0) = R(O) () < 0 < 2iT " r( L. 0) = Roo  
(1.Ii)
he side (L) in the transrormed plane will corre.,pond to the houndarj r and side (2) will be the bou"ndary roo' Such a transforrndtlon can be dOlle in several ways. [11 particular. 'the rollowing stretched one is very elfectl\e
(1.18) Hl
So that at s = 1, r( 1,0) = R( e). Demanding r( 1, e) the stretchi IIg pclrarnet('r 0(0) as
= Roo, we determine
I
0(0)= ilog(Roo/R(O)).
(1.19 )
This transformation in turn changes the Laplace equation into the form
(4.20) where a, b. C ellHl d dre functions of sand 0 obtained from chain rule. Seek approximat.e' soilltion., or the form AI
/.£N (.'3,0)
=
..
L L
Umk Tm(s)
elkS
(4.2 L)
m=O IklSN
where Tm is the rnth Chebyshev
polynomia~
Tm(cos e) and
/.£N
of the first kind, defined by
= cos (me)
is detcrmined by I'£~ at the (AI
+ 1) x 2N
(4.22) Cheby~llt~vFourier nodes
( I 23)
i = 0, ....
1\t[
j = 0.···· 2N  1. At i = ol[ the given Dirichlet condition should be imposed. At 1 = 0, iJu/as obtained from (JIL/,)r should be Ilpdated. Placing the inhomogeneous terms arising from the boundary conditions in a forcing term I, let us re\\ rite the final ... pcctral operator dS
(I. 2 I) The malri'( rormed b) {.I ~p IL'V is large and dops not have spar.,e .,tructures. Thus ill order to '\olve (1.2 t). an iterative procedure is deslrabl~. \Vt> outline only the brh.f id 0) without renecting ("pc ligllrc 8) and gOH'rll.l2) gives a perfectly nonreflecting condition at r = L it is rntht'r impractical to impose computationally. However, m the psetldodilfcrentltll operator terminology 1 VT2  ,2 IS nothmg but the symbol of the pseudo differential operator
J:rt22~
:}, ',.
Engquist and Majda proceeded
wit.h thi ... gllid(,llI1e thtlt the approximations of this symbol tv'T2  \2 )' leld a ramily or clpproximatc boundary conditions. [n particular, approxuuatlOIl"i around, = 0 (thi; has the p\tY!'ilcal meaning of near normal incidence) give" til 11)
= L to obtain
elrt+a