Bayliss Paper

AD-AO94 881 NEW YORK UNIV NY COURANT INST OF MATHEMATICAL SCIENCES F/B 12/1RADIATION BOUNDARY CONDITIONS FOR WAVE-LIKE EQUATIONS. (U)DEC 79 A BAYLISS- E TURKEL NASI-14101

UNCLASSIFIED IL

M ICRO)COPY Rf SOL I ION II SI C HARTI

)Radiation Boundary Conditions for Wave-Like /Io quations i

J~7'~ALVINbAL 1SInsite o puate tpp ons in Scice and E gineefi ng and ouran Institute

CO 1j r, /C -AND

Courant 1 of Tel Aviv7

Abstract

In the numerical computation of hyperbolic equations it is not practical to use infiniteomains. Instead, one truncates the domain with an artificial boundary. In this study we construct

sequence of radiating boundary condtions for wave-like equations. We prove that as thetificial boundary is moved to infinity the solution approaches the solution of the infinite domain

as 0(r-'-"') for the rn-tb boundary condition. Numerical experiments with problems in jetacoustics verify the practical nature and utility of the boundary conditions.

1. Introduction

In many problems of interest one wishes to solve time-dependent equa-tions in an infinite domain. Examples of such situations include jet acoustics.seismology, exterior aerodynamics, exterior ballistics, astrophysics, and laserfusion. However, for computational expediency one is required to operatewithin a bounded domain. One possibility is to map the infinite domain ontoa bounded one. However, in many circumstances this mapping can aggravatethe situation, especially when the solution is oscillatory at infinity, or whenthe mapping has a singularity (see e.g., [9]). An alternative possibility is toconstruct an artificial boundary and to impose conditions on this surface tosimulate an infinite domain, It is therefore desirable that there should be noreflections from the boundary back into the domain of interest. Unless certainrestrictions are met this will not generally be possible.

In general one cannot construct boundary conditions that give no reflec-tions. Instead one wishes to construct a sequence of boundary conditions thatare in some sense better. The notion of better can he defined in many ways.Some of these are:

(i) the reflections decrease rapidly as the position of the artificial boundarygoes to infinity;

60i the reflections decrease for shorter wavelengths;C" (iii) the reflections decrease as the incident wave approaches in a direction

D more parallel to some preferred direction;

Communications on Pure and Applied Mathematics, Vol. XXXIII. 707-725 (1980)© 1980 John Wiley & Sons. Inc. (10(0-364018010031-0707501.90

/00-7/ .... -J

708 A. BAYLISS AND E. TURKEL

(iv) the reflections are decreased in such a manner that the approach to asteady state is accelerated.

One approach to decreasing reflections is to introduce a viscous regionnear the boundary, or to introduce a sponge layer (see e.g., [19]). With thisapproach it is not clear what effect the boundary layer has on the interiordynamics. In addition it is difficult to improve these methods when one needsto decrease the reflections further. Conditions (ii) and (iii) were used byEngquist and Majda [4], [5] to construct an asymptotic set of nonreflectingconditions with the help of the theory of pseudodifferential operators. Theirhigher-order conditions require a Pade approximation for stability. Rudy andStrikwerda [20] have constructed, by heuristic arguments, a radiation bound-ary condition based on (iv). In some cases it is possible to construct boundaryconditions based on separation of variables or integral relations satisfied bythe desired solution. This method was used by Fix and Marin [6] and Marin[17] for the Helmholtz equation. Since the resulting boundary conditions arenonlocal, i.e., they couple the solution at all points on the boundary, they cannot be readily applied to a standard finite-difference solution technique forhyperbolic equations. For this reason, we concentrate on developing localboundary conditions.

Gustafsson and Kreiss [11] have shown that in general one cannotconstruct nonreflecting boundary conditions unless the behavior of the solu-tion is known in a neighborhood of infinity. We adopt their concept andconstruct boundary conditions which are based on an asymptotic expansion ofthe solution valid for large distances. As with all asymptotic expansions, weexpect reasonable results even when the artificial boundary is quite close tothe domain of interest. Extensive numerical tests confirm that the domain ofintegration can be very constricted when one uses the higher-order boundaryconditions. In fact for the Helmholtz equation, Ap+k 2p=O, exterior tosimple bodies only five to ten mesh points are generally required normal tothe body, see [2].

The boundary conditions to be developed are based on an asymptoticexpansion in l/r, where r is the distance from a fixed point. These conditionsform a sequence of differential operators B, which, for any m, annihilate thefirst m terms in the asymptotic expansion. A typical boundary condition hasthe form

Bp = 0.

This can be considered as a way of matching the solution on the boundary tothe first m terms of the expansion of the solution exterior to the boundary.The operator B~p was used by Kriegsman and Morawetz [14] for theHelmholtz equation.

The details of this construction are described in Section 2. In Section 3 we

CONDMONS FOR WAVE-LIKE EOUATIONS 709

prove that these boundary conditions yield well-posed problems and that theysatisfy (i) in that the m-th boundary condition is accurate to an error ofO(r -2 m+I). Numerical computations presented in [3] demonstrate that theseboundary conditions are also good for accelerating the solution to a steadystate, as in condition (iv).

As a specific example we consider the Euler equations in cylindricalcoordinates and with axial symmetry, linearized about a mean velocity profile(uo, Vo),

p,+(puo+u).+(pvo+v),+Pv°+V=F,,

(1.1) u, + (UUo + p) + (UVo)d = UVo~d -VUo.d + F 2 ,

v, + (vu) + (vvO + P)d = VUo.z - UVo.. + F 3 .

Here p, u, and v are the perturbation pressure and velocities, respectively,while z and d are the axial and radial coordinates. (We use the nonstandardnotation d since r will be used throughout as the spherical radius, r 2

= z 2 +

d'.) The mean pressure Po is assumed constant and the equations are scaledso that po= 1 and co = 1, where co is the ambient sound speed; F denotesource terms.

The system (1.1) with appropriate source terms and initial conditionsdescribes the propagation of acoustic waves in the mean flow (uo, v,). Italso describes the growth of small disturbances in this flow and withappropriate initial conditions the system is used to study the linear stability ofthe mean flow.

In order that it be feasible to integrate (1.1) in a bounded domain, it isnecessary to assume that the mean flow and the forcing terms tend to aconstant as r goes to infinite (see [11]). We shall assume that uo, vo, F alldecay to zero at infinity. A more general case is described in Section 5.Hence, for large r, (1.1) can be approximated by

p, + Ud + Vd + - =0,

(1.2) u,+p =0,

V,+Pd =0,

or equivalently,

(1.3) p, -Ap = 0 .

For the system (1.2) (or the equation (1.3)) an asymptotic expansion doesexist for which radiation boundary conditions can be developed.

C , I C 4 C:

ii J in


The radiation boundary conditions for (1.2) and (1.3) are constructed inSection 2. Special attention is paid to the numerical approximations to theseconditions. In Section 5 several generalizations are considered. These includethe convective wave equation, two-dimensional cartesian systems and also theHelmholtz and Laplace equations. Further discussion of some of thesegeneralizations are presented in [2] and [3].

2. Boundary Conditions

Based on a separation of variables, one can heuristically show thatradiating solutions p to the wave equation have an expansion of the form

(2.1) P (t, r, 0, 40)= i=1 -r, ,

Here r, 0, are spherical coordinates centered at a fixed origin in space. Theexpansion (2.1) is well known. Friedlander [7] has proven that if f1 (s, 0, 40) isan analytic function of 0 and 4, then the series in (2.1) is convergent. It is aconsequence of the integral representation, using the retarded potential, ofsolutions to the wave equation, that (2.1) will hold as an asymptoticexpansion in 1/r under mild smoothness conditions (see for example [1]).Given (2.1) we can derive boundary conditions which are accurate on theboundary, to any order in 1/r.

We define the operator

a aat ar

If (2.1) is multiplied by r", we obtain

(2.2) rmp(t. r,O8, tb= rk 1M t-rO, b+ rk-if(-r,O, 0).

Applying the operator L" to both sides of (2.2) we see that the first sum isannihilated and the leading term of the second sum is O(r - ' - '). Hence.

(2.3) L'(r"p)=O(r ' ).

We wish to rewrite this expression as an operator acting only on p. Therefore,

I

CONDITIONS FOR WAVE-LIKE EQUATIONS 711

we recursively define the operators

(2.4) B, - Lr

(2.5) B,, = (L+ 2 m ) B _

This can be expressed as

(2.6) B_ = (L+ 2 1-1),

where the order of the products is given by (2.5). It then follows from (2.3)that

(2.7) Bmp = 0(1/r +1),

for any function p which satisfies the expansion (2.1). In fact, B., annihilatesthe first m terms in the expansion (2.1). This property is derived by usingformal manipulations with (2.1) and does not use any other properties of thewave equation.

We propose therefore the family of boundary conditions

(2.8) B,,,p=O.

By (2.7) this is a sequence of boundary conditions for the wave equation (orfor any system for which (2.1) holds) with increasing accuracy in l/r. Theequation (2.8) holds identically for the first m terms in the expansion (2.1).The operators Bm are linear differential operators of order m which can beconsidered as a measure of the difficulty in their implementation. We showfirst that the operators B., are optimal in this measure. It is obvious from theabove derivation that the B. are unique up to a constant and that thefollowing theorem is true.

THEOREM 2.1. Let Q be any linear differential operator which annihilatesf, (t - r, A, O)/ri for j = 1,. ., m for arbitrary fi. Then the order of 0 is at leastm. If 0 has order m, then Q = cB for any function c(r, t).

A two-dimensional version based on B, p = 0 was introduced by Engquistand Majda (5] as the first approximation in a series of boundary operators.These operators are based on an expression in the angle of the wavemeasured with respect to the normal to the boundary. In this study theboundary operators are asymptotic in powers of l/r.

When the operator a/8r is used for nonspherical boundaries both normaland tangential derivatives to the boundary appear. For the normal derivatives


one-sided differences must be used. In general, one requires that thenumerical approximation to these derivatives be no less than one order ofaccuracy less than the accuracy of the interior scheme (cf. [10]). Hence, thenormal derivatives appearing in B. for large m will involve many meshpoints normal to the boundary. This may present stability and accuracydifficulties. It is therefore desirable to implement the boundary conditionswithout using high-order normal derivatives.

We can introduce a local cartesian coordinate system. The wave equation

Pn =Ap

can then be used to express all normal derivatives of order greater than onein terms of tangential and time derivatives. When working with the first-ordersystem

(2.9) p,+div w= 0,w, + grad p = 0,

the first-order normal derivative of p can also be eliminated in terms of timederivatives. With the first-order system the number of time derivativesrequired is reduced. This is desirable for computational reasons since it iseasier to calculate two k-th order time derivatives rather than a (k + 1)-st ordertime derivative.

It thus follows that when the system (2.9) is used, no normal derivativesoccur in the equation B,,p -0. The system (2.9) arises naturally in the studyof the linearized Euler equations (1.1). For this case no second-order wavetype equation can be derived when the mean flow is nonzero. The system(2.9) is also advantageous in that the boundary values for p and w can beobtained by standard finite difference algorithms. This is preferable to usingone-sided differences for a nonstandard boundary condition which wouldcomplicate a stability analysis.

The first two boundary conditions written in cylindrical coordinates z andd are given explicitly by

(2.10) Blp =p,+w, +-r

and

B2 = p. -(2w. + w.)(2.11) 4p, 2ppr - w1 "

r 7 1

where r 2 =z2 +d 2 , tan 8O=d/z, and w v cos O+v sin .


3. Welposeldess

We have constructed the operators B. given by (2.5), so that they form asequence of more accurate boundary approximations. This does not guaranteethat the ensuing problems are well posed. In [4] a different sequence ofabsorbing boundary conditions was developed, some of which gave rise toill-posed problems. When Pad6 approximations are used instead of a Taylorseries, the ensuing boundary conditions are all stable. For computationalefficiency, it is important that the initial boundary value problem be wellposed for relatively general boundary shapes.

We consider the two-dimensional case but all proofs extend to threedimensions. The region of integration is x -0, -- <y<-. We then have,along the boundary x = 0,

(3.1) a =r a +b!y8r ax ay

with

a 2 +b 2= 1.

Since the interior is x <0, we have a >0.We define the q norm for the interior solution as

jllUlllC.O= Y- I dx dr.I 1-o afI

Here, j is a multi-index, x represents all the space dimensions, C = (x, t) and0 is the domain of the problem. Similarly we define the L2 norm of theboundary data as TI

jjgjL = J g2 dy dt.

Now, at is the boundary of 01 and y represents the tangential variables.

THEOREM 3.1. Consider the problem

(3.2a) p, =,&p + F, p E 11,

(3.2b) Bmp = g, peaf0,

where

(3.2c) B.= (a+a-+b -- +*, a2+b 2 =1, a>O,-0 a0t ax cy r)/


the aj being constants that depend on the dimension (see (2.5)-(2.6) and (5.4)).We then have the estimate

(3.3) jIpIL.n + lpIl~a < K[BjgjLn + JIF14a], q =0, 1,.-., m.

Proof: The weliposedness of hyperbolic initial boundary value problemswas studied by Kreiss [13] and it is assumed that the reader is familiar withthis theory.

For the boundary condition B,,p = 0 it is sufficient to consider only thehighest-order derivatives that appear. We also restrict ourselves to thedomain x 0. For simplicity we assume that F 0. We thus need onlyconsider

(3.4a) p. = p. + p", x O, -_ < y < 0,

and

(3.4b) Lp = g along x = 0,

where

(3.4c) L = -+a --+b -8t ox Oy

If

g = eSleirg, 9Re s >s0,,q real,

then there are solutions

(3.5) p = e e"y-O(x),

provided that c satisfies

(3.6a) 0. = A24, x 5-0,

where

(3.6b) A = (sl+ 712)1/2 Re A >0,

and also

(3.7) 1al+s+ibi jb= at x=O.


Restricting 41 so that 40 E L2(--, 0), the solution to (3.6a) is

(3.8) 4i=Ae'.

Substituting (3.8) into (3.7) we have

(3.9a) d,(A, s, ni)(1 + Isj)' 45 = je , -0<x <0,

q=l.".-.

ord;(ak + s + ibl

(3.9b) dq=aA -})

According to the theory of [13] we must eliminate the possibilities of

eigenvalues or generalized eigenvalues to

(3.10) D aA+s+ibq =0.

Re s>0 implies that SieD>O and so no eigenvalues to (3.10) exist. Toconsider the possibility of generalized eigenvalues we let A = Al +iA 2, s=sl+ is2. Squaring (3.6b) it is clear that the possible generalized eigenvaluesare characterized by either

(3.1 1a) Xm= s,= 0, Is1l L, 2>0,S2

or

(3.11b) 2 =S 1 =0, Is2 1< vIf.

For the first possibility, 0m D>0 since jbI< 1, and for the second possibility,Ote D= aA1 #0. Hence, in any case we can not have solutions to (3.10) withRe s>0 or those that are limits of solutions with Rie s>0.

Remarks. 1. The proof fails when a = 0. It is readily seen that (3.2c) is notwell posed for such boundary points. This corresponds to using a ray0 = constant as part of the artificial surface. Such boundaries arise naturally inpolar coordinates. Hence, the use of cylindrical coordinates may offernumerical benefits.

2. For m = 1 we can obtain the estimate (3.3) by an energy method.

Consider system (3.4) and let

(3.12) E= (p+ p2+ p'+ 2bpp,) dx dy.


Then

dE (ap2 + gp.)dy- -2a 2 )dy.

(3.13) dt -_ (p.2a

In particular this shows that there are no eigenvalues or generalized eigen-values to d, =0 for q = 1. From the form of dq in (3.9b) it immediatelyfollows that for any m there are no eigenvalues or generalized eigenvalueswhen q=l,...,m.

We can use the results of this theorem to get an estimate of the errorgenerated by the boundary conditions. More specifically, we have

THEOREM 3.2. Let p be the solution to the infinite domain problem, i.e.,

(3.14) p,, =Ap + F, xC=R,

and let u_ be the solution to (3.2), the wave equation in the finite domain, withthe boundary condition Bmu. = 0. Let R be a measure of the size of theartificial domain f) and let e,, = p - u.. Then

(3.15) lem Lm. = O(m ).

Proof: v. satisfies the differential system

(3.16a) E,.,, = AE.,

(3.16b) BmEm= Bmp, x E Cfl.

In view of Theorem 3.1 we know that

(3.17) jlen, .- K,5 IIB,,pIL .

Since p is a solution to the wave equation in R3, p can be expanded in termsof the traveling wave expansion (2.1). From the construction of B., we sawin (2.7) that

II~mn.R 2m +1

We, therefore, have

(3.18) I1jm.I.,-K.O( r-t)•

R .+


It remains to find the dependence of Km on R. We consider a fixed domainwith size measured by R0 . As the size of the artificial domain increases weintroduce the change of variables

I'= t/Ro, x'= x/R 0 .

Using this scaling in (3.18), the estimate (3.15) follows. If the equation to beintegrated is not identically the wave equation, but approaches the waveequation sufficiently rapidly for large r, then (3.15) can be generalized.

In Theorem 3.2 we have considered the order of the boundary operatorB_ as fixed and let R increase. In practical calculations, one would like toreverse the situation, i.e., keep R fixed and increase the accuracy byincreasing the order of the boundary operator. Extensive computations withm equal to one and two confirm the superiority of B2 over B, in a fixeddomain. We have been able to bring in the artificial boundary extremely closeto the region in which the variation of the inhomogeneities is large and stillobtain good accuracy when using B2. This was done for a series of problemswith different sources modeling various physical processes. The equationshave coefficients with large gradients near the axis of symmetry and so arenot close to the wave equation in this region. Different boundary configura-tions were used to model free space or a bounding wall above the nozzle exit.In all these cases the second-order condition, B 2P =0, gave very accurateresults. For the simpler cases, the first-order approximation Bjp =0 wasadequate. The characteristic condition, p + p, =0, was not useful and tendedto give rise to large spurious reflections. This is intuitively obvious since p,and p, are both going to zero due to the spreading of waves in multi-dimensional problems. The essence of the Sommerfeld radiation condition isthe rate of decay which is not included in the simple condition that p, + p,=0.Hence, B, p = 0 is the minimal condition that is reasonable. The accuracy wasmeasured by varying the finite difference mesh as well as the position of theartificial surface. More details of the computations are presented in the nextsection.

4. Numerical Result

In this section we present some numerical results. These results areprimarily designed to demonstrate the improvements which can be obtainedby use of the second-order boundary condition B2P = 0 over the first-orderboundary condition Blp=O.

The computations will be made for the system (1.1). The computationaldomain will be a cylindrical rectangle with an axial source. (See Figure 1.)Because of the axial symmetry the problem reduces to two dimensions. Atthe near field boundary (dotted line in the figure) different boundaryconditions are imposed.


Circle ofa, p-0 Vmeosuremeni

B p=O 8,p=ONeat __

fieldboundary

- Source \-Axis of

symmetry

Figure 1. Computational domain.

For the first experiments a jet flow is assumed to exit from a pipe ofradius 4 at the near field boundary. The cylindrical rectangle is 30 units high.and 50 units in length. A source of the form

cos cot . 5(Ix - Xof)

is assumed with o = 1.055 in the above units. We plot the pressure versustime at a fixed axial point 40 units from the source. (A Gaussian function wasused to model the 8 function.) Here we compare the characteristic condition

(4.1)

with the first-order condition

(4.2) Bp = 0.

Results are shown in Figures 2 and 3. The spurious reflections generated by(4.1) are evident from these figures.

For the remaining computations we consider the system (1.2) without anymean flow so that the exact solution is known. Computations are now carriedout in a cylindrical square with each side of length 40 units. An importantcase occurs when the radius is measured from an origin which is displacedfrom the position of the source. Applications include the computation of thejet interaction with two distinct sources. This information is useful for theinverse problem, i.e., obtaining information about the sources based on farfield measurements (see Maestrello [161). Another important application is toscattering by a body where the scattered field is the result of a distribution ofsources along the body (see [2]).

Therefore we consider a problem with the origin taken four units awayfrom the source position. A time harmonic point source of nondimen-sionalized frequency 1.055 is assumed. We compare the solutions for the

CONDITIONS FOR WAVE-UKE EOUATIONS 719

'000.00 X 1O-6

714.29

428.57

r 142.86.

e

S

ss

r -142.86e

-429:.57

- ?14.P93

0 2 4 6 a 10 12 14 X I0'

Time

Figure 2. Sommerfeld condition.

cases where BIp = 0 or B2p = 0 are imposed along the near boundary (dottedline in Figure 1). Since the mean flow is zero, the exact solution is a sphericalwave centered at the source position. We measure the total power as afunction of the angle 0,

(4.3) i(8)= p2 dt.

This corresponds to the square of the amplitude of the spreading wave. InFigure 4 we plot this function using the two different boundary conditions.The solution is presented on a quarter circle of radius 20 and is normalized sothat the exact solution has an amplitude of 1. The improvement due to thesecond-order condition is evident from Figure 4. In particular, the use of thefirst-order boundary condition creates a large 0 dependence even though theanalytic solution is independent of 0. The accuracy of any of these boundaryconditions may deteriorate as w increases. This can happen, for example,


1000.00 x to-$

714.29

428.57

P

S

Tim

Figure 3. B I I =O.

1.5 0 Second order condifloi 9X First order condition 1

1.3

1.2 ?

P'rmS .

A--

0 10 20 30 40 50 60 70 s0 90

Figure 4. Relative power output for different boundary conditions.


when the number of wave lengths between the origin and the source positionis large.

As a third example we consider a quadrupole source. This is an importanttest case since the theory of Lighthill [15] indicates that the sources of jetnoise can be regarded as arising from quadrupole sources. Thus, in theright-hand side of the continuity equation (first equation in (1.1)) we impose aforcing term of the form

(4.4) cos wt8k. (Ix - X01)

where the second derivative of the delta function is obtained by differentiat-ing a Gaussian function. The frequency a is again taken as 1.055.

The solution to this problem is given by Morse and Ingard (see [18], p.314), in spherical coordinates,

p(t, r, 0) = -sin(w(t-r))cos - 0 1 cos (wr(t- r))

(4.5) o4rrr 2

(,o(t- r))

+sin () (3c O- 1)J

The solution (4.5) indicates that the leading order term in the expansion (2.1)(i.e., f,) becomes zero when 0= 1r. Thus, when the condition Blp =0 isimposed, the computed solution would be expected to be inaccurate forangles neai 90' , It is also evident from (4.5) that a phase change of 900 is tobe expected as 0 varies from zero to 21r. The problem (4.4) was chosen sothat B2p = 0 does not hold identically for the solution (4.5).

In Figure 5 the root mean square power is plotted as a function of theangle 0. The normalization is obtained by the exact solution (4.5) at 0 = 0° .From the figures we see that the first-order boundary condition is lessaccurate even for angles near 0 = 0". It has been the experience of the authorsthat localized boundary errors can introduce significant global errors for longterm integrations. The first-order boundary condition will generally giveaccurate results only if f, (in the expansion (2.1)) is significantly larger thanthe other terms.

The total power is not a good measure of the accuracy of the solution as 0approaches 2r because the solution becomes very small there. A bettermeasure of the accuracy is the phase of the solution relative to the phase at0 = 0. Examination of the exact solution (4.5) shows that as a function of timethe solution changes from a sine dependence at 0 = 0 to a cosine dependenceat 0 =-2r. Thus, there should be a time delay, i.e., phase change of JPrbetween successive peaks at 0 = 2r. In Table I this is shown for the differentboundary conditions. (For these tests the time step, At, was chosen so that


2.01.9 /1.6

1.6 X Second order boundary condition1.5 0 First order boundory condition /1.4 /1.3

9.2

1.1

.6

.5

.4

.3

.2

.i 1 I I I I

0 10 20 30 40 50 60 70 so 900o

Figure 5. Relative power for quadruple.

Af= 0.08. Hence, the phase change can only be computed with thisaccuracy.)

The fact that the computational domain has a corner can give rise toinstabilities when matching two different boundary conditions. In these testruns the problem was not encountered since the corner is far from thedomain of interest. This difficulty can be overcome by interfacing theboundary conditions as indicated by Engquist and Majda [5].

We point out that in order to obtain good accuracy it is essential to use afourth-order difference scheme for solving (1.1) (see [8]). This also requires ahigh-order approximation to the boundary operators. This is not difficult sincethe boundary condition can be reformulated so that only tangential deriva-tives are involved (Theorem 2.2). Fourth-order central differences are thenused to approximate the tangential derivatives.

Table I

hi'bae change.

Order of B.C. Phase change

1 0.77A+0.082 1.64*0.08

CONDITIONS FOR WAVE-LIKE EOUATIONS 723

5. Extenotns

In the previous sections we have concentrated on the three-dimensionalwave equation in spherical or cylindrical coordinates. The only case consi-dered so far is that of equations that tend to the wave equation in the farfield. In this section we investigate extensions to two-dimensional problemsthe convective wave equation and also to elliptic equations. We shall onlyintroduce formal manipulations. Proofs of the utility of these boundaryconditions will be presented in future papers.

We assume first that p(x, t) = e-13(x). Then #(x) satisfies the Helmholtzequation

(5.1) Al5+k 2 = 0.

The boundary operators B,,, given by (2.5) can be applied to (5.1) byformally replacing O/Ot by -ik. Alternatively, we can derive these boundaryconditions by looking at the analogue of the traveling wave expansion (2.1).We then have

(5.2) =e e-ik" me .

For the Helmholtz equation, Wilcox [21] has proven that (5.2) is a convergentseries. We then consider the boundary operators

(53) -- m ik+ +21-1

(5.3 B,=\ O ( r r/

As before this product is taken by first operating with the index 1 = 1 andcontinuing to I= m. In [2] we derive some energy estimates for the errorwhen applying the condition Bmp = 0 at a finite boundary. The operator B,was used by Kriegsman and Morawetz [14] for calculations with the Helm-holtz equation.

The condition (5.3) can be generalized to any number of space dimensionswith changes only in the lower-order terms. For two space dimensions we usean asymptotic expansion of Karp [12] to obtain

(5.4) B. - ik +1+41-3'~~=,\ r 2rI

Numerical results for this case are presented by Kriegsman and Morawetz andin [2]. Extensions to the Laplace equation are investigated in [2].


In many fluid dynamic applications one uses the time dependent equationsas a method of reaching a steady state. In the far field, away from the sourcesand viscous regions, the perturbed pressure Pi = p - p., where p- is a steadystate pressure, will satisfy a convective wave equation. Choosing a coordinatesystem with the x-axis parallel to the far field steady velocity we have

(5.5) p,, - + 2 - 2 + - ) 0,

where c- is the far field sound speed. This equation can be transformed to thewave equation where the boundary conditions B. are applied. One thentransforms back to the physical variables. For m = 1, the resultant boundarycondition is (cf. [3])

1 p -p- 2X au uav\

(5.6) (c!-)1 12 at c!- u! r (at U )

r(-t ax / 2r

and

r2

= c x2

+y2

.

This result is a subsonic outflow boundary condition which can dramaticallyaccelerate the convergence to a steady state. Computations using (5.6) withthe Navier-Stokes equations have accelerated the approach to a steady stateby a factor of two (cf. [3]).

Acknowledgments. The authors would like to thank Professor C. Morawetzfor her many suggestions. This work was supported under NASA Contract No.NAS1-14101 while the first author was in residence at ICASE, NASA LangleyResearch Center, Hampton, Va. 23665. Additional support for the secondauthor was provided under the U.S. Air Force Office of Scientific Research,Contract No. AFOSR-76-2881.

bo aphy

(1] Bayliss, A., and Turkel, E., Radiation conditions for wave-like equations, ICASE Report No.79-26, October 1979.

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[17] Main, S. P., A finite element method for problems involving the Helmholtz equation intwo-dimensional exterior regions, Ph. D. thesis, Department of Mathematics, Carnegie-Mellon University, 1978.

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7, 1955, pp. 271-276.

Received December, 1979.

Bayliss Paper

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