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Sonic-Point CapturingBra m van LeerWen-Tzong Leeand

Kenneth G. pow ell^T he University of MichiganDepartment of Aerospace Engineering

1 IntroductionThe ability of conservative, dissipative differenceschemes to capture shocks in discrete flow solutions ismuch appreciated and fairly well understood. Some un-resolved issues in shock captu ring concern the existenceand uniqueness of steady shock structures [1,2], numer-ical noise radiated by captured shocks [3,4,5], and ro-tationally invariant shock capturing [6,7]. All of thesearise from pursuing the narrowest possible shock pro-file under all circumstances, and hardly detract fromthe utility of th e technique.Less understood, but no less of a miracle, is the abil-ity of difference schemes to captu re sonic points in dis-crete solutions, especially in steady solutions, wherethe transition through a sonic point marks bifurca-tion. In quasi-one-dimensional nozzle flow, for exam-ple, the transonic solution is an isolated solution of theboundary-value problem: perturbing the inflow Machnumber renders the solution either fully subsonic or un-steady.Steady numerical solutions containing sonic pointsmay exhibit a variety of oddities. If the discrete schemedoes not recognize the entropy condition, i.e., if it ad-mits expansion shocks, an arbitrarily large jump may befound where one expects a smooth transition throughthe sonic point [8]; if it does, the solution may stillexhibit a jump of magnitude O(Ax) [9], or a twecellplateau (see below), or anything in between, in additiont o being non-unique and slow to converge. Th e small

the transonic puzzle have been carried on by others,viz. Harten [12], Roe as reported in [8]), and Good-man and LeVeque [13]; the missing piece is producedhere. Th e secret of success lies in treat ing t he sourceterm in the same way s the flux-derivative term; thishas the appearance of applying an entropy fix to thesource term. In reality all that is done is balancing theflux derivative and the source term in both forward-and backward-moving parts of the transonic expansionfan.

2 Basic upwind schemeOur s tar ting point is th e first-order upwind-differencingscheme of Roe[l4] and Van Leer [8] for a hyperbolicsystem of equations incorporating a source term:

to fix our thoughts we shall think of this equation asrepresenting th e quasi-one-dimensional Euler equationsfor flow in a duct with a variable cross-section D(x). Asteady transition from subsonic to supersonic flow canonly occur when D1(x) = 0, D1'(x) > 0, i.e., in a th roat.The difference scheme can be written as

transonic Jump, in particular, can be found in -in which the numerical flux f~l nct ion i+ is defined asous published flow solutions obtained with first-orderupwind methods [3,10]; higher-order upwind methods, f i f ( ~ i ,i + l )though, if patte rned after the MUSCL scheme [l l] , areinsensitive to the presence of a sonic point. 1= - [ f ( ~ i ) + f ( ~ i + ~ ) ]In what follows we shall derive and test a proto- 2type scheme that produces perfectly smooth transonic 1solutions to nozzle-flow problems, without recourse to [ l ~ l i + %+I u i) i l i + + ~ x ] 3)2heavy-duty artificial dissipation. Most of the pieces of the notation requires further explanation.

Professor Member AIAAGraduate Research Assistant dfldu = = RA Rd l, where A diag(ak) carriesAssistant Professor Member AIAA the characteristic speeds in th e diagonal;

Copyright American Institute of Aeronautics andAstronautics, Inc., 1989. All rights reserved. 176

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t rIAl = R ~ A ~ R - ' i t h -x - 6a 0d t a 7 a- a a

diag (Iakl) ; 4)also useful are the definitions

A* = RA*R-', A* = d ia g ( a t ) , 5)and the re la t ions

A + + A - = A , A + -A - = I A I ; (6) x 2 s XI xI 1S[ = J z k Rk follows after writing s = R z = Figure la : Geom etry of transonic expansion fan: space-z k R k , i .e. , expressing s in terms of the r ight time diagram.eigenvectors of A; other useful definitions are

withs + s- = S , s s- = IsI; (8)

A . A ( u i + l ) , w h er e u . .ir(ui, ui+') is th e+ +Roe av erage [14] of ui an d u i+ l; similarly, i A+s (Gi++) .W ith the above flux function the full scheme 2) can bewritten in a form that shows its upwind character:

for each k a t each in te rf ace x j+ l .Th e scheme thus formulated does not contain an en-tropy fix , i .e. , a special measure to break down expan-sion shocks. S uppose th at in cell I the re is a sonic pointin the k-th chara cteristic field, i.e.

then t he vk-component in UI, i .e. , the component alongR k , s no t upda ted , as seen from (10). This is the reasonwhy an initially present expansion discontinuity maynot decay. In th e next section we shall discuss howt o modify the f lux function in order to prevent suchunphysical b ehavior.

3 Satisfying the Entropy Condit ion

Note th at both the f lux difference and the source term To satisfy the entrop y condition we merely have to buildare split t o achieve upwind bias. Although a f irs t-order into the f lux formula the model of a smooth transonictime-stepping scheme, (9) becomes second-order accu- expansion w ave; this will gener ate a sm all artificial vis-rat e when the solution becomes stea dy: each flux differ- cosity of a special functional form.ence is centered on t he same in terval as the source term Assume, th erefore , we have a c harac teristic field withtha t mus t ba lance i t . Eq (9) can further be expanded a sonic point between X a nd x ~ + l ; e shall approxi-as mate the spatial variation of the characteris tic speedu;+l = by a linear distribution, as shown in Figure 1. In thefollowing derivation, the subscript k is suppressed forclarity. In th e transonic expansion fan between

E x hk b k ~ v k XI and xI+l various average characteris tic speeds canA x d k t O I be recognized:the average characteris tic speed in the entireah A v k - fan ;

8 k < O a + th e average positive characteris tic speed inhere we have introduced the characteris tic variables v the f an ;defined by dv = Rdu. (111 a- he average negative characteris tic speed inthe fan .In a ste ady st at e witho ut shocks we have Because th e averaging opera tion comes after taking the

(b k A v k 2 k A ~ ) ~ + +0 12) positive/negative pa rt of a , the quantities a* do not

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Figure l b : Geometry of transonic expansion fan: a ver-sus 2

relate to 13 as usual, i.e.,a+ 6+ max(a, 0 , a- ii- min( ,O), (14)

but we still havea + + a - = a , (15)

and we may introduce a consistent absolute value bydefiningwhich, however, does not equal the true average ab-solute characteristic speed in the fan. Introducing thespread 6a of the characteristic speed across the fan, i.e.,

we have the following expressions,

valid in the transonic case ax i ax+l, or

We shall now determine the change of the characteristicquantity v across the forward- and backward-movingportions of the fan. Conservation and symmetry re-quirements dictate the following constraint:

Assuming that v changes linearly through the fan,we take Av in proportion to the width of the for-wardlbackward par t of the fan, i.e.,

Figure 2: Functions of and 6a.

2a- Av.6aUsing (21) to eliminate (AV)*, we may rewrite thesplitting (20) as

this, by construction, is an identity. The quantitiesa* * may be called the effective forward and backwardcharacteristic speeds in the fan. s seen from (22), theystill add up t o 2,

and we may again introduce an effective absolute value:

This is precisely the form of the modified absolute valueproposed by Harten [12] as a replacement for 161 in Eq(4), in order to circumvent the vanishing of 161 in atransonic expansion wave, which would mean vanishingdissipation and the possibility of an expansion shock.Harten's choice was simply a smooth fit to the standardabsolute-value function (see Figure 2); the uniquenessof this choice, suggested by the above derivation, ap-parently has not been appreciated until now.

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Even more surprising is that Godunov's [15] fluxfunction, known to satisfy the entropy condition, canbe implemented exactly, or in close approximation, byusing (25). To understand this, we do best to considerthe special case of the inviscid Burgers equation,

in which the characteristic speed a equals u itself.Godunov's flux is taken from the solution of Rie-mann's problem at the cell interface; in a transonicexpansion this yields

Using our modified flux function

with

( ~ + i UI+I UIwe find the same result:

This flux function, as mentioned in the Introduction,produces a small jump across the sonic point. In anattem pt to remedy this, Roe, as reported in [8], deriveda transonic flux for Burgers' equation that is negative:

This formula, combined with the standard upwind fluxaway from the sonic point , achieves that

which is exactly how a transonic expansion wave wouldevolve in reality, in the absence of a source.

The flux function (33) reaches its minimum when thesonic point falls exactly at x I , i.e., when

the minimum flux value equals

The same level of dissipation can be reached by themodified flux (28) if we r t i f i c i l l y double the spread ofthe expansion 'fan, by redefining

This leads to the transonic flux

its minimum value is reached in the case of (35) and isgiven by 36).

At this point it is instructive to investigate how thestandard upwind flux (3), without entropy fix, com-pares to (27) and (33) or (38). In the absence of asource term it reads

1min ( :, 5u:+l) 40)and reaches a maximum in case (35):

If we measure the amount of dissipation in a transonicflux as the difference between that flux and the above,dissipation-free reference flux (39), in the worst case(35), we see that Roe's formula (33), and the smoothversion (38), have twice as much dissipation as Go-dunov's formula (27).

The amount of dissipation in (33) or (38) is still notenough to get a smooth transition through the sonicpoint; it needs to be doubled once more, as shown byGoodman and LeVeque [13]. In their analysis, the cru-cial point is to make the scheme for the transonic regionconsistent with a higher-order equation derived fromthe inviscid Burgers equation:

Expanding the second term yields

the second right-hand term can be neglected near thesonic point:

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Calling the unknown transonic flux ft, and using thestandard upwind flux at 21 and XI+;, we find thatthe first-order upwind scheme approximates (43) as

Numerical Fluxes for Inviscid Burgers Equationupon comparing (45) with (44) we see that the formerwill be a consistent approximation to the latter if thetransonic flux is chosen as 2.00

ft -,(.? :+I) u1.1+1. (46) 1.00The minimum of this flux again is reached in the case 0.00of (35), and equals

3 2 -1.00ft(-W+1 UI+I)= - p + 1 , (47)-2.00four times as far away from the dissipation-free flux

(39) as Godunv s flux (27), and twice as far as (33)or (38). Th e same dissipation level can be reached -3.00with our generic formula (28) by artificially doublingthe spread of the transonic fan once more, i.e., by re- 0.00 0defining

( ~ u) I ++ 4(ur+i UI). (48)A compact, scale-free way to represent flux formulas Figure 3: Dimensionless numerical fluxes for the invis-for Burgers equat ion, due to Roe,is to define the angle B~~~~~~ quation.variable 0,

1. Standard upwind flux, Eq. (39), away from tran-(494 sonic expansions and compressions2. Godunov, Eq. (27), equivalent to Eqs. (28-30)

Ui+l with 6u Ausin(Oi++) = ,I- (49b) 3. Osher4 . Roe, Eq. (33)and normalize the upwind flux by the central- 5. Modified transonic flux, Eqs. (28-30) with 6u =differencing flux: 2Au

f (ui, ui+l) 6. Goodman and LeVeque, Eq. 46)4(0i++) 4 ( 4 +u?+1) (50) 7 Modified transonic flux with 6u = 4Au8. Dissipation-free transonic flux, Eq. (39)Figure 3 summarizes all the transonic fluxes mentioned

before, the standard upwind flux, and two distinctfluxes used in a transonic compression shock (ur > 0 >u ~ + ~ ) ,amely, Godunov s and Osher s [lo]. From theviewpoint of implicit differencing i t is desirable to havea numerical flux function that is differentiable with re-spect t o the input states; this condition is met through-out by Osher s flux, and by the transonic fluxes of thefamily (28).

It has been known for some time [16] that Godunov sflux represents the border-line of fluxes satisfying theentropy condition. To understand this , consider a flux

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of the form (28), with an artificially reduced spread ofthe expansion fan:

Then there is always some location of the sonic pointwith respect to the cell boundaries for which the cellcontain ing the sonic point does not receive any flux con-tributions; specifically, this is the case when

This will lead to an expansion shock of some finitestrength.

Treatment of the Source TermNone of the transonic fluxes derived above will yield agood approximation of a steady transonic expansion ifone fails to tr eat the source term properly. Th e mostdissipative scheme, i.e., the one including (48) s thefan's spread, merely seems to smooth an underlyingroughness of the solution over a few cells (see numericalexample in next section).

The question is, how to properly split the sourceterm. To answer this, we must go back to the geometryof Figure 1. Assume th at z, just as a, varies linearlyin the transonic region; we may then define a linearmapping of z onto a:

where z, is the source value in the sonic point (see Fig-nre 4). We may define averages i z+ and Z- on thefull fan, its forward and its backward par t, in the samen nway as previously a , a + and a . This yields, in thefirst place,

to be used to eliminate z,; furthermore,

note that hz+ + 2 = z, + 2, (58)unlike anything we have seen before.

In the steady state, the flux difference must be bal-anced by the source integral, approximated by i A x (seeEq. (12)); we shall split this te rm in analogy to (20):

Here (AX) denotes the portion of the mesh (x I, XI 1)from which the forwardlbackward characteristics in thefan depart. We therefore have

and we may rewrite the splitting (59) asiAx z+*(Ax)+ z- (Ax), (61)

The quantities zf * may be called the effective sourceterms for the forwardlbackward characteristics in thefan; as usual, they add up to i

z+* z-* i (63)We may now define Izl* s needed to replace li[ n theflux function 3):

Note that if z, = 0, which means that the sonic pointalready lies in the throa t of the channel, we have

therefore.

which is the same formula s for lal*, cf. Eq. (25).The significance of the general formula (64), for

z, 0, is illustrated in Figure 4. As long s the steadystate has not been reached, one will in general find thesonic point away from the th roa t, i.e., z does not vanishwhere a vanishes. Th e splitting of i according to (64)guarantees th at , when th e sonic point approaches its fi-nal position, the flux difference and source integral overthe positive/negative par t of the fan smoothly approacha perfect balance. Another way to achieve detailed bal-ance is to split the source term similarly, but only nthe throat, regardless of the position of the sonic point.This also works bu t is somewhat clumsy, as one still hasto define some kind of splitting of i near a sonic point,for use in the flux formula (3). Moreover, this approachwill be harder to generalize to multi-dimensional flow,as it requires examining the shape of streamtubes.

Numerical VerificationTh e combination of Eqs. (25) and (64) in the flux func-tion (3), i.e.,

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Figure 4: The Entropy Bypass: synchronous splittingof source and characteristic speed near a sonic point.Grid points on the x-axis are suppressed; the grid canbe arbitrarily shifted with respect to the sonic point.

with 6ak 4A ak , 6rk 4 k, performed flawlessly incomputing steady solutions of both the inviscid Burgersequation and the one-dimensional Euler equations withsource terms. The choice 6ak 2A ak , 6zk 2Az k,performed equally well in the simple cases we studied,and yielded almost the identical solutions. Using 6akAu k, 6zk A rk , however, made the scheme lose agreat deal of robustness: unless starting from initialvalues very close to the correct steady sta te, the resultstended toward an incorrect solution featuring a sonicplateau.

Figure 5 shows the steady solution obtained for Burg-ers' inviscid equation with a source term, namely,

1 7rUt (-u2), --sin [271.(x , ) 1 (68)2 2

and periodic boundary conditions on the interval (0 ,l ).The grid is uniform and has 32 cells. The sonic point isgiven by x 0, which coincides with a cell face. Th enumerical values in the adjacent cells yield a discretevalue for d u l d z of 3.1482, close to the correct value 7rRepeating the calculation on finer grids confirms thepoint-wise second-order accuracy of the solution. Th econvergence history for the computation of Figure 5 isshown in Figure 6. In all the computations reported inthis section, local time-stepping was used.

Figures 7 and 8 show solutions of similar quallity, ob-tained for different sub-cell locations of the sonic point,namely, x, Ax14 (Figure 7 and x, Ax12 (Figure8 .Figure 9 shows what happens in using Godunov's flux

Inviscid Burgers Equation with Source TermComputed versus Exact Solutions1 Computedxact

1

Figure 5: Steady transonic solution of Burgers' equa-tion with source term. Modified transonic flux withSa 4Aa, Sr 4Az. The sonic point, at z, 0, liesexactly a t a cell face.

Convergence History1 0

Iterations

Figure 6: Convergence history for the computation ofFigure 5.

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Inviscid Burgers Equation with Source TermComputed versus Exact Solutions1 Computedxact

1

Inviscid Burgers Equation with Source TermComputed versus Exact Solutions

Figure 9: Same as Figure 5, but with Sa = Aa, 6.2 =Az. The results converge slowly to an incorrect solutionwith a sonic plateau.

Figure 7: Same as Figure 5, but with x, = Ax/4.with proper treatment of the source term, i.e., 6ak =Auk, 6zk= Az k, star ting from zero initial values. Thesteady solution with the plateau is very hard to reach,as seen from the convergence history in Figure 10; thissuggests that there may actually not be an exact steadydiscrete solution. The computation may end in a limitcycle not checked).

Some Euler results are shown in Figures 11-17, forflow in a channel with cross-section. .Computedxact 3 1D x) = - - os wx), -1 x 1; 69)2 2

Inviscid Burgers Equation with Source TermComputed versus Exact Solutions1

he cross-section is constant =2) outside the interval- 1 , l ) . Figures 11 and 12 show the Mach-number dis-tribution and convergence history for the flux functionincorporating bak = 4Aak , 6zk = 4Azk, for the non-linear characteristic families, i.e., = 1, 3. Again, theagreement with the exact solution is point-wise second-order.

The robustness of the Entropy Bypass is shown inFigures 13 and 14, in which oscillatory initial con-ditions, with multiple sonic-points, are given. Themethod converges rapidly to the exact solution.

10 Figure 15 shows what one loses if the source term isnot treated according to Eq. 64). In this case we used

Figure 8: Same as Figure 5, but with x, = Ax, . with considerable loss of accuracy.Finally, Figure 16 is based on the Godunov-like

flux, with proper source-term treatment, i.e., 6ak =Auk, bzk = Ark, = 1,3. The linear initial-value distribution indicated in the figure ended up,

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Convergence History

Iterations

Convergence History

Iterations

Euler Equations for Channel Flow

.Computedxactnitial

Figure 11: 'llansonic solution of the Euler equationsfor flow in a converging-diverging channel; shown are Figure 13: Same as Figure 11 but with multiple sonic-Mach-number distributions. Modified flux, with Sak points in the ini t ial distribution.4 h k , 6zk 4A. k,k 1 3.

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Euler Equations for Channel FlowMach Number in Channel

Convergence HistoryComputed

xactInitial

i

Figure 16: Same as Figure 11, but with 6ak =Aak, 6zk = Azk, = 1, 3. Starting from straight-lineinitial values, an incorrect solution with a sonic plateau

Figure 14: Convergence history for the computation of is slowly established.Figure 13.

Euler Equations for Channel Flow

s for Burgers equat ion, in a solution with a sonicplateau. The associated slow convergence can be seenfrom Figure 17. If, on the other hand, one providesthis Godunov-like scheme with the steady solution fromFigure 11 s initial-value distribution, it will essentiallypreserve it.

Computed6 Conclusions nd PerspectiveInitial The analysis in Sections 2-4 has, among other things,yielded the following results:

a numerical flux function for use near a sonic point,based on a full model of a transonic expansionwave;a matched treatment for the source term.

Figure 15: Same as Figure 11, but with incorrectsource-term splitt ing, viz.: l l = i(2iilSa).

The flux formula turns out to be identical t o Harten s[12] ad hoc formula. I t contains a free parameter bywhich the rate of spreading of the expansion fan, i.e.,the dissipation, can be controlled. For optimal robust-ness in time marching, this parameter should be givena value equal to four times the nominal value that fol-lows from the model; this follows from the analysis ofGoodman and LeVeque [13].

Numerical results confirm the correctness of the anal-ysis. With the proper flux splitting and source split-ting, the first-order upwind scheme of Roe [14] andVan Leer [8] produces pointwise second-order-accuratesteady transonic solutions of the Burgers and Eulerequations, without transonic jumps or plateaus.

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Iterations

Figure 17: Convergence history for the computa tion ofFigure 16.

It must be emphasized that the above analysis ap-plies exclusively to finite-volume schemes for comput-ing steady solutions. For such schemes, what mattersmost is the proper balance, near a sonic point, of flux-differences and source terms. This yields the desiredaccuracy in the steady state; for the sake of robustnesswe've had t o concern ourselves to some extent with tem-poral accuracy near a sonic point. Th e transon ic fluxderived, however, is ot the full answer t o the questionof designing a scheme that maintains temporal accu-racy in the vicinity of a sonic point; this was pointedout by Roe [17]. Future publications of Roe and thepresent authors may address this question in detail.

The extension of the one-dimensional analysis tomulti-dimensional flows appears straightforward, if anauxiliary flow-aligned coordinate system is adopted.Th e two-dimensional Euler equations, for instance , canbe written as

u f rl 0, (71)where measures length in the streamwise directionand in the normal direction; the 77-derivative of thenormal flux may be regarded here as source termthat determines the cross-section of the streamtube,and may be treated as in the one-dimensional case.One thus ends up with fluxes in a coordinate systemrotated with respect to the grid lines; the fluxes neededfor advancing the solution on the grid are then obtainedby the inverse of this rotation. Implementation of thisquasi-one-dimensional technique is presently under way.

It has been observed that multi-dimensional flows aremore forgiving as regards the numerical treatment ofsonic points than one-dimensional flows. An expan-sion shock, for instance, is easily generated in a one-

Figure 18: Convergence history for two-dimensionalcomputations of transonic channel flow, (a) using anentropy-violating flux, (b) using an entropy-satisfyingflux.

dimensional calculation through the use of a numeri-cal flux function like (39), violating the entropy condi-tion. In two dimensional calculations, though, expan-sion shocks appear to be destroyed by transverse waves.Yet, the desire of the numerical solution to derail maybe evident from a deterioration of the convergence to-ward the steady s tat e. An example of loss of conver-gence because of an entropy-violating flux functionis given in Figure 18. Th e flow computed is the two-dimensional equivalent of the case of Figure 11, i.e.,transonic flow in a converging-diverging channel. Theentrance Mach number for the case was M 0.47.The channel had a 20% constriction, and the grid was45 15. Th e stagnati ng residual-convergence curveis for the entropy-violating flux; the fully convergingresidual corresponds t o the numerical flux function dis-cussed in Section (no source term included).

Th e example of Figure 18 suggests tha t there is a lotto gain from extending the present analysis to multi-dimensional flows. That, of course, will be the subjectof a sequel paper.

cknowledgementsThe research reported in this paper was funded in partby the National Science Foundation under Grant EET-8857500, monitored by Dr. George Lea, and by theNational Aeronautics and Space Administration underGran t NAG-1-869, monitored by Dr. James Thomas ofNASA Langley Research Center.

The authors are indebted to Phil Roe for the flux

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normalization defined in Equations (49) and (50) and [ 2]used in Figure 3.

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A. Har ten , High-resolution schemes for hyper-bolic conservation laws, Journal of Cornputa-tional Physics, vol. 49, pp. 357-393, 1983.J B. Goodman and R. J LeVeque, A geometricapproach to high resolution TVD schemes, SIAMJournal on Numerical Analysis, vol. 25, 1988.P. L. Roe, Characteristic-based schemes for theEuler equations, Annual Review of Fluid Mechan-ics, vol. 18, pp. 337-365, 1986.S K. Godunov, A finite-difference method forthe numerical computat ion and discontinuous solu-tions of the equations of fluid dynamics, Matem-aticheskii Sbornik, vol. 47, pp. 271-306, 1959.S Osher, Riemznn solvers, the entropy conditionand difference approximations, SIAM Journal onNumerical Analysis, vol. 21, pp. 217-235, 1984.P L. Roe, A note on sonic-point fluxes. PrivateCommunication, 1989.