On the existence of the entropic solution for the transonic flow problem

Nonlinear

Ano/ysis,

Theory,

Methods

Pergamon

& Applications, Vol. 22, No. 4, pp. 467-480, I!394 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/94 $6.00+ .OO

ON THE EXISTENCE OF THE ENTROPIC SOLUTION TRANSONIC FLOW PROBLEM PETR School

of Mathematics, (Received

University

10February

Key words and phrases: Full potential asymptotic solution, generic solution.

FOR THE

KLOUEEK

of Minnesota,

Minneapolis,

1992; received for publication equation,

quasilinear

parabolic

MN 55455, U.S.A. 18May 1993) equation,

entropic

solution,

1. INTRODUCTION RESULTS contained in the presented paper are the further development of the paper by Kloueek and Netas [l]. The spirit of the analysis for the psuedo-time-dependent approach is taken from the paper of Friedman and Netas [2]. The “nearest” mathematical model bringing the problem of transonic flow within the framework of well-established theory, is the so-called unsteady transonic flow problem. The analysis of the existence and eventually the uniqueness of the solutions of the resulting nonlinear second order hyperbolic equation is still an open problem. Most of the numerical methods for the calculating of transonic flow can be rewritten in a form, from which a successive iteration could be derived, using an artificial time coordinate (see, e.g. Jameson [3,4], Hafez et al. [5]). Typically, one ends up with

aim,,

+ a2uyt + (YOUR = -div(p(lVU(2).

VU),

where u is the velocity potential. Implementing the code on parallel computers, one has to avoid the sequential updating and this requirement calls for introducing the term u,, (see Jameson and Keller [6]). Further comments on the time-dependent approach can be found in Zanetti et al. [7], on the pseudo-unsteady methods in Veuillot and Viviand [8]. All the indications show that the time-dependent or pseudo-time-dependent approaches to the transonic flow problems have advantages from the point of view of the physical framework, mathematical model and implementation. Though it is not clear whether there is uniqueness or not in the original stationary problem (cf. Steinhoff and Jameson [9]), we can use these models to capture the steady-state. Here, we deal with a similar situation. We start from the assumption of adiabatic, isentropic, nonviscous, irrotational steady flow (i.e. full potential equation) and we introduce an artificial time coordinate and we add the damping 4,

-

Au,,

into the original equation. Then we solve the quasilinear obtain a solution which converges to the entropic solution as time goes to infinity. 461

parabolic equation. We expect to of the original stationary problem,

P. KLOUEEK

468

2. FORMULATION

OF THE

PROBLEM

Prior to the further development of the theory we briefly mention the formulation of the problem and results from [ 11. We consider the steady, inviscid, irrotational, adiabatic and isentropic flow of an ideal gas in two or three dimensions. This situation is covered by the following system of equations -div(p

. v) = 0,

v. (Vv) = -fvp,

conservation

of mass,

(2.1)

conservation

of momentum,

(2.2)

adiabatic,

vxv=o

isentropic,

ideal gas flow,

and irrotationality.

(2.3) (2.4)

Here v is the velocity vector, p the density, p the pressure and K the adiabatic exponent (K = 1.4 for air) and po, p. are the stagnation values of the density and pressure. We will consider the system to be in a bounded domain 0 C a”, n = 2, 3. We assume, that the boundary %2 is smooth enough and, thus, there exists a function u E C2(Q) such that Vu = v which is called velocity potential. From (2.2)-(2.4) we can obtain the density as the function of the velocity potential P=Po

(

l-

?I$

,..,2)“y

(2.5)

0

where a is the local speed of sound and a, corresponds density p(s) is defined for (S = IVu1’)

Here, we will consider conditions hold

the density

to be extended 0 < PI 5 PW P’(S)

-p’(s)s Equation

(2.1) with the density

to the stagnation

smoothly

(2.7) (2.8)

5 0,

5 c,

- 2u,u,u,

such that the following

5 p2 < a,

< a.

(2.5), in two dimensions

(0’ - G)u,

to all of a+,

values of p and p. The

(2.9) can also be written

+ (a2 - 24,‘)~~~= 0,

as (2.10)

(the indexes x and y denote the derivatives with respect to the x- and y-direction), has clearly the The equation is elliptic (and the flow subsonic) in a region elliptic/hyperbolic character. 0, C 0, if Ivu12 <

2a,2 K+1’

in a,,

(2.11)

Transonic

and hyperbolic

(and the flow transonic)

Q2, c L2, if

in a region

lvul > ~

469

flow problem

2a,2

Kf

To keep in mind the physical origin of the problem, we usually class of the solutions for which there exists a certain constant, /vu12 5 c,, This condition ensures the density is naturally

Instead function

of the steady-state

velocity

OF

potential

We perturb

the equation

(2.12) to obtain

to the (2.13)

can only be in that part of 6I+ for which

STABILIZATION

u = u(x), we will consider

a time-dependent

t E [O, 7-I.

x E fi,

24 = 24(x, t),

have to restrict ourselves say C,, , such that

< 00.

that the norm of the solution defined. 3. METHOD

(2.12)

in &&.

1’

its regularized

parabolic

ii - Vti - div(p()Vuj*)Vu)

(3.1) counterpart

= 0.

(3.2)

For this equation, we formulate the initial boundary-value problem. Let Q be a bounded domain in a”, n = 2,3, with C2 boundary KJ; for any T > 0 we set fi2, = 1(x, T) 1x E Q2) Let us consider

the equation

and the boundary

denotes

Qr = I&t) 1x E fi2,t E LO,Tll.

(3.2) in Qr, with the initial

conditions

u(x, 0) = u,(x),

for x E 0,

(3.3)

C(x, 0) = u,(x),

for x E Q,

(3.4)

condition

g (x9 t) + Here a/an

and

Pdwx,

tm

the differentiation

for x E LX& t E [0, T].

g (x, t) = g(x),

with respect to the outer normal

Ig(x)l 5 const.

of XJ. We assume

for x E at2

< 00,

(3.5) that (3.6)

and

g(x) dS = 0. The last requirement

is natural; 4. WEAK

Definition 4.1. A function

it corresponds FORMULATION

(3.7)

to the conservation OF

ic is called a weak solution

THE

of mass.

PROBLEM

to (3.2)-(3.5)

if

ti E L”([O, T], W1*2(sZ)),

ii E L2(Qr)

(4.1)

470

P. KLOUEEK

and (3.2)-(3.5)

are satisfied

in the following

sense T

Q,(ii'ql + vii* v ~1 +

p(lV~[~) Vu - vcp) dxdt

= is0

1.l

an

gcp dS dt,

(4.2)

for any v, E L2([0, T], W1*2(sZ)), and any T > 0.

Remark 4.2. We assume that ug E w’*2(n), We use the following defined as follows

24, E W’~2(sz).

(4.3)

Lp([O, T], X) is a space of time-dependent,

notation:

X-valued

functions,

L’(]O, Tl, XI

=

x 1x(t) E X a.e. on [0, T], x is a measurable L

In the case p = 00, we replace

the last condition sttyoe;

We identify LP(QT) compact subdomain compact support in We observe from

function

of t,

;iktlii;

dt < 00 . 1

1

in the above definition

by

IIW )Ilx < 00.

with Lp([O, T], LP(Q)) and write u E W,kd,p(fi), if u E Wkpp(O’) for any a’ of a. By a>(a) we denote the set of all functions from C”(a) with a n and by D’(n), the functions from D(n) which are positive. (4.1) that ti E C(]O, Tl, L2(a))

thus,

(4.4)

u( - , 0) and ti( *, 0) are well defined. 5. GLOBAL

The following mentioned paper

EXISTENCE

theorems [ 11.

and

AND UNICITY

lemmas

can

OF THE STRONG SOLUTION

be found

with

detailed

proofs

in the above-

THEOREM 5.1. Assume that the conditions (2.7)-(2.9) for the density, the bounds (3.6), (3.7) for the boundary function g and the regularity requirements (4.3) for the initial values, are satisfied. Then, there exists a uniquely determined function ti which is the weak solution of (3.2)-(3.5), for any T > 0. THEOREM 5.2.

Let fi be a bounded domain with a Lipschitz boundary and let the assumptions for the density (2.7)-(2.9) and (3.6), (3.7) for the boundary function g, the regularity requirements (4.3) for the initial values, be valid. Let U E L”([O, T], Wiy2(Q)) be the weak solution asserted by theorem 5.1. Further, let o be a smooth positive function, zero on the boundary of Sz. Then

(5.1) for 1 I i, j I n.

Transonic

Under somewhat stronger assumptions, compared tee the global existence of the strong solution. LEMMA 5.3. Let all the assumptions

471

flow problem

of theorem

with those of theorem

5.2 be satisfied

5.2, we can guaran-

and, moreover,

let

(5.2) i

IVu12 dxdt

LEMMA 5.4. Under the assumptions does not depend on T, such that

of the preceding

lii12 dxdt QT

+

I C < ao.

(5.3)

lemma 5.3, there exists a constant

C which

Ivtil%lX I c < 00. RT

(5.4)

THEOREM 5.5. Let Sz be a bounded domain with a Lipschitz boundary and let the assumptions for the density (2.7)-(2.9) and (3.6), (3.7) for the boundary function g, be valid. Let moreover, (5.2) and (4.3) for the initial solutions, be true. Then there exists the unique global strong solution of the problem (3.2)-(3.5). 6. ADDITIONAL Using the lemmas

REGULARITY

of the preceding

section,

OF THE WEAK SOLUTION

we can derive additional

regularity

results,

THEOREM 6.1. Let fi be a bounded domain with a Lipschitz boundary and let the assumptions for the density (2.7)-(2.9) and (3.6), (3.7) for the boundary function g, be valid. Let (5.2) for the initial solutions, be true. Let us assume, moreover, that ug E W2’2@), Then there exists a constant

UI E W2,2(sz).

C which does not depend jar/tii2ti

on T, such that

+ jlQ,lVti]2dxdt

5 C.

(6.1)

Proof. We approximate the initial values u0 and u, by the smooth initial value uh, u:, as I -+ 0, in C2(sZ). By a fixed point argument we can prove (since the density p is a smooth positive bounded function) that the problem (3.2)-(3.5) with the initial data uh, uf has a classical solution in Qr,, if T, is small enough. The strong solution, asserted by theorem 5.5, has to coincide with the classical solution for any t < T, and, as L -+ 0,

ux 4 ll We now work with finite differences

in C’([O, T,], W1p2(Q)).

(6.2)

in time. Let us denote

&ti(x, t) = -&(U(x, t + At) - ti(x, t)).

(6.3)

P. KLOU~EK

412

In the interval [0, T - At] which is again written as [0, T], we have the existence of the weak of the solution a,ti”, for the initial values uk, u:. Because the function g from the definition boundary condition (3.5) is constant in time, we have

where &p(lvul*)

VU = (l/At)((p(lV~1*)

VU)[,+~~ - (p(lVu12) VU)~,). We introduce

w(t) = ru’(t

(6.5)

(6.6)

Pow~)12)VW(T).

S(r) = In view of the above notion

+ At) + (1 - r)ux(t)

it holds

=

* at vu’)

‘(p(lV~1~)L?~ VU’) + 2p’(lvw(*)(Vw

Vwdr.

(6.7)

i .0 Thus, 1

we can rewrite

(6.4), substituting

(v$.ixl* d_X+ ’ \Ii Qr

&.P\*

2 .i 0, = ; I

cp = a;ti”, as

dxdt

l&ri~~%l_x no

_ .is

QT al ~Pdwmt~x i0

. Va,ti”)

+ 2p’((Vwl*)(Vw

* Va,u’)(Vw

* V8,ti’)) dr dxdt. (6.8)

This yields ;

I?&tiAl%

&Al*

JzP~2dX+C i nfl

(6.9)

!I
side of (6.9) converges ;

Iv&ux12 dwdt.

dx + c

i 00 the right-hand

tidt

il; Qr

I ; As At -+ 0,)

Iv&ti’l*

+ (1 - cm&)

1 fir

to

1 I~ri’/~ dxdt. S! QT

(6.10)

In view of lemma 5.3, the second term in (6.10) is bounded independently of T and, therefore, independently of 1. Since, in view of the assumptions for uo, u, , we can choose the approximaof I, it thus follows from tions U,“, u: such that the first term in (6.10) is bounded independently (6. lo), that l&LPI* dX + er

IV&ti’l* dxdt ss QT

5 C

(6.11)

Transonic

and C does not depend

on A and T. That is liixl’ dxdt

IViix[2 dxdt QT

+

1 0, Letting

473

flow problem

A -+ 0, and using (6.2) we get the assertion

I C.

n

of this theorem.

THEOREM 6.2. Let Q be a bounded

domain with a Lipschitz boundary and let the assumptions for the density (2.7)-(2.9) and (3.6), (3.7) for the boundary function g, be valid. Let (5.2) for the initial solutions be true. Let us assume that u0 E W2*2(sZ), U, E WzP2(sZ), and that the weak solution ic satisfies IIVz.illEzcnj5 const.

Then there exists a constant

Iviil%

Proof. If we use the weak formulation substituting cp = a,ii, we end up with l&iil’dwdt QT

v&i. QT

+

(6.13)

I c.

0,

QT.

is

on T, such that

C, which does not depend ICI2 dwdt +

(6.12)

< co.

for the finite differences

v&ii dxdt

+

&zi, introduced

a,(p(lvu12)

Vu) V&iidxdt

in (6.4) and

= 0.

(6.14)

It holds

is

v&tie v&ii dxdt = ;

Q7

s QT.

s

Iv&til” dx.

(6.15)

02,

We can write

ii

fi (&(p(jVu12) Q,dl

(2,(p(lVu12) Vu) V&ii dwdt = QT -

Now, using c(r) defined

in (6.6)

1 (&p(lvu12) ij

si

dQ7z G’,(~(lVul~)

’

we have Vu) * V&t; dxdt

QTdt

p(lvw12).

Vu) Va;ti) dxdt

IV&til’dsdxdt

Vu)) V&c kdf.

(6.16)

P. KLOU~EK

414

sss 1

QT

Using

u- -$$idxdt.

4

+ (6.15)-(6.17)

0

(6.17)

I

and the inequality

Ilv&&(a) 5 IIV4lLZ(n)9 from equation

(6.17) we get d(at(p(l~ul~)vu)).V~~lidudt QP

Summing

up (6.13,

_

1 -= C ( 1 + [[QTlv&~l

hdt)-

(6.18)

(6.17) and (6.18) we obtain

+;s

la,ii12dxdt

Iva,tiI’dxdt QT

07

QT

>

+ ;

s%

)v&ti12 dx

at(p(~vu~2)vu)va,tidx

&(p(Jvu12)

VU) v&ti dxdt.

(6.19)

Also because

IS

a,(p(pu12)Vu) v&ri

dx

QT

=

1

ISSC I

PW12)

DT o

I

c +

Iv&ti12dx,

&

) (6.20)

9,

we get from (6.19)

ss

l&iil’d_xdt

QT

+ nr

(6.21) Now, similar to the above theorem, we assume that u = ux is considered as the weak solution of the initial boundary-value problem with the initial data ui, z.$ E C2(L2), as I + 0, . Considering again the short time interval [0, T,], we have (6.2) from the proof of the preceding theorem.

475

Transonic flow problem

the first bracket

In fact, as At -+ 0,)

on the right-hand

side of (6.21) converges .

IVii’(* dxdt

Cl+ (

to (6.22)

>

ss QT

We can choose uk and u: in such a way that, in view of theorem 6.1 and lemmas other integrals in (6.21) are bounded independently of T and A. the last three integrals in (6.21) converge to AsAt+O,,

5.3, 5.4, the

(6.23)

IviiA12 du + ~.,IVti+).

C (1 co Similar to the preceding assertion follows. n

proof,

7. GENERIC

The following

Definition problem

(6.22), (6.23) can again be bounded

SOLUTION

definition

OF THE

and theorem

TRANSONIC

can be found

p((vu,(*)

- vy, dx =

vu,

1’an

of ,I and the

PROBLEMS

in [l].

7.1. We say that a sequence (u,)“,= 1 is a generic if there exist mappings F,,, E (W’**(Cl))*, such that

sn

FLOW

independently

g-cpdS

solution

of the transonic

+
flow

(7.1)

for all a, E W’,*(sZ), and ~~Frn~~~w~~~~n~~* -+ 0, Here,

( *, * > denotes

duality

asm+oo.

(7.2)

pairing.

THEOREM7.2. Let Q be a bounded

domain with a Lipschitz boundary and let the assumptions for the density (2.7)-(2.9) and (3.6), (3.7) for the boundary function g, be valid. Let (4.3), (5.2) for the initial solutions be true. Then there exists a sequence (t,); =, , such that {u(e) t,)]; = 1 forms a generic solution of the transonic flow problem. As an immediate consequence of this theorem we obtain (theorem 7.3) that the stable limit of u(t), as t + 00, is the solution of the subsonic flow, in which case the stable limit has to satisfy (7.3)

This constraint

implies

(s = lVu)*)

p(s) + 2p’(s)s 2 THEOREM 7.3.

for the density for the initial

P3

>

0,

Let Sz be a bounded domain with a Lipschitz boundary and let the assumptions (2.7)-(2.9) and (3.6), (3.7) for the boundary function g, be valid. Let (4.3), (5.2) solutions be true. Let, moreover, (7.3) be true. Then there exists a function

476

P. KLOU~~EK

ii E W’*‘(sZ) such that u(t) -+ u and U is the solution

of the subsonic

in w1v2(CJ) as t + co

strongly flow

p(]VU12) vu * va, dx = a

an

Proof. Using the estimates

already

u(t) -+ u Vu(t) -+ vu From

definition

obtained,

weakly-star

JV(u(t) - U)12d_xI a

(7.4)

we have

in L2(Q_), as t + 00 and

weakly

4.1 of the weak solution

p3

v $9E Fe2(cl).

m dS

in L”([O, co], L2(S)),

with the test function ii(ti - u(t)) dx +

as t -+ 00.

9 = u(t) - U, we get v2.i * V(U - u(t)) dx

.n i 1

/I(

-

IVU12) vu - V(u(t) - U) dx.

(7.5)

\n Because of u(t) -+ U a.e. in CL!and ii E L’([O, 001, L2(n)), to zero, as t -+ 00. Thus, Vu(t) + vu Because

of the continuity

of the density

function

dwt)12) 4 PdVfi12),

OF THE

ENTROPIC

SOLUTION

side of (7.5) converges

a.e. in C2.

(7.6)

p, we have a.e. in fi for t -+ co.

We know (theorem 7.2) that there exists a sequence solution, thus, the proof is complete. H 8. EXISTENCE

the right-hand

(t,]

OF THE

such that (u(t,)j

TRANSONIC

forms

FLOW

the generic

PROBLEMS

We have already pointed out (in (2.11) and (2.12)) that, with respect to the value of the velocity, the domain of computation can be divided into two parts: Q, and s1,. The boundary between these two subdomains I contains possible shocks on its part IsHoCK with jumps in v, p andp, or equivalently: v, p andp are not continuously differentiable across the “shock/sonic line”. It can be shown using the standard arguments that the following (so-called RankineHugoniot) conditions hold across the shock vu_ - t = vu, p(IVu_l’)

. t,

Vu_ * n = p(IVu+12)

(8.1)

Vu, - n,

(8.2)

where +, - denotes the quantities in front of the shock and behind the shock, respectively, t, n are the tangential and normal vectors to the shock. While these conditions are automatically satisfied by the weak solution, there is a so-called entropy condition, which has to be satisfied if the solution is a physically correct one. It can

Transonic

be formulated

(cf. Landau

and Lifschitz P(lVU-I?

477

flow problem

[lo]) as follows

5 P(lVU+12)

(8.3)

on IsHoCK.

shocks are the only acceptable This can be expressed as “compression (8.1)-(8.3), this requirement can be also formulated as vu_ - n > vu, This condition

is usually

v $7 where K is a positive

E

satisfies

-

CD+(n):

In view of

on LOCK.

et al. [ll,

used (cf. Bristeau

Definition 8.1. We say that u E IYiP’

*n

ones”.

(8.4)

121) in the weaker

the entropy

condition

form. if (8.5)

D

fixed constant.

Remark. (i) If u satisfies the entropy condition (8.9, then it can be shown that u also satisfies (8.4), provided that uln, E C2(Qi) n C’(!?&), i = 1,2 and that the boundary IsHoCK is smooth enough. (ii) There is also another possibility to formulate the entropy condition 1

v (See Feistauer

q7 E

-

D’(sz):

!

,R

and Necas [13, theorem

~‘(Jvu~~)]VU]~ Vu * VP dx I K 4.11) and the following

i .n

~1dx.

(8.6)

remark.)

8.2. Let us assume that there exists a constant C > 0 such that IVu(t)(’ I C, for any t > 0 and assume that all the assumptions which guarantee the global existence of the strong solution are valid (assumptions of theorem 5.5). Let u0 = u(. ,0) E C2(sZ) and, moreover,

THEOREM

weakly in 1&,,(M) as T + m.

> dt + f (4 Then there exists a function

U E I+‘1*2(Q) such that u(t) + u,

and U is the weak solution

(8.7)

strongly

of the transonic

in w’,2(fi)

flow problem

which is entropic

in the sense

AU
such that _ i

Remark.

In virtue

vu0 *va,dx5A40 0

of theorem

cpdx, i

vu,

E

D+(a).

0

5.2, we have ti E L2([0, co]; W:;d,2(sZ)), thus,

u E L2(10, Tl; K?@N,

for any T > 0.

(8.8)

478

P. KLOUEEK

proof.

For the sake of simplicity

we will assume

that C(O) = 0. We already

~~~Uy,dx+~*~V~.V~dX+~~p(lVul”)vu.vy,dX=O, Integrating

the above equation

know that

vy,EzD+(Q).

over the time interval

10, Tl, we get

I’~~updx+~~~vu~vrgdr+~~,~~lv~l2~v~~v~~df=j~~v~~Owrlr, This yields the inequality -

vu*vy,dxs

(8.9)

v-+$;&

t/ v, E 9’(Q)

-

Vu*Va,du+

jerti,dx+

3‘Se(~(p(ia,l’)Vu))-.pdxd~.

(8.11)

%I

RT

Now, in view of lemma

5.3 there exists a sequence u(t,)

Because of the bound

weakly in E”,2(Q),

+ u,

of the gradient

a bounded,

closed,

= 1, such that as I, + 00.

of u, we get, by the application

&P(n)5 c, IlWm)ll and because

(t,]:

convex

(8.12)

of the result of Meyers [14],

for some p > 2,

set (u(t,J I t,,, 2 O] is weakly

(8.13) closed,

also

5 c. II4 @P(Q) Let us define (G(T),

G(T)

E (FK,r9’(Q’))* as vu-va,dx

ul> = -

+

-

tiy,dx+2

*cpdxdt

0,

00 vu*va,dx,

+

(8.14)

where M’ = C& = {x E Sz I dist(x, &2) 2 h]. {T,‘,): = , such that tiy,dx+

0,

we claim

Now,

that

there

exists

a sequence

t/ $0 E a+(a).

T, +*,

(8.15)

.r n%I To show this, we estimate CY,k 1.r nrln Using the result of theorem

5

. bU,nll~~(n)5 II~,1l~~(a)llVti(T,)lI~~(n). lIdlL~(n)

7.2 we get a subsequence ]Vc(T,,)]

Thus,

if we define

(8.16)

I

+ 0,

of (TM]; =, , such that

as Tm, + CQ.

(8.17)

G E ( wc,rF2(CY))* (8.18)

(G, p> = Qo

Transonic

479

flow problem

we see that G(T)

(G(T), Using a stronger

version

G(T)

theorem

Let us denote w(t) = (u(t) - @ai, Moreover, we have w(T))

(see Feistauer

strongly

+ C?

and

(8.19) (8.20)

V p E a>‘@).

cp> 2 0,

of Murat’s

lVw(T)1* = (G(T),

weakly in (w:**(D))*

-+ G

et al. [15]) we obtain

in ( wdpP(CY))*.

with oh defined

(8.21)

in (6.1)-(6.3).

Then

w(t) E W~~“(sZ’).

- (G, w(T))

In view of the compact imbedding of FV’,*(Q) into L*(Q), (8.21) and (8.7), the right-hand of (8.22) converges to zero. The continuity of the density function p and (6.1) give AtVu(T)1*)

Acknowledgement-The flows and who has always

a.e. in Sz as T + 00.

-+ AlVUl*)

We already know that there exists a sequence follows. n author is grateful been interested

side

(TkJ so that (u(T,)j

is generic,

thus,

the proof

to Professor Dr J. Necas, who introduced him to the problems in his theoretical work and numerical results.

of transonic

REFERENCES KLOUEEK P. & NEEAS J., The solution

of transonic

flow problems

by the method

of stabilization,

Applicable

Analysis 37, 143-167 (1990). FRIEDMAN A. & NEEAS J., Systems

of nonlinear

wave equations

with nonlinear

viscosity,

Pacif. J. Mafh. 135,

29-55 (1988). JAMESONA., Iterative solutions of transonic flows over airfoils and wings, Communspure uppl. Math. 2, 283-309 (1974). JAMESON A., Accelerated finite-volume calculation of transonic potential flows, in Numerical Methods for the Computation of Inviscid Transonic Flows with Shock Waves (Edited by A. RIZZI and H. VIVIAND). Notes in Numerical Fluid Mechanics III. Vieweg, Braunschweig (1986). methods for numerical solutions of transonic full 5. HAFEZ M., SOUTH J. & MURMAN E., Artificial compressibility potential equation, AIAA J. 17, 838-844 (1979). study of the use of the STAR-100 computer for transonic flow calculations, 6. JAMESONA. & KELLER B., Preliminary

AIAA 78 (1978). 7. ZANETTI L., COLLASURDOG., FORNASIER L. & PANDOLFI M., A physically consistent time-dependent method for the solution of the Euler equations in transonic flow, in Numerical Methods for the Computation of Inviscid Transonic Flows with Shock Waves (Edited by A. RIZZI and H. VIVIAND). Notes in Numerical Fluid Mechanics III. Vieweg, Braunschweig (1986). method for the computation of transonic potential flows, AIAA J. 8. VEU~LLOTJ. P. & VIVIAND H., Pseudo-unsteady

17 (1979). 9. STEINHOFF J. & JAMESON A., Multiple

solutions

of the transonic

potential

flow equation,

AIAA J. 20, 347-353

(1982). 10. LANDAU L. D. & LIFSCHITZ E. M., Lehrbuch der theoretischen Physik VI-Hydrodynamik. Akademie (1981). 11. BRISTEAU M. O., PIRONNEAU O., GLOWINSKI R., P~RIAUX J., PERRIER P. & POIRIER G., On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element method (II). Application to transonic flow simulations, Comput. Meth. Appl. Mech. Engng 51, 363-394 (1989).

480

P. KLOU~EK

12. BRISTEAU M. O., PIRONNEAU O., GLOWINSKI R., P~RIAUX J., PERRIER P. & POIRIER G., Application of optimal control and finite element methods to the calculation of transonic flows and incompressible flows, in Numerical Methods in Applied Fluid Dynamics (Edited by B. HUNT), pp. 203-312. Academic Press, New York (1980). 13. FEISTAUER M. & NEEAS J., On the solvability of transonic potential flow problems, Z. Anal. Anw. 4, 305-329 (1985). 14. MEYERS N. G., An LP-estimate for the gradient of the solutions of second order elliptic divergence equations,

Annafi Scu. norm. sup. Piss 3(17), 189-206 (1963). 15. FEISTAUER M., MANDEL J. & NE~AS J., Entropy regularization Univ. Carol. 25, 431-443 (1984).

of the transonic

potential

flow, Communs Marh.