- Email: [email protected]
Some estimates for the eigenvalues of a perturbation operator
25. RAI M.M. and CHAKRAVARTHY S.R., An implicit form for the Osher upwind scheme, AIAA Journal 24, 5, 735-743, 1986. 26. CHAKRAVARTHY S.R., The versatility and reliability of Euler solvers based on high-accuracy TVD formulations,AIAA Paper 243, 1986. 21. OBAYASHI S., MATSUSHITA K. and FUJI1 K., Improvements in efficiency and reliability for Navier-Stokes computations using the LU-AD1 factorization algorithm, AIAA Paper 338,1986. 28. COAKLEY T.J., Implicit upwind methods for the compressible Navier-Stokes equations, AIAA Journal, 23, 3, 374-380, 1985. 29. STEGER J.L. and WARMING R.F., Flux vector splitting of the inviscid gas-dynamic equations with application to finite-differencemethods, J. Comput. Phys. 40, 2, 263-293, 1981. 30. WORNOM S.F. and HAFEZ M.M., Implicit conservative schemes for the Euler equations, AIAA Journal. 24, 2, 215-223, 1986. 31. KOLGAN V.P., Application of the principle of minimal values of the derivative to the construction of finite-differenceschemes for calculating discontinuous solutions of gas dynamics. Uch. Zap. TsAGI, 3, 6, 60-77, 1972. 32. CUFFEL R.F., BACK L.H. and MASSIER P.F., Transonic flowfield in a supersonic nozzle with small throat radius of curvature. AIAA Journal, 7, 7, 1364-1366, 1969. 33. IVANOV M.YA. and KORETSKII V.V., Flow calculation in two- and three-dimensionalnozzles by the approximate factorizationmethod. Zh. Vychisl. Mat. mat. Fiz., 25, 9, 1365-1381, 1985. 34. CHIMA R.V. and JOHNSON G.M., Efficient solution of the Euler and Navier-Stokes equations with a vectorized multiple-grid algorithm. AIAA Journal, 23, 1, 23-32, 1985.
Translated by D.L.
U.S.S.R. Comput.Maths.Math.Phys .,Vo1.27,No.6,pp.88-91,1987 Printed in Great Britain
0041-5553/87 $10.00+0.00 01989 Pergamon Press plc
SHORT COMMUNICATIONS SOME ESTIMATES FOR THE EIGENVALUES OF A PERTURBATION OPERATOR* S.V. KUROCHKIN
Suppose 10is a simple isolated eigenvalue and x0 the corresponding eigenvector of the operator A,1 and x the same for the perturbation operator A+B. Estimates and algorithms for calculating L and x are obtained that explicitly use the distance from BxO to a straight line that is taut at XO.The results are partially listed in the case when l0 is multiple. Suppose ho is an isolated simple eigenvalue of the operator A, and x0 the corresponding eigenvector. Consider the perturbation operator AfB and its eigenvalue 5 that corresponds to the linear approximation obtained by perturbation tneory. The question arises of the estimates for eigenvalues and vectors which explicitly contain the distance from the vector to a straight line that is taut at XP. In this paper we obtain estimates of this kind B4 and numerical algorithms that use them to find eigenvalues and eigenvectors of a perturbation operator.
1. Suppose A is a closed operator in a Hilbert space X, b. an isolated point of its spectrum that is a simple eigenvalue, F a contour that does not contain bo and does not contain F and does not contain other points of the spectrum, E is the unit operator in X, We introduce the bounded operators and R(t)=(A-;E)-’ the resolvent of the operator A.
(see /l/j. with the properties: PLP, ImP is one-dimensional,(A-A&)P=O,PS-SP-0. (A-A&S-E-P Let KerP and ImP denote the kernel and image of the operator P. For the operator T:X-X the notation TIRNP will mean the restriction of T to KerP (with domain of values X). Let us take a normed vector r0 in Im P. Then Px-(s,$)x0 for some functional $. Consider the perturbation operator AcB, and for B assume that D(B)aD(A).We shall that corresponds to the eigenvalue h, near to look for an eigenvector of the operator A+B P is a small error correction. Then lo* in the form XO+I,where IeKer
’ 27,11,1736-1739,1987 'Zh.vychzsl.Mat.met.F~z.,
89
(1)
(A+B)(Xp+l)=~.(X~tl).
I.-ethi--(BG$).Then projecting (1) onto
Im P
we obtain
XDLL,+(BI, l$)=%. KerP we obtain
Putting I in projection (1) onto
(d-1~E)l~(B-1,E)~,+(E-P)(B-A,E)I=(BI.
(2) $)I,
(3)
and on the right-hand side instead of El we can write (B-&I for any constant p. in particular with ~-7.~. Furthermore we assume that either the operator B or ES is bounded. Consider the first case. Multiplying (3) on the left by S we obtain I--S(B-h&)x, -S(B-~,~)l+((s-x,~)I,rp)SI. (4) We have an equation of the form I-F(I)in KerP, where F : lier P-b P is a mapping of class C', and li~(l)IlGllS(B-A& [xel pil* 2!lP(B--h,E)is*r pll~llSll~lllll. V(O)il-liS(B-Z*E)X*il, We introduce the notation F'(r)=P(P"-I(*)), f'(r)-P(r). From known facts about contraction mappings (for example /2, Chapter XVIII, Sect.1, Theorem 1) it follows that if the conditions IIS(B-;i,E)xullCa,, 7=~IIP(B-h,E)lK.,~!l+il(E-P)(B-b,E)~K.,~:I!llSll6at
(Be) (5b)
are satisfied, Eq.(4) has a unique solution 1' close to zero, and III'II
@a) (6b)
where C-(l--rrz)-'. The requirements that the constants a,.oz are small is stronger than the requirements that guarantee the separability of the perturbed eigenvalue from the remaining spectrum. Substituting (6) into (2) gives an estimate of the error of the linear approximation and an algorithm for finding the eigenvalue. Eq.(4) can be transformed thus: I-6((),c(I)-(LE-S(B-A,E)l]K:i pL-S(B-~.,E)x,-t((B-blE)I,
$)Sl],.
(7)
Similar reasoning for the condition IIS(B-A,E)l~.rpnCdrci, IIS(E-h,E)xallllPB~r,rPIIIISIICdr,
(6)
dz/(i-d,)*<‘l,, gives
estimates
for
a solution of (7); for
111’11 this is estimate (6a) and
I~~‘-C”(O)II~C~~S(E-A~E)X~~~“+‘(C’IIPB~~.~ pllllSil)*, where
(9)
C-2/(1-d,), C’=4/(1-d,)‘. From the above we obtain the following theorem.
Theorm 1. For the eigenvector of problem (1) with a bounded operator B under conditions (5) or (81, estimate (6a) holds and for an eigenvalue A an estimate of the error of the linear approximation is IA-A~-I.I/cCIIPEI~.,~~II~S(B-A~E)X~~I.
conditions (5) thereis an algorithm bF”(0) (F as in (4)) with convergence estimate (6b). For conditions (B), the algorithm l,=-@(o)(c as in (7)) with estimate (9). Using (2) gives the algorithms for calculating eigenvalues and estimates, respectively, For
IA-ho-L-A(k)
I =WPBI,c.r
pllllS(B-A,E)x.,lly~.
IA-ho-J.i-A(L)
[
piln+‘ilS(B-~.,E)zpil”+‘(C’IISII)”.
Note, In the case when II(B-AIE)xoll or IlS(B-A,E)xJl is small, algorithm (9) converges faster than (6b). At the same time if n-1, (6b), as distinct from (9), gives a quadratic approximation for A. It should be noted that the iterational process L-P"(O)(P from (4)) has already been considered, see /3, p.340/. NOW let us consider the case when B is not bounded but suppose that BS is bounded (later there will also be some conditions of smallness imposed on it). We require a solution of (3) in the form I-5 that gives 2=1;(z). !iO) F(s)=-(E-ij*E)Ip-(E-P)(B-hlE)St~((B-~‘~)SI.1P)S~.
or 2=6(z), G(E)-(E+(E-P) and,
(II) (B-~~E)SJIK;:P[-(B-A,E)XOC((B-A,E)S~,~)S.~.
exactly the same as Theorem 1, we establish the following. Theorem
2.
Under the conditions IISIIII(B-A,E)xoll
7’[ll(E-P)
(B-A,h)SIx.,
pII+ llP(B-1,E)SI ELIpll]C~~
(1%)
90
~~(E-P)(B-L,E)SIE,I.IIC~,. :I(B-AIE)x~;I~IPBSIE~~ pll!lSll~dl
(I?b)
for the eigenvector of problem (1) the estimate '!l'~i~CIISII!I(B-hlE)xai(. holds, and for the eigenvalue the linear approximation estimate is (b-).0-A, I~CII(B-i.,E)xoilllPBSIY.r pii. With condition (12a) the algorithm r,=F‘(O) (F as in (10)) holds with the estimate Ill'-l.,li~CIISllll(B-h,E)xa;l:', and with condition (12b), the algorithm z,=G"(O) (C as in (11)) with the estimate n~~-~.~~sCil(8-A,~)xo!l~+~ll~ll~+~(C’llPB~~~er pil)” The dependence of C,C' on Q,d, is the same as in Theorem 1. 2. The results obtained transfer in part to the case of a multiple eigenvalue. Suppose that near the point &the isolated part of the spectrum of the operator A is located, separated from the remaining spectrum by the contour F. The operators P and S are defined as in Part 1. Below it is assumed that 6=ll(.4-A0E)~rm~ll is small. This includes the case of a semisimple eigenvalue (i.e. of finite multiplicity and without adjoint vectors) and the case when A is selfadjoint and has a group of eigenvaluescloseto lo. suppose that B is bounded (by analogy with Part 1 it is also possible to study the general case; in this case the boundedness Blrmp which is sued afterwards follows from the fact ,thatB is closed and S(B)=B(A)). Suppose that for some\eigenvalue A of the operator d+B, close to J.0.there corresponds an eigenvector of the form k+l,ksImP, Ilk!i=l, l=KerP.illil is small (the existence is not proved as nothing is assumed about the operator A in this relation). In the projection onto ImP and IierP we obtain, respectively dk+PBk+PBI=Ak,
(13s)
AI+(E-P)Okl(E-P)W=A~
(13b)
For an arbitrary bounded operator T we introduce the quantity R(T)-inflIT-AElI. A and k,the number that realizes this minimum (it is not difficult to Let C denote Q(PSl1u.P) show that it exists and is unique). Theorem 3.
Suppose that 6+Q+IIPB[x,, ~lllilll~c~/l!SII, IIS(B-AZ) where C=(i-c,-cz)-'.
lricr PIIS ~2, c,+c~ci.
Then
illll GCilSBl no PII.
IA-ho-~.,~c;d+~+CIIPB[..,pllllSB~rm~ll
Proof. From (13a) we have (d-h,E)k-P(o-?.,E)k~PEl=(i.-).,-).,)k. from which
~A-Ao-A,~&6+R+iIPB~
~~~~41111.From
(13b)
we get
I=-(E+S(B-i.lE)-(h-i~-;.,)~]l~~~~(B-A,~)k,
which the necessary estimates follow. The following theorem, of interest in its own right, shows that Theorem 3 cannot be strengthened by using analogous "local" characteristicsof the operator PS/rrnpinstead of Q. Tor the arbitrary bounded operator T put from
o(T)-supinfllTr-Axll. 11111-1 * Theorem 4. Suppose X is a real or complex Hilbert space and T a bounded operator in X. Then G(T)--o(T). we consider Proof. It is clear that U(T)CR[P). In the real case for each x=X, !Ixli=l the interval II==(A6R: IITx-AxllCo(T)) with centre A.-((TX, x).We shall prove that Lfil, is nonempty for any x, y. Ilsl]=llyll=l. We take a point p~e[L,h,] such that Il~x-A~xllz’~A,-~~‘=;;Ty-A~yll’+~i.,and a (if for example IIF~-h.~l12+~A.-A,~Z~llT~-A,yil*, then AleI,). Then r(T-pE)xll=II(T-~E)]'II PI’ such that II(T-Wzil>ll(T-pE)x! and (Ir,z)=jt. Therevector I can be found of the form ax+~~.l!z11=1, for o(T),li(T-pE)x!l and @el.nl,. Applying Helly's theorem on the intersection of convex bodies for dimension 1 we obtain a point7 that belongs to the whole interval I. and, thus, III'-yEIIc m(F). In the complex case the proof is similar with the following changes: a standard realization is made, instead of sections circles are constructed on a plane, the non-emptiness of the intersection of any three circles is used in the proof for the geometric transformation and the convexity of a numerical image of the operator, and Helly's theorem is applied with dimension 2. The author thanks A.A. Abramov for suggesting the problem and for his interest, and V.B. Lidskii for valuable remarks. REFERENCES 1. EAT0 T., Perturbation theory for linear operators, Mir, Moscow, 1972. 2. KANTOROVICH L.V. and AKILOV G.P., Functional analysis, Nauka, Moscow, 1984.
91
3. BAUMGARTEL Ii.,Analytic perturbation theory for matrices and operators, Birkhauser, Basel, 1985. Translated by S.R.
U.S.S.R. Comput.Maths.Math.Phys .,Vo1.27,No.6,pp.91-94,19&7 Printed in Great Britain
OC41-5553/87 510.00+0.00 01989 Pergamon Press plc
NUMERICAL SIMULATION OF A CIRCULATED GAS FLOW BASED ON THE COMPLETE AND SIMPLIFIED NAVIER-STOKES EQUATIONS*
YU.P. GOLOVACEEV and N.V. LEONT'YEVA Using the example of stationary axisymmetric flow near any surface of a sphere situated in a supersonic wake region, the applicability of simplified mathematical models for the description of gas flows with a developed recirculation zone is studied. A comparison is made of the results of numerical solutions of the complete and simplified NavierStokes equations with two different boundary conditions at the head end shock wave. At the present time, the simplified ("parabolized")Navier-Stokes equations are used to simulate viscous gas flows. These include all terms of Euler's equations and the boundarylayer equations and do not contain second dervivatives of unknown functions with respect to the coordinate which coincides with the basic direction of flow. Mathematical models were formulated in /l-3/ for problems of supersonic flow over solids, as a result of evaluating a number of terms of the Navier-Stokes equations describing the molecular transfer of momentum and energy. These evaluations were based on concepts of classical boundary layer theory and hold for continuous flow at large Reynolds numbers. Practical experience in using the models under examination has shown that the region where they are applicable is wider than the formal limits expected from the evaluations specified above. Simplified equations are used, in particular, for calculating flows with flow separation from the surface of the solid and with the formation of closed regions of backward-circulationflow (see, for example, /4-6/I. The study of errors connected with the description of their simplified equations in the simulation of such flows is of interest. With this object in view, in this paper the results of a numerical solution of the complete and simplified Navier-Stokes equations are compared for one example of gas flow with a developed recirculation sane. 1. Statement of the problem and numerical method. Stationary axisymmetric flow over any surface of a sphere by a non-uniform supersonic stream, whose parameters correspond to flow in a remote trace is examined. The distributions of the gas-dynamic functions in the oncoming stream are described by formulae taken from /7/: V(y)=V(~)[l-oeap (-bu’)l,
h~~)-~(o)(~+c~i-~‘(y)/~‘(O)11, p(y)-const.
(0
Here, y is the distance from the axis of symmetry referred to the radius of the sphere, V is the modulus of the velocity vector, h is the specific enthalpy, p is the pressure, and a, b, and c are numerical parameters describing the shape of velocity and enthalpy profiles in the oncoming stream. using (l), flow in the wake is simulated by a parallel axisymmetric stream with constant pressure, axial minimum velocity and maximum enthalpy. It is assumed that the density of the gas is sufficient to form a head end shock wave in front of the sphere. Flow between the head end shock wave and the surface of the solid is described by the simplified Navier-Stokes equations /3/:
au a0 bP v an u--f---+P r -&-fd8+2u+vctg6 or r86 T( ea ” 80 ll= OP ;i;-+---+--0, Pa ( rai3 r ) ar av Y av pi-+---+( ar rae
ah
D
uv
tap a +---_-p-~_o, r ) F a8 ar (
1
au
-0,
(2)
ar j
Bh
s,+--r de ) Eqs.(Z) are written usingthegenerally accepted notation in a spherical system of coordinates which has its pole at the centre of the sphere in the flow. +Sh.vychisl.Mat.mat.Fiz.,27,11,1739-1744,1987
Translated by D.L.
U.S.S.R. Comput.Maths.Math.Phys .,Vo1.27,No.6,pp.88-91,1987 Printed in Great Britain
0041-5553/87 $10.00+0.00 01989 Pergamon Press plc
SHORT COMMUNICATIONS SOME ESTIMATES FOR THE EIGENVALUES OF A PERTURBATION OPERATOR* S.V. KUROCHKIN
Suppose 10is a simple isolated eigenvalue and x0 the corresponding eigenvector of the operator A,1 and x the same for the perturbation operator A+B. Estimates and algorithms for calculating L and x are obtained that explicitly use the distance from BxO to a straight line that is taut at XO.The results are partially listed in the case when l0 is multiple. Suppose ho is an isolated simple eigenvalue of the operator A, and x0 the corresponding eigenvector. Consider the perturbation operator AfB and its eigenvalue 5 that corresponds to the linear approximation obtained by perturbation tneory. The question arises of the estimates for eigenvalues and vectors which explicitly contain the distance from the vector to a straight line that is taut at XP. In this paper we obtain estimates of this kind B4 and numerical algorithms that use them to find eigenvalues and eigenvectors of a perturbation operator.
1. Suppose A is a closed operator in a Hilbert space X, b. an isolated point of its spectrum that is a simple eigenvalue, F a contour that does not contain bo and does not contain F and does not contain other points of the spectrum, E is the unit operator in X, We introduce the bounded operators and R(t)=(A-;E)-’ the resolvent of the operator A.
(see /l/j. with the properties: PLP, ImP is one-dimensional,(A-A&)P=O,PS-SP-0. (A-A&S-E-P Let KerP and ImP denote the kernel and image of the operator P. For the operator T:X-X the notation TIRNP will mean the restriction of T to KerP (with domain of values X). Let us take a normed vector r0 in Im P. Then Px-(s,$)x0 for some functional $. Consider the perturbation operator AcB, and for B assume that D(B)aD(A).We shall that corresponds to the eigenvalue h, near to look for an eigenvector of the operator A+B P is a small error correction. Then lo* in the form XO+I,where IeKer
’ 27,11,1736-1739,1987 'Zh.vychzsl.Mat.met.F~z.,
89
(1)
(A+B)(Xp+l)=~.(X~tl).
I.-ethi--(BG$).Then projecting (1) onto
Im P
we obtain
XDLL,+(BI, l$)=%. KerP we obtain
Putting I in projection (1) onto
(d-1~E)l~(B-1,E)~,+(E-P)(B-A,E)I=(BI.
(2) $)I,
(3)
and on the right-hand side instead of El we can write (B-&I for any constant p. in particular with ~-7.~. Furthermore we assume that either the operator B or ES is bounded. Consider the first case. Multiplying (3) on the left by S we obtain I--S(B-h&)x, -S(B-~,~)l+((s-x,~)I,rp)SI. (4) We have an equation of the form I-F(I)in KerP, where F : lier P-b P is a mapping of class C', and li~(l)IlGllS(B-A& [xel pil* 2!lP(B--h,E)is*r pll~llSll~lllll. V(O)il-liS(B-Z*E)X*il, We introduce the notation F'(r)=P(P"-I(*)), f'(r)-P(r). From known facts about contraction mappings (for example /2, Chapter XVIII, Sect.1, Theorem 1) it follows that if the conditions IIS(B-;i,E)xullCa,, 7=~IIP(B-h,E)lK.,~!l+il(E-P)(B-b,E)~K.,~:I!llSll6at
(Be) (5b)
are satisfied, Eq.(4) has a unique solution 1' close to zero, and III'II
@a) (6b)
where C-(l--rrz)-'. The requirements that the constants a,.oz are small is stronger than the requirements that guarantee the separability of the perturbed eigenvalue from the remaining spectrum. Substituting (6) into (2) gives an estimate of the error of the linear approximation and an algorithm for finding the eigenvalue. Eq.(4) can be transformed thus: I-6((),c(I)-(LE-S(B-A,E)l]K:i pL-S(B-~.,E)x,-t((B-blE)I,
$)Sl],.
(7)
Similar reasoning for the condition IIS(B-A,E)l~.rpnCdrci, IIS(E-h,E)xallllPB~r,rPIIIISIICdr,
(6)
dz/(i-d,)*<‘l,, gives
estimates
for
a solution of (7); for
111’11 this is estimate (6a) and
I~~‘-C”(O)II~C~~S(E-A~E)X~~~“+‘(C’IIPB~~.~ pllllSil)*, where
(9)
C-2/(1-d,), C’=4/(1-d,)‘. From the above we obtain the following theorem.
Theorm 1. For the eigenvector of problem (1) with a bounded operator B under conditions (5) or (81, estimate (6a) holds and for an eigenvalue A an estimate of the error of the linear approximation is IA-A~-I.I/cCIIPEI~.,~~II~S(B-A~E)X~~I.
conditions (5) thereis an algorithm bF”(0) (F as in (4)) with convergence estimate (6b). For conditions (B), the algorithm l,=-@(o)(c as in (7)) with estimate (9). Using (2) gives the algorithms for calculating eigenvalues and estimates, respectively, For
IA-ho-L-A(k)
I =WPBI,c.r
pllllS(B-A,E)x.,lly~.
IA-ho-J.i-A(L)
[
piln+‘ilS(B-~.,E)zpil”+‘(C’IISII)”.
Note, In the case when II(B-AIE)xoll or IlS(B-A,E)xJl is small, algorithm (9) converges faster than (6b). At the same time if n-1, (6b), as distinct from (9), gives a quadratic approximation for A. It should be noted that the iterational process L-P"(O)(P from (4)) has already been considered, see /3, p.340/. NOW let us consider the case when B is not bounded but suppose that BS is bounded (later there will also be some conditions of smallness imposed on it). We require a solution of (3) in the form I-5 that gives 2=1;(z). !iO) F(s)=-(E-ij*E)Ip-(E-P)(B-hlE)St~((B-~‘~)SI.1P)S~.
or 2=6(z), G(E)-(E+(E-P) and,
(II) (B-~~E)SJIK;:P[-(B-A,E)XOC((B-A,E)S~,~)S.~.
exactly the same as Theorem 1, we establish the following. Theorem
2.
Under the conditions IISIIII(B-A,E)xoll
7’[ll(E-P)
(B-A,h)SIx.,
pII+ llP(B-1,E)SI ELIpll]C~~
(1%)
90
~~(E-P)(B-L,E)SIE,I.IIC~,. :I(B-AIE)x~;I~IPBSIE~~ pll!lSll~dl
(I?b)
for the eigenvector of problem (1) the estimate '!l'~i~CIISII!I(B-hlE)xai(. holds, and for the eigenvalue the linear approximation estimate is (b-).0-A, I~CII(B-i.,E)xoilllPBSIY.r pii. With condition (12a) the algorithm r,=F‘(O) (F as in (10)) holds with the estimate Ill'-l.,li~CIISllll(B-h,E)xa;l:', and with condition (12b), the algorithm z,=G"(O) (C as in (11)) with the estimate n~~-~.~~sCil(8-A,~)xo!l~+~ll~ll~+~(C’llPB~~~er pil)” The dependence of C,C' on Q,d, is the same as in Theorem 1. 2. The results obtained transfer in part to the case of a multiple eigenvalue. Suppose that near the point &the isolated part of the spectrum of the operator A is located, separated from the remaining spectrum by the contour F. The operators P and S are defined as in Part 1. Below it is assumed that 6=ll(.4-A0E)~rm~ll is small. This includes the case of a semisimple eigenvalue (i.e. of finite multiplicity and without adjoint vectors) and the case when A is selfadjoint and has a group of eigenvaluescloseto lo. suppose that B is bounded (by analogy with Part 1 it is also possible to study the general case; in this case the boundedness Blrmp which is sued afterwards follows from the fact ,thatB is closed and S(B)=B(A)). Suppose that for some\eigenvalue A of the operator d+B, close to J.0.there corresponds an eigenvector of the form k+l,ksImP, Ilk!i=l, l=KerP.illil is small (the existence is not proved as nothing is assumed about the operator A in this relation). In the projection onto ImP and IierP we obtain, respectively dk+PBk+PBI=Ak,
(13s)
AI+(E-P)Okl(E-P)W=A~
(13b)
For an arbitrary bounded operator T we introduce the quantity R(T)-inflIT-AElI. A and k,the number that realizes this minimum (it is not difficult to Let C denote Q(PSl1u.P) show that it exists and is unique). Theorem 3.
Suppose that 6+Q+IIPB[x,, ~lllilll~c~/l!SII, IIS(B-AZ) where C=(i-c,-cz)-'.
lricr PIIS ~2, c,+c~ci.
Then
illll GCilSBl no PII.
IA-ho-~.,~c;d+~+CIIPB[..,pllllSB~rm~ll
Proof. From (13a) we have (d-h,E)k-P(o-?.,E)k~PEl=(i.-).,-).,)k. from which
~A-Ao-A,~&6+R+iIPB~
~~~~41111.From
(13b)
we get
I=-(E+S(B-i.lE)-(h-i~-;.,)~]l~~~~(B-A,~)k,
which the necessary estimates follow. The following theorem, of interest in its own right, shows that Theorem 3 cannot be strengthened by using analogous "local" characteristicsof the operator PS/rrnpinstead of Q. Tor the arbitrary bounded operator T put from
o(T)-supinfllTr-Axll. 11111-1 * Theorem 4. Suppose X is a real or complex Hilbert space and T a bounded operator in X. Then G(T)--o(T). we consider Proof. It is clear that U(T)CR[P). In the real case for each x=X, !Ixli=l the interval II==(A6R: IITx-AxllCo(T)) with centre A.-((TX, x).We shall prove that Lfil, is nonempty for any x, y. Ilsl]=llyll=l. We take a point p~e[L,h,] such that Il~x-A~xllz’~A,-~~‘=;;Ty-A~yll’+~i.,and a (if for example IIF~-h.~l12+~A.-A,~Z~llT~-A,yil*, then AleI,). Then r(T-pE)xll=II(T-~E)]'II PI’ such that II(T-Wzil>ll(T-pE)x! and (Ir,z)=jt. Therevector I can be found of the form ax+~~.l!z11=1, for o(T),li(T-pE)x!l and @el.nl,. Applying Helly's theorem on the intersection of convex bodies for dimension 1 we obtain a point7 that belongs to the whole interval I. and, thus, III'-yEIIc m(F). In the complex case the proof is similar with the following changes: a standard realization is made, instead of sections circles are constructed on a plane, the non-emptiness of the intersection of any three circles is used in the proof for the geometric transformation and the convexity of a numerical image of the operator, and Helly's theorem is applied with dimension 2. The author thanks A.A. Abramov for suggesting the problem and for his interest, and V.B. Lidskii for valuable remarks. REFERENCES 1. EAT0 T., Perturbation theory for linear operators, Mir, Moscow, 1972. 2. KANTOROVICH L.V. and AKILOV G.P., Functional analysis, Nauka, Moscow, 1984.
91
3. BAUMGARTEL Ii.,Analytic perturbation theory for matrices and operators, Birkhauser, Basel, 1985. Translated by S.R.
U.S.S.R. Comput.Maths.Math.Phys .,Vo1.27,No.6,pp.91-94,19&7 Printed in Great Britain
OC41-5553/87 510.00+0.00 01989 Pergamon Press plc
NUMERICAL SIMULATION OF A CIRCULATED GAS FLOW BASED ON THE COMPLETE AND SIMPLIFIED NAVIER-STOKES EQUATIONS*
YU.P. GOLOVACEEV and N.V. LEONT'YEVA Using the example of stationary axisymmetric flow near any surface of a sphere situated in a supersonic wake region, the applicability of simplified mathematical models for the description of gas flows with a developed recirculation zone is studied. A comparison is made of the results of numerical solutions of the complete and simplified NavierStokes equations with two different boundary conditions at the head end shock wave. At the present time, the simplified ("parabolized")Navier-Stokes equations are used to simulate viscous gas flows. These include all terms of Euler's equations and the boundarylayer equations and do not contain second dervivatives of unknown functions with respect to the coordinate which coincides with the basic direction of flow. Mathematical models were formulated in /l-3/ for problems of supersonic flow over solids, as a result of evaluating a number of terms of the Navier-Stokes equations describing the molecular transfer of momentum and energy. These evaluations were based on concepts of classical boundary layer theory and hold for continuous flow at large Reynolds numbers. Practical experience in using the models under examination has shown that the region where they are applicable is wider than the formal limits expected from the evaluations specified above. Simplified equations are used, in particular, for calculating flows with flow separation from the surface of the solid and with the formation of closed regions of backward-circulationflow (see, for example, /4-6/I. The study of errors connected with the description of their simplified equations in the simulation of such flows is of interest. With this object in view, in this paper the results of a numerical solution of the complete and simplified Navier-Stokes equations are compared for one example of gas flow with a developed recirculation sane. 1. Statement of the problem and numerical method. Stationary axisymmetric flow over any surface of a sphere by a non-uniform supersonic stream, whose parameters correspond to flow in a remote trace is examined. The distributions of the gas-dynamic functions in the oncoming stream are described by formulae taken from /7/: V(y)=V(~)[l-oeap (-bu’)l,
h~~)-~(o)(~+c~i-~‘(y)/~‘(O)11, p(y)-const.
(0
Here, y is the distance from the axis of symmetry referred to the radius of the sphere, V is the modulus of the velocity vector, h is the specific enthalpy, p is the pressure, and a, b, and c are numerical parameters describing the shape of velocity and enthalpy profiles in the oncoming stream. using (l), flow in the wake is simulated by a parallel axisymmetric stream with constant pressure, axial minimum velocity and maximum enthalpy. It is assumed that the density of the gas is sufficient to form a head end shock wave in front of the sphere. Flow between the head end shock wave and the surface of the solid is described by the simplified Navier-Stokes equations /3/:
au a0 bP v an u--f---+P r -&-fd8+2u+vctg6 or r86 T( ea ” 80 ll= OP ;i;-+---+--0, Pa ( rai3 r ) ar av Y av pi-+---+( ar rae
ah
D
uv
tap a +---_-p-~_o, r ) F a8 ar (
1
au
-0,
(2)
ar j
Bh
s,+--r de ) Eqs.(Z) are written usingthegenerally accepted notation in a spherical system of coordinates which has its pole at the centre of the sphere in the flow. +Sh.vychisl.Mat.mat.Fiz.,27,11,1739-1744,1987