Numerical solution of equations with a power boundary layer

133

Similar expressions can also be written for VM- V, VM”- V. Since K, , a~_&’(H(A), If) (or what amounts to the same thing, K,, c_$?(II(A), H), see Theorem 4), we obtain from t(t)~W,*(O, 0~; H), 1IAz(t)IIvt=[O, -) the relations IIK,,s(t)Illc;c,l~z(t)ll~c,lIP(t) [I<=J Vt= [0, m) . Using these facts, we can easily prove the following relations ( v 1 w co, “)):

Y$-v

MJ=O

V,--vzO,

VM>N,

(3.8)

V&--V Jf=L M_ 0.

Theorem 13. Let the conditions of Theorem 12 hold. Then, for the optimal triple (x, V, u) of the 1.q.p. problem, we have limit relations (3.6x3.8). Note 5. It can be seen from the methods of proving Theorems 12 and 13 that the limit relations in them as n - (Q, M - =, can be refmed, i.e. expressed as inequalities. In addition, if the condition of Note 3 holds, we can indicate in these relations the nature of the decrease of the difference of the solutions as I - =. The author thanks V. A. Il’in and A. I. Egorov for useful discussions. Translatedby D. E. B.

RKFERKNCKS

4. 5. 6. I. 8. 9. IO. II. 12.

EGOROV, A. I., and RAKHIMOV, hf., About the problem of synthesis of optimum control by elastic oscillations, in: &cr. Notes in Compur. Sci., Vol. 27, Springer,Berlin, 197.5. EGOROV, A. I., Optimal Confrol of l&ma1 undD~&7iukm hcesses (Optimal’noe upnvlcnie teplovymi i diffuzionnymi protsmsami), Nauka, Moscow, 1978. RAKHIMOV, M., On optimal stabilization of systems with distributed parameters, IN. AN i’i~km SSR, Ser. Fil.., r&ha. khim.. geol MI&, 3,3-9, 1978. RAKHIMOV, hf., On a generalizationof a theorem of optimal stabilization theory, IN AN Turkm SSR, Ser. ?Yz.,fek~kh~., geoi. nmrk, 1,3-q, 1979. IL’IN, V. A., On the unconditional basis property in a closed interval of systems of eigen and associated functions of a second-order differential operator, DaW A&ad No& SSSR, 273.5, 104g-1053,1983. SHKALIKOV, A. A., Bounduy value problems for ordinary diierential quations with a parameter in the boundary conditions, in: It: seminara im. I. G. Pewovskogo, 9, Izd-vo MGU, Mtxcow, 1%3. MALKIN, I. G., Titeory OfSmbUiry of Mo:bm (Teoriya ustoichivoati dvi&eniya), Nauka, Moscow, 1966. LIONS, J.-L., Oprinrai Cwmol of Systems &scr#ed by Pa&al D~~~t~~ti~ Mr. Moscow, 1972. BA~RISHNAN, A. V.. ~~~ti~~~ (Prikiadnoi f~~io~nyi an&z), Nauka, b&cow. 1980. GOKHBERG, I. TX.. and KREIN, M. G., inrroclLeton to the Z’&eoryof E&ear Nowse@@oin~ Operamrs (Vvedenie v teoriyu lineinykh nesamosopryazhennykb operatorov), Nauka. Moscow, 1965. TIKHOMIROV, V. V., Exnct estimates of regular solutions of the one-dimensional Schrklinger quation with a spectral parameter, Do&L Akad Nauk SSSR, 273,4,807-810,1983. KANTOROVICN, L. V., and AKILOV, 0. P., FWIC&M/ Anaiysi~ (Funkuional’nyi an&), Nauka, Moscow, 1977.

U.S.S.R. Comput Maths. Marh Phys.. Vol. 26,No. 6,pp. 133-139, 1986 Printed in Great Britain

NUMERICAL

SOLUTION

OF EQUATIONS

0041JS53/86 $10.Ooto.00 0 1988 Pergamon Journals Ltd.

WITH A POWER BOUNDARY

LAYER*

V. D. LISEIKIN An algorithm is devised for solving numerically a problem with a power boundary layer. For certain assumptions regarding the coeffkicnts of the equation, the uniform convergence of the numerical solution to the exact solution is proved. The usual difference schemes for solving boundary value probkms give a poor approximation on unwon meshes of the initial equation, in the zones where the required function has singularities. The approximation can be improved if we introduce new independent variables, with respect to which the function-solution has either no singularities, or their order is reduced. The problem can then be solved with respect to the new variables on a uniform mesh, or in the old variables, on a non-uniform mesh, constructed by transformation of the independent variables. This approach is used below for problems with a power boundary layer.

*Zk. vychisl Mar. mot. Fis., 26.12, 1813-1820,1986.

134

1. Estimates of derivatives; the solution of the problem Consider the boundary value problem ~~u~-(E+~~u”+a(s)u’-c(x)u~f(r), u(i)=+,

u (0) -10,

a(O)-a>O,

OCxCl, c&)%0,

Problems of this kind arise when modelling processes in which the (1.1) with a > 1 was obtained in [I]. To solve (1.1) numerically, we find a transformation x(q) such that uniformly bounded derivatives [2]. For this, we obtain estimates of the Zikeorem 1. Let r&r) be the solution of (1.1); then, if a(z), ~(a$, Iu”(z)jGf(E+X)~-~~, while if a(z), c(z), j(z)@?[O,

a<$,

l&00.

(Lla) (l.lb)

viscosity is variable. The asymptotic expansion the dependent variable u, (q) = u [x(q)] has derivatives of the solution (1.1). f(t)~C*fO, i}, we have k-l,

2, 3,

(1.2)

i] , we have (k = 1,2,3)

Here and below, M and m are positive constants, independent oft, x, and i.

Pm$ Solving (1.1) for u’(x) and transforming the integrand, we obtain (1.4a) (1.4b)

Notice that lg, (x)1 5 IU, and since, by the maximum principle for the operator L, we have ju(x)l S M, we find that lrs~~~l<~(e+~~“jb+U”-l~~~. s Let u C 1. Obviously, there is a point x,, in the interval (y2, 1) such that lu’ (x0)1 I M. Putting x = x0 in (i.4), we obtain j~‘(O)IGlfs-~ and hence, }u’(s)l~M(e+z)-‘. It also easily follows from (i.1) with k = 2,3. Estimate (1.2) is thus proved for D < 1. that lab(z) l
Io=u‘(O)

i

eln(e+

f)8xp[-gr(i)]-elne (1.5)

0 Since la(E)-al-=la(E)-a(O) IcAfe, the expression in the braces is e g, (f) - e ln e, where ]gl (01 5 M. For sufftciently smaI1 c 5 tn. m > 0, we have eg,($)-c In e> Ie In e l/2. It therefore follows from (1.5) with e I m that lu’(0) I
eCm,

i.e. (1.3) holds fork = 1, QI: m. Since I&x)] 5 Mfor c L m, then (1.3) clearly holds fork = 1.1 L t > 0. We will now estimate u”(x). On diffe~ntiating (1.1) and solving the resulting equation for u”(x), we obtain

(1.6)

We will estimate g,(x) with u = 1. For this, we use (1.3), which has been proved for k = 1, II = 1:

135

Further, since, by (l.l),

Iu”(0) /
If
we obtain from (1.6), using the estimate of g,(x):

lu”(z) jQM[i+(e+r)-Pln-L Similarly, on differentiating u “(x), we obtain

e-l],

(1.1) twice and solving the resulting equation for u”‘(x), then using the estimates of u’(x),

Let u > 1. It follows from Hal 5 M that there is a point x0 in the interval (0, e) such that [u’(z~) fume-*. Putting x =x, in (1.4), we obtain I&(O) j&We-‘, and hence, by (1.4), /u’(z) ~
k--2,3,

l~M[l+E”-‘(E+s)‘-“-~f ,

is proved in the same way as for a = 1. The theorem is proved. By (1.2) and (1.3), the derivatives of the solution of (1.1) may be estimated by a power function with argument (c + x), so that we in fact call (1.1) an equation with power boundary layer [ 1J.

2. Numerical algorithm For the numerical solution of (1.1) we introduce the new independent variable q by means of the nondegenerate transformation x(q) .= C 2 [0, 11, x(O) = 0, x(l) = 1. Problem (1.1) for the variables q, u, (q) = u [x(q)}, takes the form L,(u,frIefz(p)]ui”(p)+a~(gfu~(q)-c~(p)ut(q)~ft(q), O
ut (0) -20,

(2. la)

U‘(i)==ll,

(2.lb)

~~(~~-~ls(~)l~‘(q)-_[e+~(q)ls”(q) IdG$l-1, c,(q)~I2’(~)l’cI~(q)J, f1(9)““[r’(q)l’f[s(p)l. We approximate (2.1) on the uniform mesh q, = ih, i = 0, 1, . . . , N, h = l/N, by the scheme with directed differences + ni-r +-[a1 (q,) f 1at (4,) I]v

LI” (u) ca (e + z~) u’+’ - 2 +~~l(!zr~-l~l(q3II-I=&2

Ut - vi-x 2h

(2.W

Cl b) ui = fl Q),

,..., N--I, (2.2b)

I.cmmu. Assume that, for some m > 0, the mesh function n, (q 3, i = 0, I, . . . , N, satisfies one of the two conditions: %(%)(O, or

then, (2.3) where &] is the solution of (2.2), and u1 (q),f, (q) are functions of (2.1). Proof. We introduce the mesh function w = {wJ, i = 0, 1, . . . , N, by putting wt=M,-M,iha,(0)/la,(O) M,=#+ll,l+M,+M,,

l--ili,(ih)k, JK==aQI+Q*,

i-0, 1,. . . , N, M,=2”‘(Q,+Q,),

We have L?(w)=

_Mo[e+zt+h

x (fh f

hlk-

[

al(W;lalWl]

2(G)” + (fh - k)k _ Msal (fk) ffNk - (fk - kP

?t’

h

-

(2.4 cont’d over)

136

j)f,k

(thy”[(k

- 1)(E+

zd (2.4 cont’d)

-

(ih)

101(ih) II-

Mm w*

*

M&

o
With (I, (ti) < 0,O 5 ih I m. we have from (2.4): aloh)

[MS

-

(m/V*],

(Ih), m/2 < ih B m, (k - I) (e + 2,) > Ia1P) I. - M&m”-’ + Mr 101Oh)1. m
(2.5)

Meal LP

(ro)<

(k - 2)(e+ 24 > ta W I-

I

By the conditions of the lemma, la, (i/r)1 5 M(t + x,) for (I, (ik) < 0, ih 1 m/2. Hence, for sufticiently large k, independent of t, x, i, for the mesh function I( = {u,},i = 0, 1, . . . , N, which is the solution of (2.2), we obtain from (2.5): LF( w*u)GO,

W”fU~20.

w.fuo20,

Hence, by the maximum principle [3], for the operator L’ (L /), i = 1,2. . . . , N - 1, we have

(2.6) i.e. (2.3) holds for al(q,)O, O
L: (w) <

1 >qt>fh

1 and for sufficiently large k, irrespective oft, x, and i, L:(ILtstIz)co,

w&a,20,

wan*u,>o.

Hence we obtain (2.6) from the maximum principle for L’. The lemma is proved.

3. cnastructIoa of the trMsformstion aQ) We shall construct the transformation (1.3). Let (I < 1. In this case we put

x(q) with the condition lu ‘, (q)l I M, for which we use estimates (1.2),

~(q)-(ei-p+pq)t~~t4~-e, l3-(a+2)/3.

O
p-(l+e)‘-p-el-p,

(3.1)

It is easily verified that

IulYq) IG+f,

k<3,

(3.2)

where ~,(q)=u(z(q)], U(Z) is the solution of (1.1). Let us prove this, for example, for k = 3. We have u,‘“(q)-I&rJt’(q)lJ+3Ud~(q)z’(q)+UIZ~(q). Hence, using (1.2) and (3.1), we obtain IL,M(q)
Further,

a,~q~-(aIt~cl~l-~c+~~q~l~“~~~r~‘~~~l-’~~~~~-~~rt~~~l-_B}z’~q~. Since a(O) -acp,

z(O) -0,

It’(q) ( 0 such that u[z(q)] -p&m,

OGgdm<‘/,

i.e. (3.3)

In addition,

la*(q)l/re+4q)lcK

qsml2.

We will now estimate the error of difference scheme (2.2) with u C 1. Theorem 2. Assume that o
(3.4)

137

where u(x) is the solution of (1. l), II = (u,), i = 0, 1, . . ., N, is the solution of (2.2) with x(q) given by (3. l), and x, = x(qc). Pro& We put

On expanding Y, (q) in a Taylor series in the neighbourhood of q, = 31 and substituting the expansion into (2.2), we obtain

From this, using (3.2), we have

I~~Ic~hrs+s+la,~ih)Il,

I--i, 2,. . . , N-l.

Since, by (3.3), (3.4), the conditions of the lemma hold, it then follows from estimates (2.3), (3.3), (3.4) that Ir+=ju~-z&) [l.~n~~~~~~q~~

z(q)=

I

Ts [[i - Z#t(l*) - f], 0<4<%* T (8’ - 8) f 2Tpd (u - i)-+@-sr/d (q - a/,) + 2Tp’d (a - 1)-s (/I + d- f) &d-r)ld(q - ‘/#+k(q-y*)‘,

Va
I

p-&e’“‘, z-(a-i)/(u+2), 9>dXt, T-((o-i)?[(p-~)‘f(o-i+d)‘c”U”‘]-*,

1.

(3.5)

kl;8[2-T(s’-e)-Tpd(a-i)-‘e’(d-*“/d -Tp*d(a--i)-‘(a+d-i)

P-‘)“/2].

The function r(q) belongs to the class C? [0, 11,and is infinitely ~ffe~ntiable in the intervals [0, $J and [X, 11. Let us show that the transformation x(cr) is nondegenerate. From (3.5) we have 2Tpd (4 - I)-’ s (1 - 2gq)@+d-l)i(l-@ OGqd%t 2Tpd (o - i)-” az@-*)ld + 4TpBd (a _’ I)-* x I’ tq1= x (a + d - 1) tPd+d (q - I/%) + 3k (q l/a)? vaa;q 0 for 0 C=q I X. Let q > &. Since 12 2 (qz’(q)>@Tpd(a--i)-‘6

X)

(3.9)

for (q - X) , we obtain from (3.6):

‘~~-*~~~+8Tp’d(a-i)-*(a+d--l)e”d-”’4+3k]

(q-‘/r)*:

-8{3+3T~-(T/2)e’(“‘“(a-l)-*~6(a-i)~~”’~~ +4pd(cl-i)e’“+p*d(afd-1) >8(3-3T(a-i)-*(aSd--t)*e”“*“~

1) (q--‘l,f’ (q--‘/&d?.

Thus x’(q) > 0, i.e. the expression for ut (q) in (2.1) is meaningful. We will now estimate the error of scheme (2.2) when (I > 1. Theorem 3. Let a(t), C(I), f(t)&T[O, i], in (1.1); then, 1UPu (21)Iefh

i-0,

f,.

. . , N&N,,

where (u,), i = 0, 1, . . . , N, is the solution of (2.2) with the x(q) given by (3.5), and u(x) is the solution of (1.1). Proof: From (1.3) and (3.5). WCeasily obtain, for Iu1 (q)l , 0 I q I 1, (3.7a) f3.7b) Next, we estimate o,(q) Let q % X; then, using (3.5), we arrive at

138

(II

Since T -

(q) =

I

a[s (q)] - (a +. d - 1) T + (I - “&

2H)d’(cl)) 2’ (q).

1 as c - 0, while O(0) = a > 1, it follows for sufficiently small m > 0 from the expression for n, WW, o X. In this case,

ac (9) an&(q),

(q) that

Ie+r(q) lz”(q)=-2a~a,+2(ala, +3&k) (q-‘/,)+z((r,‘+3a,k)(q-‘/*)* +8ka~(q-‘/,)*+6k*(q--‘ln)‘.

(3.8)

Here, a&I’(e*--e) + e, al=2Tpd(u-i)-te’(‘-*)‘“, 02 - 2Z’p*d(a-l)-‘(a +d-i)~‘+‘~“‘. Since 6 Cd< 9, for suffkiently smail m > 0 we have &z& X, that [s~z(q)]S~q)c~~+3cr,a*~q-‘~*~+302~q-’~*)*~ +8k~(q-1/,)‘+6k*(q-*/,)‘
Hence, for sufficiently small m > 0, we have a,~q~-(a~s~q~l-~a+z~q~lz”~q~rt’~~~l-’~~’~q~~mz’~~~, e
Consequently, there exists m > 0 such that e
a,(q)*mz’(q),

O
(3.9)

We now put r-{rl=~t--U1(q+)f, i-0, i, . . . , Nd?hf~, &=L~(r), i-i, 2,. . . , N-i, where fuJ is the solution of (2.2), u, (q) is the solution of (2.1) By (3.9), we have a,(qt)>O, OCq~C%+m. Further, obviously, ia, I/(e +t,)tM, rn+‘/,
ma ‘R’I+ [ Pi%+‘/, Ql (a)

maa lRIl r*>m+Y.(a + 2‘) 1 ’

e(m.

(3.10)

From (2.2) and (3.7). we have

Substituting these expressions into (3.10), we obtain, with t 5 m, m > 0, Ird+-i&n)

ICMk.

(3.11)

Since, for (II m,

then (3.11) obviously also holds for t L m. The theorem is proved. Let a = 1. In this case we define x(4) by eCP--

fl,

O
I (q) =r

II + 2epT (9 - l/d I 2spF (q - ‘r*)* ( + ‘APT’ 14- w + ks (q - w. %
By arguments similar to those of sections 1 and 2, we can show that, for CT = 1, we have l~u(z3 ICIhta(l/s)h, where u(n) is the solution of (1.1) with a(z), c(a), f(r)ee’C[O, i], (IZ,} is the solution of (2.2) with the x(q) given by (3.12), X,-Z(U), r-0, 1,. . , , N, h-i/N.

4. Numerical experiment To confirm our results we solved numerically the boundary value problem L(u)-(tfz)u”f’l,tr’-_e+z)‘u--((e+s)’, rr(i)~(f+s)~, u(0) -a’&,

(3.12)

139

whose solution is

TABLE 1

I:

I

2.8435.10-3 5.3347~104

I

7.Owioi.?tB?.iO-*

I

~~:~~~~

f

i.O141*10-* I 2.5!305.10-’

1.0688.10-* I 2.7005.10-I

Expressions (3.1) and (2.2) become respectively p”(e+i)“‘-e”.,

r(g)-(e’WgF-et

(4.1)

L,(~,)-(e*‘+pq)~,“-2pu,‘-36pf(e”~pq)*~u~~-36p*(e”~pqJ~~. Row A Of Table

(4.2)

1 ShOwS the CITOIS of the solution of (2.2) with N = 80 and x(q) given by (4.1). For comparison, gives the errors of problem (2.2) for x(q) I q.

I. 2.

3.

row B

LOMOV, S. A., The powerboundwy&yerin problems with small parameter, Dokl. Akua! Nawk %!iR, 184, No. 3,516-519.1969. LISEIKIN, V. D., and YANENKO, N. N., Ou a uniformly convargant algorithm for the numerical wlution of an ordinary second-order diicrantial aquation with a small paramctcr in laading derivative, in: Nameri& Mefrkodcof the Me&a&$ ofu Coar~aoa~ Mcdiw,Vol. 12, No. 2, ITPM SO AN SBSR, Novoaibink. 198I. SAMARSKII, A. A., andANDREYEV, V. V., D#%tcncehfethodtfor EllipticEqnotiow~ (Raznoatnye metody dlya cllipticheskikh uravnenii), Nauka, Moscow, 1976.

l.ES.SR. Compn~ hfds.

Printedia Graat Britain

Ma& Pays.. VoI. 26,No. 6, pp. 139-147,1986

0041&53/86 SIO.~.~ 0 I988petsrmonJournals Ltd.

THE ACCURACY OF THE SOLUTION OF A DIFFERENCE BOUNDARY VALUE PROBLEM FOR A FOURTH-ORDER ELLIPTIC OPERATOR WITH MIXED BOUNDARY CONDITIONS I. P. GAVRILYUK,V. G. PRIKAZCHIKOV and A. N. KHIMICH An estimate of the accuracy of a conservative diierence scheme in an energy net metric is established, when the solution of the initial problem belongs to Sobolcv spaces. The accuracy of ditfertnce schemes for fourth-order elliptic differential operators when free-boundary type boundary conditions are imposed were investigated in [ 143 assuming that the solution and the coefficients belong to the spaces Cc*)(Q) and C”)(Q) respectively. A new approach, proposed in [S], when investigating the accuracy of difference schemes uses special averaging operators and enables one to obtain an estimate of the accuracy of the net method when there are requirements regarding the smoothness of the coeffients and the solution of the problem in terms of Soboiev spaces. Estimates of the accuracy of these schemes for certain problems with fourth-order elliptic operators were obtained in [6-Q In this paper we investigate the accuracy of a difference scheme from [4] assuming that the solution and the coeffkients of the initial problem belong to the spaces W,‘“’(Q) , k= [ 3, 4) and W8(I)(B) ,m~ (1, 23 respectively. 1. Initial problem

In the rectangle n- (z=(z,, rr), OCX,+%, a-i,

5% vychisl.Mar. mar. F&T.,26, 12,1821-1830, 1986.

2) we consider the problem