On the numerical solution of a singularly-perturbed equation with a turning point

U.S.S.R.

Printed

Comput.Maths.Math.Phys., in Great Britain

0041-5553/84 $lo.oo+o.oo 01986 Pergamon Press Ltd.

Vo1.24,No.6,pp.135-139,1351

ON THE NUMERICAL SOLUTION OF A SINGULARLY-PERTURBEDEQUATION WITH A TURNING POINT* V.D. LISEIKIN A special difference net is constructed which 1s compressed in the boundary layer region of a singularly-perturbed equation with a turning point. It is proved that the simplest monotonic scheme on the resulting net converges regularly with respect to a parameter wrth first-order accuracy. Suppose

it is necessary

to solve the boundary

value problem

~[U]=EZU”+ZU(Z)U’-C(Z)U=j(l). a(z)

C(Z)

>ao>O,

Oc,>o,

U(o)=U(i)==o,

(1)

I>m>E>O

numerically. Problems of this type arise in the solution of boundary layer equations, heat transfer equations, andinthe investigation of the flow of liquids, the rate of which changes sign. In the literature (/l/, for example),(l) is referredto as a~. equation with a turning point. The numerical solution of (1) has been considered in :l, 2/. In /l/, a difference scheme on a regular net was constrcuted for (11 with the czndltion If(0)ICMefl and theestimate IU(si)-U,l~Mh(l-tEb-‘),

Pi,

]lt(s,)-u,]<.Uh]ln e],

B=l,

paper, constants which are where ~=c(O)la(O), was proved. Here, and in the remainderofthls independent of E,i, and z will be denoted by ill, m, M,,m,, etc. In /2/, a numerical algorithm was constructed for problem (1) which made use of a special coordinate function z(q) and, when pZ=3, the estimate ]u(zi)-u,]
Let

-l
of the function

u(*l)=b*l. u(5)

c(r), /(z)d?[O,11,a(z)d'[O,l] in (1) ,

are required

tk.en, when

Me-~,

in order

k=l,Z,

to construct

the estimate

OG~rn,&,

(2)

moe
~(E-keXp[-_cF(Z,E)/2]+~=-L}.

I

cp(z,E)=E-*

SEaWE,O~acc(O)la(O),

O
0

holds. Proof. The estimate ]n"'(s)]<~~-~, k=O,1,2,3 is eas:iy proved Let us therefore prove (2) for z>mQe. It follows from !1) that

(see /2/, for example).

u'(z)=u'(O)exp[-cp(z,e)]+

(3)

* s-'exP[--cp(r, s) 1 s

0

Let us estimate L[cu+fl

and, having

the quantity

L[cu+f](r)

(cu+f)e+etdi, when

E) 14.

r>m,~:

(z)=E*C”(Z)a(2)+2&zc’(z)u’(z)+~ta(z)c’(r)u(~)+~~“(z)+sa(z)~(s):

made use of the

fact that

f(Z),C(Z)ECZIO,l] an?

]L[cu+fl(z)]
the barrier

IuIa) (2)]
rn,&
function

fi(z)=ilf,exp[-cp(z,&)]illllz*, O~~ra
136 m,&Gx
Then, when

Lfg*(cu+f)](x)=-M,[a(x)+xa’(x)+ c(x)]exp[--cp(x,~)]+M~(e'a(a-1)x"-*+ x"[aa(x)-c(x)]}*~[cu+f](x)~ Mx-M*x"[c(x)-au(x)]-M,xa'(x). Since aa(x)>m,

c(O)-aa(O)>O, constants m,and m,, which are rndependent of e,exist such that when m,eix
It is now clear that it is possible [g*(cu+f)

lX(m,e)>O,

(z)GMx--M2m2xa--Mixa’(

to choose

[g*(ca+f)

c(x)-

I(w)20

M, and #f, such that L[gzt(cu+f)

and

l(x)<0

when

m,eGxGm,.

The estimate Icu+fl

cxp [-cp(x,

(x)Gg(x)=M,

then follows from the maximum principle is obvious that

the estimate

obtained

for

Since

]cu+f](x)GM, OGxGi,

into

(31, we obtain

lu’~x~l~lu’~O~IeXP~--(P~~,~~l+~e-’exP~-cp~x,~~l~ r. =o* SexP[dE,e) d

ldE+Jfe-‘exP[-dx,

It

rn,E~X=zl.

&)]fZ’I), cu+f

m,eGx~m, L.

for the operator

]Cu+f](x)
~)]+M~xa,

(4)

e) I j {I+ mot

~aexp[cp(&,e)]}d~~M(e-‘+ze-a)exp[--cp(x,s)]+ ille-2 jr -1

exp [ w’(r-x2)

1

dgG%f(e-‘+xe-z)expI-g(x,s)]+

L dEsM((s-'+xe-')exp[-g,(x,.s)]+x=-I}.

By virtue of the fact that

XE-‘esp[--cp(X,E)/?]~:ll,

we

lSAf(e-‘exp[

122’(x)

obtain

*]+xa-‘)

Hence, thelemmais proved for k=l. (4). Let US now consider the case when k=?.. It follows u'f[cu+f]'. From this and from (4) we obtain that from

JuU(x)IGIu”(O)

from

(11 that

Iexp[-‘P(x,e)]+E-*exp[-cp(x,~)]X

s

~[~+~~‘(E)~lexp[cp(~,e)ldE4~f~-*erp[-~(x,~)l~

0

MC-Ze~p[-~(z,

E)] s {~-~+~~-I+;~-‘esp[cp(~,

M(e-L+xe-J+x’~-‘)~~p[-~(x, x

xe-‘+x*e-‘) and since

e) ]+

exp[ --cp (x,6)

E)/Z]GM,

(~e-‘+sZE-z)exp[-(p(I(

E) ])d:G

] +xa-‘},

then

The lemma is proved. The estimate

we note that exp

[

~]~e~p[-m~e-‘x(~-i)]

exp(-m,e-‘x)GAf

exp(--m,&-lx),

X

E*u'"+.rn(x)u'I=[-xa(r)]'

137 of E and 2, follows from the lemma. of (1) can be written in the form

where rn3is an arbitrary constant which is independent Hence the estimates of the derivatives of the solution

OGr
k=1,2,

me-?r
(5bJ

rn,EGdl. It follows estimated

from

(5) that, as z increases P-'exp (-m,e-‘x),

by the quantity

the quantity

Y-'.

may be specified

s(q)

The function

by linking

together

from

until

which

Oto

1, the derivative

it is equal

is defined

the two functions

of

u(z) is initially

*a-l , and subsequently

to

by

by the condition

z*(9) which

z,(9) and

are the inverse

However, to do this, it is necessary to know the of exp(-m,e-Is) and x' respectively. location at which the functions are joined, i.e. to solve the equation z'-'=e-'exp (-rn,e-‘x) which it is impossible to do exactly. In view of this let us define z(q) as a composition of the two functions Let us put Zz(Z,)=&"=. z,(9) and

z(a)-tzl(q)l”“,

(64

G(9) -

(6b) 0GqG’/2r

-e’ln(l-pq), - (a’/2) e” In e+pe”“-Q’z’ (q-l/,)

‘/,GqG 1,

+k,(q-I/,)‘,

where

i>mae=-0.

The function

z,(l)=l.

z,(O)=O,

z,(9) is smooth and

a’ - - e”ln e+ps Q(‘-a’r) (q-‘/%) +4( 2

Jf_eO”-““’ t,(9) is a non-negative

function

8’ In e-

1+ f

V,CqGi,

(q-‘/>)‘>4(P-‘I,)‘,

2

then

Since

and the expression

ep[-ln(l--pq) $ We note that, when

(*)==(

I”“-‘la(l-pq),

From

osqwr,

[5(q)]L-a[pe’(‘-a’*1+2k,(q-~/l)]/a,

'I,Cq~l

is meaningful.

[r1(4)1""

we have

(61,

(7)

‘/,GqGi.

,

pe”‘~-“/“+2k,(9--‘/,)=pe”‘~-““‘+~

1+ ;

e= In 8 _ $

e”‘*-““’

(q-‘/*)5

We therefore obtain from (7) that s’(q)>O, i.e. z(q) is a monotonic function. Let us specify the difference net for the solution of (1) in the form ri=.r(ih),i=O,l,..., N, h-l/N, N=2N,, where z(9) is specified by formula (6). Let us approximate (1) on the irregular net 2, using the following montonic scheme: uc-nt_l

(8)

h-i

i=l, 2 , Theorem

when

1.

Let

c(z),f(z)d*[O, Proof.

uo=uN=O,

hi=x,+,-xt

Il=(U<} be the solution of (8) and let 11, a(z)~?[O, 11. Then ]u(zc)--u,(
Let us put

JY[r]=R,, i=i, 2,..., (8) that

, N-l,

N-l.

u(z) be the solution

r={ri=u(z,)-a,), i-0, 1,. ., N, R,=L,“[u(z))-J(z,), r.,-r,=O,

Hence, to prove this theorem,

R=

and it follows

it suffices E*[

from the maximum

to show that

(hi)*u”‘(gc’)-(h,-,)“u”‘(5,(-,)1 3(h‘i-L,)

)R,]GMh.

i=l,2,. principle

From

, hdL”(C5,~) 2

,I’f-1.

of (1) we have

/4/ for scheme

: 8) , ‘de obrai:. r:hat

138

Mhie-‘,

W( It follows

from

then

+ (z,_,)~-‘1,

(9)

m,e=Zz,G 1.

(6) and (7) that h&(

Let idNl4, i&V/b.

OGZ‘im,E,

Mh,[ (E-‘+5,e-‘)exp(--m,e-‘z,_,)

O
[I-p(i+l)h]~(l+e”“*)/Z-ph>O

Oiz,dm,e,

Let N/4
Meh[l-p(i+i)h]-“a, Mh(z,+,)‘-“,

then

7!Z,&
and,

from

(6),

(10)

from

(9),

(9), and (lo),

we obtain

IR,lcMh,

0-c

we have

IR~I~Mh{(~+~,~-‘)[l-p(if~)h]-“~[l-p(i-~)h]~~’~+e~[~-p(ifl)h]-”~[-ln(l--pih+ph)]~~-~~~~}~ MhU

1 1 + In”“---[l-p(i+l)h]-““[l-p(i-i)h]ma’~S I-pih )

e”[l--p(i+l)h]-~~“}~Mh([l-p(i+l)h]-z~=[l-p(i--l)h]m~‘~+ e”“)~Mh([l-p(i+l)h]““r-a’“f(2ph)m~’~[~-p(i+l)h]-*‘a+~), &= [-ln( I-pih-tph) ]r’a-‘, According to the condition, N/4m3{ln[1.'(I'

where

e=“*) ])““~‘>~/cz,

~R,~~~h{l+h”“[l-p(i+l)h]-z’“),

whence

IR,lf.Nh[l+h”=(e

=r’i+ph)-*‘a]Gfh,

jRNIZ-,1dfh[ Now

let

NIZGi
l+h”“e-=I

1R,j
e&h?‘”


In this case we obtain

.b’/4
from

(6),

(9), and

(10) that

(e-‘+Zie-‘)E’+(z,-,)=-‘I~ hf.2

(ih+h) [~,(ih-h)]--l]“-=J’=]~

,~k{e’-~+~+[p~““-“Z’h+

(ih+h-‘/,)

k,(ih-h-‘/,)*I”-I”‘},

h]“-a”“[

e=+pe=~‘-a’zi (ih-h-I/,)

$

N/2
) RNil 1< Mh (.T~,~+J*-~ [(e-l + e3.rN,J (e@” + ph)m”s2 + (5N,1_1)=-1] < Mh [E@'~*)m~tN/1-2 + jl"'*'N/2 E-2,(Pe=(~-="' h + h2)a-@/ae-'=j, where S=m,(-In that s-2>0,

E)“am’. At a sufficiently large value of m,,which IS independent (a’/2)m,t,,,-220 and it follows from the preceeding estimate that

IR,j<.lfh,

of e,we have

N/2+26&

~R~,,l,~~:~~~[l+e”~‘(he=‘L~=‘Z’+hZ)~’~-a~’a]
&ah’ =. when i=N/2-1,

JR,l<.llh

remains to prove the estimate

N/2, N/2’l

and

e
we have that R,=L”(il(z)]-f(s,)=~*u”(~,‘)+~,a(~,)u’(~,’)~e’u” (I,) --~,a (z,) u’(z),

z,-,<‘5>‘GL+,. 2,9~,‘~3,+,. -^

and hence

1R,I G2e’ / ,_~=
max *I~‘-,+!

M[exp(-m,e-Lz~-,)+e2(zl-,)a-r+~re-’ Here estimate

(5) has been used.

Now, when

Iu’Lr) Ii

exp(-m,E-‘li)+(zi)“].

i=N/Z-1,

Ni’?,

.\'/l+l , we obtain In E],

jR,j<.V[ (~““‘+h)mJ’r+ea+(l-pih)“J~‘~~-~‘a-~a i=X/2, RN;.‘_l,

I, <

(114

N/2, j,,f {E(a’/z)

“‘.‘N/W

,_

$

+_ Em.‘N;2t

1

(llbl

i_

I(-1ne)"Qf P]"a), where

t,=ln”=-‘(I--pih+ph).

When m,,which

is independent

from (il) that m3(a’/2)t,>2/a and IRiI~M(e”‘*+h-&“ln~)dMh theorem is proved. Let us now consider the boundary value problem e’a”+sa(z)a’-c(z)u=f(z),

n(r) 2n,XL

c(5) 3c,>o,

-lm>e>O.

of &is when

sufficiently

large, we have

ed h2f",i=N/Z- 1. iv/Z. :V/Z<-1. T!.e

u(--I)=u(l)=O.

Il?.;)

In /5, 6/, a numerical the estimate

algorithm

on a regular

net was constructed

for this problem and

fl=min('/,, c(O)/a(O)}

lu(x*)-u,I
was proved. we shall solve (12) by the same method as was used for (1) on the irregular net y,= is-N ,..., -1, 0, I,..., which is defined by the mapping y(q) and y,=y(ih),h= N, N=2N,, y(ih), Let us put l/h'. y(q)= where

--i
z(q) is the mapping defined in (6). Let us approximate (12) using the monotonic L,h[U]P-

2e’ h*+h‘-,

(z‘-ll‘l)a(+C)i=-N+l,. where

OGq=Gl,

m(4) { -d-q),

(

uc+1-uc -h‘

7)

I 1

scheme (13)

+(z,+Iz,I)a(zJ”*+ 1

urut-1

w-i - C(4u‘=f(.Zi),

. . , -i,O,l,...,

N-l,

u-~=uN=o,

ht==yi+,-y,

Then !u(x,)--ii,/G.Mh. Theorem 2. In (121, let c(.r),f(s)EC’[-1, 11, a(z)EC’[--I, I]. U(Z) is the solution of (12) and U; is the solution of (13). The proof of Theorem 2 is analogous to the proof of Theorem 1 and is therefore omitted. we note that the following estimates for the derivatives of the solution of (12) were obtained in /l/:

where

1

c(o)=

E

a (0)

Id"'(Z) Il, yields a convergence with first-order accuracy y,=ih

1,

the difference scheme (13) on the net which is regular with respect to P.

REFERENCES 1. BERGER A.E., HAN H. and KELLOG R.B., On the behaviour of the error in a numerical solution and Asymptotic of a turning point problem, in: Boundary and Interior Layers-Comput. Methods. Math. Proc. I. Conf. Dublin, p.5-20, 1982. 2. LISEIKIN V.D., On the numerical solution of an ordinary second-order differential equatlon with a small parameter accompanying the highest deterivative, in: Numerical Methods of the Mechanics of a Continuous Medium (Chisl. metody mekhan. sploshnoi sredy), Na;ka, Novosibirsk, 13, 3, 71-80, 1982. 3. BAKHVALOV N.S., On the optimization of methods for solving boundary value problems with a boundary layer, Zh. vychisl. Mat. mat. Fiz., 9, 4, 841-859, 1969. 4. SAMARSKII A.A. and ANDREEV V.V., Difference Methods for Elliptic Equations (Raznosznye metody dlya ellipticheskikh uravnenii), Nauka, Moscow, 1976. eu"+ra(r)u'-~(r)u=f(z),in: Difference 5. EMEL'YANOV K.V., A difference scheme for the equation Methods for Solving Boundary Value Problems with a Small Parameter and Discont;r<_sus Boundary Conditions (Raznostnye metody resheniya kraevykh zadach s malym parametrom i razryvnymi kraevymi usloviyami), Inst. Mat. i Mekhan., Sverdlovsk, 5-18, 1976. 6. FARREL P.A., A uniformly convergent difference scheme for the turning point problt:), in: Boundary and Interior Layers-Comput. and Asymptotic Methods. Math. Proc. I. Conf. Dublin, 270-274, 1980. Transiated

b:; E.L.S.