Higher order schemes and Richardson extrapolation ...

BULL. AUSTRAL. MATH. SOC.

VOL. 39 (1989)

65BO5, 6 5 L 1 0 , 65L50

[129-139]

HIGHER ORDER SCHEMES AND RICHARDSON EXTRAPOLATION FOR SINGULAR PERTURBATION PROBLEMS DRAGOSLAV HERCEG, RELJA VULANOVIC AND NENAD PETROVIC

Seniilinear singular perturbation problems are solved numerically by using finite—difference schemes on non-equidistant meshes which are dense in the layers. The fourth order uniform accuracy of the Hermitian approximation is improved by the Richardson extrapolation. 1. INTRODUCTION

We consider the following singularly perturbed boundary value problem: (1)

-e V

+ c{x,u) = 0, x e I = [0,1], w(0) = u(l) = 0,

with a small parameter e, e G (O,£o)- Our assumptions are cGC8(/xR),

(2.1) (2.2)

g{x) < cu(x,u)

< G{x), ( x , « ) e l x R ,

(2.3)

S := nnn{5g{x) - 2G(x): x 0,

(2.4)

0 < 7 2 < g{x), \g'(x)\ < L, |G'(x)| < L, x £ I.

It is well-known that under the given conditions there exists a unique solution, it € C'10(7 x R) , to the problem (1), and that the following representation holds: (3.1)

u{x) = vo{x) + v1(x) + y{x),

where (3.2)

»o(a0 = exp(-7z/e),

(3.3)

v1{x) =

exp{i(x-l)/c),

and (3.4)

M, s = 0, 1, . . . , 8, x € J,

Received 14 April 1988 This research was partly supported by NSF and SIZ for Science of SAP Vojvodina through funds made available to the U.S.-Yugoslav Joint Board on Scientific and Technological Cooperation (grant JF 799). Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/89

129

SA2.00+0.00.

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D. Herceg, R. Vulanovic and N. Petrovic

[2]

(see [6, 7]). Here and throughout the paper M denotes any positive constant independent of e. As well as in [6] and [7], problem (1) was solved numerically in [1, 5, 3, 4] - just to mention some of the papers. For other references see [3, 4]. Our aim is to solve (1) numerically by using classical finite difference schemes on special non-equidistant meshes which are dense in the layers of u, located at x — 0 and x = 1. The same approach can be found in the papers we have mentioned. In this paper we combine the methods from [7] and [4] to obtain high order convergence uniform in e. In [7] Richardson extrapolation was applied to the central difference scheme and high accuracy uniform in e was proved. In [4] (see [3] as well) the Hermite scheme was used and fourth order uniform convergence was proved. Here we shall apply Richardson extrapolation to the Hermite scheme. We shall give a proof of sixth order convergence uniform in e . We believe that a general theory can be developed in the same way as in [7] and that even higher order uniform convergence can be obtained. Numerical experiments confirm this. Conditions (2.2) and (2.3) are the same as in [3, 4] and they guarantee stability uniform in e. In a forthcoming paper we shall avoid these constraints on the function c. In Section 2 the discretisation is given and stability uniform in e is proved. In Section 3 we give a representation of the consistency error, which justifies the use of Richardson extrapolation. We end the paper by giving some numerical results in Section 4. The constants M will be independent of the discretisation mesh as well. 2.

DISCRETISATION

Let If, be the discretisation mesh with the points: (4.1)

Xi = \(ti),

ti=ih,i

= 0,l,...,n,h=-,n-

2m, m e N, n

(4.2)

( « ( < ) = £!*, \{t) = I n(t), {i _ m _ t),

*€[,«], t 6 [a, 0.5], te [0.5, l],

where

(4.3) w(t) = A(t - a) 4 + w"'(o)(< - a) 3 /6 + w"(a)(< - af/2 + u'(a){t - a) + u(a). The parameter a is (4.4)

a = t k,

[3]

Semilinear singular perturbation problems

131

for some k 6 {1, 2, . . . , m — 1} , (4.5)

q = a + $1.

and the coefficient A is determined from (4.6)

TT(0.5) = 0.5.

Moreover, the coefficient a should satisfy ( B+ 2£-1/49(0.5 - a ) ^

(4.7)

^ a ^ B'1,

where (4.8)

B = 2 ( e 3 / 4 a + c^'^O.S - a ) + e 1 /« 9 (0.5 - a ) 2 + g(0.5 - a ) 3 ) .

We have A: I -> J, A £ C ^ I ) , A € C 3 [0, 0.5], A G C°°[0, a], A € C^a, second inequality in (4.7) implies A ^ 0, so ir'" is nondecreasing, and ir((a) = w(f)_(a) > 0, a = 3,2,1, « £ [a, 0.5]. At the same time w(')(t)>OJa = l , 2 ) . . . >

te[O,q),

and taking (4.5) into account we get 0 < A(J)( (see (5.1)). Now if k + 6 < i(< m) we have

Let = S V ( i ) - ff'(< + 2/0, (< = ti_i). It follows that p(5)( 0, s = 3, 2, 1,

a + 6ft, from (5.1), (5.2), (7), (3.1)-(3.4) and (16) we have

+ £~5 exp(- T A(a + 6h)/e)],

) D1 < Mh6[l + e~13/4 exp{ and

l + £-s exp (-7u;(a)/e)] ^ Mh6. If i