Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

xy(estim.) y(exact)

− 0 .5 (1.648) —

0 (1.000) (1.000)

0 .5 0.648 0.607

1 .0 0.352 0.368

1 .5 0.296 0.223

2 .0 0.056 0.135

2 .5 0.240 0.082

3. 0 − 0. 184 0.050

Table 27.11 The solution of differential equation (27.61) using the Milne

central difference method withh=0.5 and accurate starting values.

more accurate, but of course still approximate, central difference. A more accurate

method based on central differences (Milne’s method) gives the recurrence relation

yi+1=yi− 1 +2h

(

dy

dx

)

i

(27.64)

in general and, in this particular case,

yi+1=yi− 1 − 2 hyi. (27.65)

An additional difficulty now arises, since two initial values ofyare needed.

The second must be estimated by other means (e.g. by using a Taylor series,

as discussed later), but for illustration purposes we will take the accurate value,

y(−h)=exph, as the value ofy− 1 .Ifhis taken as, say, 0.5 and (27.65) is applied

repeatedly, then the results shown in table 27.11 are obtained.

Although some improvement in the early values of the calculatedy(x)is

noticeable, as compared with the corresponding (h=0.5) column of table 27.10,

this scheme soon runs into difficulties, as is obvious from the last two rows of the

table.

Some part of this poor performance is not really attributable to the approxi-

mations made in estimatingdy/dxbut to the form of the equation itself and

hence of its solution.Anyrounding error occurring in the evaluation effectively

introduces intoysome contamination by the solution of

dy

dx

=+y.

This equation has the solutiony(x)=expxand so grows without limit; ultimately

it will dominate the sought-for solution and thus render the calculations totally

inaccurate.

We have only illustrated, rather than analysed, some of the difficulties associated

with simple finite-difference iteration schemes for first-order differential equations,