Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODSx y(estim.) y(exact)−0.5 (1.648) —0 (1.000) (1.000)0.5 0.648 0.6071.0 0.352 0.3681.5 0.296 0.2232.0 0.056 0.1352.5 0.240 0.0823.0 −0.184 0.050Table 27.11 The solution of differential equation (27.61) using the Milnecentral difference method with h =0.5 and accurate starting values.more accurate, but of course still approximate, central difference. A more accuratemethod based on central differences (Milne’s method) gives the recurrence relation( ) dyy i+1 = y i−1 +2h(27.64)dxiin general and, in this particular case,y i+1 = y i−1 − 2hy i . (27.65)An additional difficulty now arises, since two initial values of y are 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 of y −1 .Ifh is taken as, say, 0.5 and (27.65) is appliedrepeatedly, then the results shown in table 27.11 are obtained.Although some improvement in the early values of the calculated y(x) isnoticeable, 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 thetable.Some part of this poor performance is not really attributable to the approximationsmade in estimating dy/dx but to the form of the equation itself andhence of its solution. Any rounding error occurring in the evaluation effectivelyintroduces into y some contamination by the solution ofdydx =+y.This equation has the solution y(x) =expx and so grows without limit; ultimatelyit will dominate the sought-for solution and thus render the calculations totallyinaccurate.We have only illustrated, rather than analysed, some of the difficulties associatedwith simple finite-difference iteration schemes for first-order differential equations,1022