Unformatted text preview: amine this possibility
using the simple example (cf. Burden and Faires 6], Chapter 5) y0 = y ; t2 + 1
which has the exact solution y(0) = 1
2 0<t 2 y(t) = (t + 1)2 ; e :
2
t Burden and Faires 6] calculate solutions using the fourth order AdamsBashforth
(5.3.13a) and AdamsMoulton (5.4.8a) formulas. Since this problem is linear, the AdamsMoulton solution may be obtained without iteration. Results with h = 0:2 are shown in
22 Table 5.4.3. The four starting values (y0, y1, y2, and y3) that are needed for (5.3.13a)
and the three (y0, y1, and y2) that are needed for (5.4.8) are obtained from the exact
solution. The global errors found when using the classical fourthorder RungeKutta
method (3.2.4) with the same step size are also shown in Table 5.4.3. tn y(tn) Adams Adams RungeBashforth Moulton
Kutta
Error
Error
Error
0.0 0.5000000 0.0000000 0.0000000 0.0000000
0.2 0.8292986 0.0000000 0.0000000 0.0000053
0.4 1.2140877 0.0000000 0.0000000 0.0000114
0.6 1.6489406 0.0000000 0.0000065 0.0000186
0.8 2.1272207 0.0000828 0.0000160 0.0000269
1.0 2.6408227 0.0002219 0.0000293 0.0000364
1.2 3.1798942 0.0004065 0.0000478 0.0000474
1.4 3.7323401 0.0006601 0.0000731 0.0000599
1.6 4.2834095 0.0010093 0.0001071 0.0000743
1.8 4.8150857 0.0014812 0.0001527 0.0000906
2.0 5.3053630 0.0021119 0.0002132 0.0001089
Table 5.4.3: Global errors for Example 5.4.1 obtained by fourthorder AdamsBashforth,
AdamsMoulton, and classical RungeKutta methods ( 6], Chapter 5).
Errors with the AdamsMoulton method are about ten times smaller than those with
the AdamsBashforth method. The ratio of their error coe cients is 251=19 13 (cf.
(5.3.13b) and (5.4.8b)). Thus, the results are close to their expected values. Accuracy of
the RungeKutta method is almost twice that of the AdamsMoulton method however,
the RungeKutta method requires approximately four times the work of the two Adams
methods. 5.5 Implicit Methods: Backwarddi erence Methods
While we haven't analyzed stability of multistep methods, our experience...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, The Land, yn, Tn, Numerical ordinary differential equations

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