42 coe cients of the adams moulton method 549 and

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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 Adams-Bashforth (5.3.13a) and Adams-Moulton (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 fourth-order Runge-Kutta 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 fourth-order Adams-Bashforth, Adams-Moulton, and classical Runge-Kutta methods ( 6], Chapter 5). Errors with the Adams-Moulton method are about ten times smaller than those with the Adams-Bashforth 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 Runge-Kutta method is almost twice that of the Adams-Moulton method however, the Runge-Kutta method requires approximately four times the work of the two Adams methods. 5.5 Implicit Methods: Backward-di erence Methods While we haven't analyzed stability of multistep methods, our experience...
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