Unformatted text preview: nce methods are illustrated in Figures 2, 3, and 4, respectively 1]. Some
observations and conclusions follow.
1. The AdamsMoulton methods of orders 1 and 2 are the backward Euler method
and the trapezoidal rule. The absolute stability regions of these methods are the
exterior of a unit circle centered at (1 0) and the entire lefthalf of the h plane,
respectively. Neither region is shown in Figure 5.6.2. Ascher and Petzold's 1]
de nition of k di ers from ours for AdamsMoulton methods. The regions labeled
k = 2 to 4 in Figure 5.6.3 correspond to our k = 3 to 5.
2. Explicit methods always have a nite region of absolute stability.
36 Figure 5.6.2: Absolute stability regions for AdamsBashforth methods of orders 1 to 4.
A method is stable inside of its shaded region 1]. Figure 5.6.3: Absolute stability regions for AdamsMoulton methods of orders 3 to 5. A
method is stable inside of its shaded region 1].
37 Figure 5.6.4: Absolute stability regions for backward di erence formulas of orders 1 to 3
(top) and 4 to 6 (bottom). Methods are stable outside of the shaded regions 1].
38 3. Implicit methods have a larger region of absolute stability than a corresponding
explicit method of the same order.
4. The absolute stability region typically becomes smaller as order increases.
5. The backward di erence formulas shown in Figure 5.6.4 have unbounded absolute
stability regions as Re(h ) ! ;1.
6. We have noted that the error coe cients of AdamsMoulton methods (5.4.5  5.4.8)
are smaller than those of AdamsBashforth methods (5.3.10  5.3.13) having the
same order. Figures 5.6.2 and 5.6.3 indicate that the absolute stability regions of the
AdamsMoulton methods are larger than those of the AdamsBashforth methods
of the same order. Therefore, the AdamsMoulton methods can be used with much
larger step sizes than required for the AdamsBashforth methods having the same
accuracy. Typically, this increase in step size o sets the cost of solving an implicit
di erence equation.
The ma...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, The Land, yn, Tn, Numerical ordinary differential equations

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