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67a satisfy j h j 1 when applied to the test problem

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Unformatted text preview: nce methods are illustrated in Figures 2, 3, and 4, respectively 1]. Some observations and conclusions follow. 1. The Adams-Moulton 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 left-half 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 Adams-Moulton 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 Adams-Bashforth methods of orders 1 to 4. A method is stable inside of its shaded region 1]. Figure 5.6.3: Absolute stability regions for Adams-Moulton 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 Adams-Moulton methods (5.4.5 - 5.4.8) are smaller than those of Adams-Bashforth 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 Adams-Moulton methods are larger than those of the Adams-Bashforth methods of the same order. Therefore, the Adams-Moulton methods can be used with much larger step sizes than required for the Adams-Bashforth 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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