The Computable Differential Equation Lecture ... - Bruce E. Shapiro

128 CHAPTER 6. LINEAR MULTISTEP METHODS
Figure 6.1: Outer border of regions of stability for Adams Methods. Left: Adams-
Bashforth methods AB1 (forward Euler, solid black); AB2 (Thick solid red); AB3
(Thick dashed purple); and AB4 (thick, short blue dashed). Right: Adams-Moulton
Methods AM3 (thin, blue); AM4 (thin, long dashed purple); AM5 (short-dashed
green); AM6 (thick, solid, orange).
Solution. From equation 6.104
hf n =
2∑
k=1
1
k ∇k y k (6.106)
= ∇y n + 1 2 ∇2 y n (6.107)
= y n − y n−1 + 1 2 (y n − 2y n−1 + y n−2 ) (6.108)
= 3 2 y n − 2y n−1 + 1 2 y n−2 (6.109)
2hf n = 3y n − 4y n−1 + y n−2 (6.110)
6.6 Nyström and Milne Methods
These methods consider the integral
y(t n ) = y(t n−2 ) +
∫ tn
f(t, y(t))dt
t n−2
(6.111)
Substitution of Newton’s backward difference formula yields Nyström’s Method
∑ν−1
y n = y n−2 + h k j ∇ j f n−1 (6.112)
j=0
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007