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

126 CHAPTER 6. LINEAR MULTISTEP METHODS
formula is evaluated, the limits of integration are [−1, 0] rather than [0, 1]. Hence
∫ 0
y n = y n−1 + h
= y n−1 + h
−1
(
f n + s∇f n + 1 )
2 (s2 + 2)∇ 2 f ds (6.94)
[
sf n + 1 2 s2 ∇f n + 1 2
= y n−1 + h
[f n − 1 2 ∇f n − 1 ]
12 ∇2 f n
[
= y n−1 + h
= y n−1 + h
( 1
3 s3 + 1 2 s2 )
∇ 2 f n
] 0
−1
f n − 1 2 (f n − f n−1 ) − 1
]
12 (f n − 2f n−1 + f n−2
( 5
12 f n + 2 3 f n−1 − 1 )
12 f n−2
(6.95)
(6.96)
(6.97)
(6.98)
None of the Adams’s methods are A-stable except for AM1 (the Forward Euler
Method) and AM2 (Trapezoidal Rule). The outer boundaries of the regions of
stability are plotted for several of these methods in figure ??, which are also listed
in tables 6.1 and 6.2.
Table 6.1: Explicit Adams-Bashforth Methods.
Method
AB1
AB2
AB3
AB4
AB5
AB6
Formula for y n − y n−1
hf n−1 (Euler’s Method)
h
2 (3f n−1 − f n−2 )
h
12 (23f n−1 − 16f n−2 + 5f n−3 )
h
24 (55f n−1 − 59f n−2 + 37f n−3 − 9f n−4 )
h
720 (1901f n−1 − 2774f n−2 + 2616f n−3 − 1274f n−4 + 251f n−5 )
h
1440 (4277f n−1 − 7923f n−2 + 9982f n−3 − 7298f n−4 + 2877f n−5 − 475f n−6 )
6.5 BDF Methods
An alternative multistep method use backward differentiation. In these methods,
the polynomial approximation is applied to y ′ rather than f. If sh + t n = t then
y ′ (t n ) = d dt
ν∑
( ) ∣ ∣∣∣∣t=tn
(−1) k −s
∇ k y
k n (6.99)
k=0
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007