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

CHAPTER 6. LINEAR MULTISTEP METHODS 127
Method
AM1
AM2
AM3
AM4
AM5
AM6
Table 6.2: Implicit Adams-Moulton Methods.
Formula for y n − y n−1
hf n (Backward Euler’s Method)
h
2 (f n + f n−1 ) (Trapezoidal Rule)
h
12 (5f n + 8f n−1 − f n−2 )
h
24 (9f n + 19f n−1 − 5f n−2 + f n−3 )
h
720 (251f n + 646f n−1 − 264f n−2 + 106f n−3 − 19f n−4 )
h
1440 (475f n + 1427f n−1 − 798f n−2 + 482f n−3 − 173f n−4 + 27f n−5 )
Since s = 0 corresponds to t = t n ,
d
dt
Hence
( )∣
−s ∣∣∣t=tn
= 1 k h
d
ds
= 1
hk!
( )∣
−s ∣∣∣s=0
⇐= (6.100)
k
d
ds [(−s)(−s − 1)(−s − 2) · · · (−s − k + 1)] ∣
∣∣∣s=0
(6.101)
k (k − 1)!
= (−1) = (−1) k 1
hk!
hk
y ′ (t n ) =
ν∑
k=0
(6.102)
1
hk ∇k y n (6.103)
Substituting this in the equation y ′ = f gives us the BDF formula or Gear’s
Method is given by
ν∑ 1
k ∇k y n = hf(t n , y n ) (6.104)
k=0
Thus there are constants a 0 , a 1 , a 2 , . . . , a ν and b 0 , with a 0 = 1, such that
ν∑
a k y ν−k = hb 0 f(t n , y n ) (6.105)
k=0
The first several BDF methods using this formula are shown in Table 6.3, BDF
methods are more stable than Adam’s-Moulton methods and have the advantage of
being explicit like Adams-Bashforth. Unfortunately they are unstable for all values
of ν ≥ 7.
Example 6.4. Derive the BDF formula for ν = 2.
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge