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

124 CHAPTER 6. LINEAR MULTISTEP METHODS
6.4 Adams Methods
Definition 6.3. An Adams Method is a linear multistep method with a 0 =
1, a 1 = −1, and a k = 0 for all k > 1. The explicit Adams-Bashforth Methods
are given by
y n = y n−1 + h(b 1 f n−1 + b 2 f n−2 + · · · + b k f n−k ) (6.72)
while the implicit Adams-Moulton Methods
y n = y n−1 + h(b 0 f n + b 2 f n−1 + · · · + b k f n−k ) (6.73)
Adams methods are derived by integrating the differential equation over two grid
points
∫ tn
y n = y n−1 + f(s, y(s))ds (6.74)
t n−1
and approximating the integrand with an interpolating polynomial over the past several
mesh points. The interpolating polynomial of choice is the Newton backward
difference formula (BDF),
where
P k (t n ) = f(t n ) +
( s
=
k)
k∑
( )
(−1) j −s
∇ j f(t
j
n ) (6.75)
j=1
s(s − 1) · · · (s − k + 1)
k!
(6.76)
t = t n + sh (6.77)
∇p n = p n − pn − 1 (6.78)
)
∇ k p n = ∇
(∇ k−1 p n , k ≥ 2 (6.79)
The resulting Adams method coefficients are then given by
∑k−1
( ) ∫
b j = (−1) j−1 i
1
( )
(−1) i −s
ds (6.80)
j − 1
i
i=j−1
Example 6.2. Derive the 3-stage Adams-Bashforth method.
Solution. The method will be given by
y n = y n−1 +
∫ tn
0
t n−1
f(u, y(u))du (6.81)
where f is estimated using
( )
( )
−s
−s
f ≈ f n−1 + (−1) ∇f
1 n−1 + (−1) 2 ∇
2
2 f n−1
( ) (6.82)
−s
+(−1) 3 ∇
3
3 f n−1
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007