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

118 CHAPTER 6. LINEAR MULTISTEP METHODS
corresponding to a method with coefficients {a 0 , a 1 , . . . , b 0 , . . . } as
Lu(t) =
k∑
[a j u n−j − hb j u ′ n−j] (6.4)
j=0
for any arbitrary continuously differentiable function u with discretiztion u n . If y(t)
is the exact solution of the differential equation y ′ = f then
k∑
Ly(t) = [a j y n−j − hb j f n−j ] (6.5)
j=0
Expanding the terms y n−j and f n−j right hand side in a Taylor series,
y n−j = y n − jhy ′ n + (jh)2
2 y′′
n + · · · + (−jh)k
k!
f n−j = y n ′ − jhy n ′′ + (jh)2
2 y′′′ n + · · · + (−jh)k
(
k!
a j y n−j − hb j f n−j = a 0 y n + (ja 1 − b 0 )hy n ′ a2
)
+
2 j2 − jb 1
Hence we have
where
Ly(t) =
C 0 =
+ (−1) k (
aj j k
k∑
j=0 i=0
k!
− b jj k−1
(k − 1)!
∞∑
(
(−1) i aj j i
i!
y (k)
n + · · · (6.6)
y (k+1)
n + · · · (6.7)
h 2 y ′′
n + · · · (6.8)
)
h k y (k)
n + · · · (6.9)
(6.10)
− b jj i−1 )
h i y n (i)
(6.11)
(i − 1)!
= C 0 y n + C 1 hy ′ n + C 2 h 2 y ′′
n + · · · (6.12)
k∑
a j (6.13)
j=0
C i = (−1) i ⎡
⎣ 1 i!
k∑
j i 1
a j +
(i − 1)!
j=1
Hence the order of the method is p if
k∑
j=0
j (i−1) b j
⎤
⎦ (6.14)
C 0 = C 1 = · · · = c p = 0, c p+1 ≠ 0 (6.15)
and therefore to obtain a method of order p, we need to set successive values of C
to zero:
0 = a 0 + a 1 + · · · + a k (6.16)
0 = (a 1 + 2a 2 + · · · + ka k ) + (b 0 + b 1 + · · · + b k ) (6.17)
0 = 1 2 (a 1 + 4a 2 + · · · + k 2 a k ) + (b 1 + 2b 2 + · · · + kb k ) (6.18)
. (6.19)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007