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

CHAPTER 6. LINEAR MULTISTEP METHODS 129
Table 6.3: The first several Backward Differentiation Formulae.
Method
BDF1
BDF2
BDF3
BDF4
BDF5
BDF6
Formula for the method
hf n = y n − y n−1 (Backward Euler)
2hf n = 3y n − 4y n−1 + y n−2
6hf n = 11y n − 18y n−1 + 9y n−2 − 2y n−3
12hf n = 25y n − 48y n−1 + 36y n−2 − 16y n−3 + 3y n−4
60hf n = 137y n − 300y n−1 + 300y n−2 − 200y n−3 + 75y n−4 − 12y n−5
60hf n = 147y n − 360y n−1 + 450y n−2 − 400y n−3 + 225y n−4 − 72y n−5 + 10y n−6
where
k j = (−1) j ∫ 1
−1
( ) −s
ds (6.113)
j
To obtain the Milne-Simpson method our interpolation includes f n
ν∑
y n = y n−2 + h k j ∇ j f n (6.114)
j=0
where
∫ 1
( )
k j = (−1) j −s + 1
ds (6.115)
j
−1
Table 6.4: Coefficients k j for the Nyström Methods.
j 0 1 2 3 4 5 6 7 8
k j 2 0 1/3 1/3 29/90 14/45 1139/3780 41/140 32377/113400
Table 6.5: Coefficients k j for the Milne-Simpson Methods.
j 0 1 2 3 4 5 6 7 8
k j 2 -2 1/3 0 -1/90 -1/90 -37/3780 -8/945 -119/16200
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge