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

CHAPTER 6. LINEAR MULTISTEP METHODS 131
Table 6.6: Milne-Simpson Methods
Method Formula for y n
MS0 y n = y n−2 + 2hf n (Backward Euler Method, step size 2h)
MS1 y n = y n−2 + 2hf n−1 (Midpoint Rule)
MS2 3y n = y n−2 + h (f n + 4f n−1 + f n−2 ) (Milne’s Method)
MS3 90y n = y n−2 + h(29f n + 124f n−1 + 24f n−2 + 4f n−3 − f n−4 )
where g = f t + f y f. If we use a weighted combination of values at multiple grid
points the result is Enright’s multistep second derivative method, which is
ν∑
a i y n+i = h
i=0
ν∑
b i f n+i + h 2
i=0
ν∑
c i g n+i (6.122)
It can be shown by using a Taylor series approximation that this method is of order
p if and only if
ν∑
a j (j) q = q
j=0
ν∑
b j (j) q−1 + q(q + 1)
j=0
i=0
ν∑
c j (j) q−2 , ∀j = 0, 1, 2, . . . , p (6.123)
j=0
The following simplifications are usually made:
• Set a ν = 1, a ν−1 = −1 and the rest of the a k = 0. This ensures stability in a
neighborhood of the origin.
• Set c i = 0 for all i < ν to ensure stability at infinity.
The simplified method is then
y n = y n−1 + h
ν∑
b k f n+j−ν + h 2 c ν g n (6.124)
j=0
Formulas for the first several methods are given in table 6.7. 1
Blended Multistep Methods 2 attempt to take advantage of the features of
both Adams methods, which are good at non-stiff problems, and BDF methods,
which are good for stiff problems. The methods are blended in the sense that
[Adams Method Expression] (ν+1) = γ (ν) hf y [BDF Expression] (k) (6.125)
1 See Enright, W.H., “Second Derivative Multistep Methods for Stiff Ordinary Differential Equations”,
SIAM Journal of Numerical Analysis, 11(2):321-331 (1974) for a table of values up through
ν = 7 (which is 9 th order).
2 See Skeel, R.D. and Kong, A.K., Blended linear multistep methods, ACM Transactions on
Mathematicaal Software, 3:326-343 (1977).
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge