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

132 CHAPTER 6. LINEAR MULTISTEP METHODS
Figure 6.3: Borders of stability region for Milne-Simpson MS3 and MS3 methods.
The stable region is outside the indicated boundary.
where the superscripts denote the order ofmethod and are not exponents are derivatives,
γ is a free parameter, and hf y is a weighting factor (for systems, we replace f y
with the Jacobian matrix). Weighting factors are chosen based on the observation
that for nonstiff problems, −hf y is small, while for stiff problems, −hf y is large.
Here the “Adams Method Expression” is
and the BDF expression is
−y n+1 + y n + h(b ν f n+1 + b ν−1 f n + · · · + b 0 f n−ν+1 ) = 0 (6.126)
− (a ν y n+1 + a ν−1 y n + · · · + a o y n−k+1 ) + hf n+1 = 0 (6.127)
For example, if we chose ν = 2, this method gives
y n+1 = y n + h
( 5
12 f n+1 + 8 12 f n − 1 12 f n−1
− γ (2) hf y
(
− 3 2 y n+1 + 2y n − 1 2 y n−1 + hf n+1
)
)
(6.128)
(6.129)
For autonomous linear equations, we have f = y ′ = f y y and hence
• if γ (2) = 1/6, cancellation of f n−1 and f y y n−1 terms leads to the ν − 1-step
Enright method;
• if γ ( 2) = 1/8, we obtain the ν-step Enright method.
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007