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

130 CHAPTER 6. LINEAR MULTISTEP METHODS
Figure 6.2: Boundaries of stability regions for BDF methods BDF1 (inside curve)
through BDF6 (outside). The stability region is outside the bounded area shown.
6.7 Other Types of Multistep Methods
Enright’s Second Derivative Method is based on the observation that a generalization
of the method
gives
y n = y n−1 + hf n (6.116)
y n = y n−1 + hy ′ n + 1 2 h2 y ′′
n (6.117)
= y n−1 + hf n + 1 2 h2 y ′′
n (6.118)
by using an additional term in the Taylor expansion. To get an expression for y ′′
n
we observe that if
y ′ = f(t, y) (6.119)
then
y ′′ = f t + f y f (6.120)
Hence
y n = y n−1 + hf n + 1 2 h2 g n (6.121)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007