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

CHAPTER 6. LINEAR MULTISTEP METHODS 133
Table 6.7: Enright Second Derivative Multistep Methods. The methods have order
ν + 2.
Method (ν) Iteration Formula
1 y n = y n−1 + h ( 2
3 f n + 1 3 f )
n−1 −
1
6 h2 g n
2 y n = y n−1 + h ( 29
48 f n + 5
12 f n−1 − 1
48 f n−2
)
−
1
8 h2 g n
3 y n = y n−1 + h ( 307
540 f n + 19
40 f n−1 − 1
20 f n−2 + 7
1080 f n−3
)
−
19
480 h2 g n
4 y n = y n−1 + h ( 3133
4760 f n + 47
90 f n−1 − 41
480 f n−2 + 1
45 f n−3 − 17
5760 f n−4
)
−
3
32 h2 g n
6.8 Prediction-Correction (P(EC) n E) Techniques
At each iteration step in an implicit linear multistep method such as
a 0 y n + a 1 y n−1 + · · · + a k y n−k = h(b 0 f n + b 1 f n−1 + · · · + b k f n−k ) (6.130)
where y and f are m-vectors (for systems, or scalars for a scalar equation), and
f p = f(t p , y p ), we are faced with the problem of knowing what value of y n to use in
the term in f n in the right hand side of the equation. More specifically, we have
a 0 y n = hb 0 f(t n , y n ) − a 1 y n−1 − · · · − a k y n−k + h(+b 1 f n−1 + · · · + b k f n−k ) (6.131)
where y n appears on both sides of the equation.
When the system is not stiff, it is generally possible to find a step size h such
that
∥ ∥∥∥ ∂f
hb 0 ∂y ∥ ≤ r < 1 (6.132)
for some number r, in which case the method is a contraction (for scalar equations,
|hb 0 f y | ≤ r ), the fixed-point theorem applies, and one can use fixed point iteration
to find y n given a reasonably good starting guess. The general heuristic is
to use an explicit method of the same order for the first guess. This is called the
prediction (P) step. This first guess for y n is used to estimate (E-step) the value of
f(t, y n ); then the implicit method is used to calculate a corrected (C-step) estimate
of y n . This corrected estimate is substituted back into the method for one final
calculation (E-step). This type of method is often called a PEC, PECE, P(EC) n , or
P(EC) n E method, depending on the number of estimation - correction iterations.
These prediction-correction are essentially explicit methods even though they
use an implicit formula, because the method is based on an explicit evaluation of
the function at the previous step, rather than using a solver to determine the volume.
If the equation is stiff, then a PECE method is not sufficient, because it is extremely
expensive to find a value of h such that |hbf y | ≤ r < 1. Instead, the value
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge