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

CHAPTER 6. LINEAR MULTISTEP METHODS 119
Theorem 6.1. A linear multistep method is consistent if it has order greater than
or equal to 1. Thus
0 =
0 =
k∑
a j (6.20)
j=0
k∑ k∑
ja j + b j (6.21)
j=1 j=0
In terms of the characteristic polynomial, the method is consistent if and only if
ρ(1) = 0 (6.22)
ρ ′ (1) = σ(1) (6.23)
The proof follows immediately from the definitions of the C i and the characteristic
polynomials.
6.2 The Root Condition for Linear Multistep Methods
If we apply the general k-step linear multistep method
k∑
k∑
a j y n−j = h b j f n−j (6.24)
j=0
j=0
to the test equation y ′ = λy, we obtain the difference equation
k∑
k∑
a j y n−j = z b j y n−j (6.25)
j=0
j=0
where z = hλ. Solutions to this difference equation include r k where r is any root
of the characteristic polynomial
φ(r) = ρ(r) − zσ(r) (6.26)
where ρ and σ are as defined in equations 6.2 and 6.3. Because stability relates to
what happens in the limit h → 0 (z = 0 in the above equation) it turns out that
only the roots of ρ matter.
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge