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

CHAPTER 6. LINEAR MULTISTEP METHODS 121
Theorem 6.2 (First Root Condition). The linear multistep method 6.1 is stable
if all the roots r i of the characteristic polynomial ρ(r) satisfy
and if |r i | = 1 then r i must be a simple root.
|r i | ≤ 1 (6.33)
Any roots that violate the root condition are called extraneous roots. One of
the objectives of designing a good method is to eliminate the extraneous roots.
A method is said to be strongly stable if all of the roots are within the unit
circle, except possibly for a root at r = 1.
A method is said to be weakly stable if it is stable but not strongly stable.
The region of absolute stability is given by that part of the Complex plane
bounded by
z = ρ(eiθ )
σ(e iθ (6.34)
)
because |e iθ | = 1.
Theorem 6.3 (Second Root Condition). The linear multistep method 6.1 is
absolutely stable if all the roots r i of the characteristic polynomial φ(r) = ρ(r)−zσ(r)
satisfy
|r i | ≤ 1 (6.35)
Theorem 6.4. An explicit linear multistep method can not be A-stable.
Theorem 6.5 (First Dahlquist Barrier). The order p of a stable linear k-step
multi-step method satisfies
p ≤ k + 2, if k is even; (6.36)
p ≤ k + 1, if k is odd; (6.37)
p ≤ k, if b k /a k ≤ 0 or the method is explicit. (6.38)
Theorem 6.6 (Second Dahlquist Barrier). An A-stable linear-multistep method
must be of order less than or equal to 2.
6.3 Backward Difference Formula
The BDF formula is obtained by seeking a polynomial of the form
P (t) = a 0 +a 1 (t − t n ) (6.39)
+a 2 (t − t n )(t − t n−1 )
+a 3 (t − t n )(t − t n−1 )(t − t n−2 )
.
+a n (t − t n )(t − t n−1 ) · · · (t − t 1 )
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge