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

76 CHAPTER 4. IMPROVING ON EULER’S METHOD
Theorem 4.3. Let δ and δ ∗ bet two different perturbations of a method y n and
let the ỹ n and ỹ ∗ n bet the corresponding perturbed methods. Then the method y n is
zero-stable if and only if for any ɛ > 0 there exist constants S and h 0 such that for
all h ∈ (0, h 0 ] whenever
‖δ n − δ ∗ n‖ ≤ ɛ, 0 ≤ n ≤ N (4.68)
then
‖ỹ n − ỹ ∗ n‖ ≤ Sɛ, 0 ≤ n ≤ N (4.69)
Definition 4.8 (Consistency). A one-step method is said to be consistent if
φ(t, y(t), 0) = f(t, y(t)). A method is said to be consistent (or accurate) to
order p or O(h p ) if the leading term in the local truncation error has order p,
d n = O(h p ) (4.70)
Theorem 4.4 (Convergence = Consistency + Stability). A one-step method
converges if it is consistent of order p and 0-stable.
Proof. By zero-stability
|y n − y(t n )| ≤ K
[
]
|y 0 − y(0)| + ‖N y n − N y(t n )‖
(4.71)
= K‖N y n − d n ‖ (4.72)
= K∥ y n − y
∥
n−1
∥∥
− φ(t n , y n ) − d n (4.73)
h n
= K‖d n ‖ = O(h p ) (4.74)
where the last step follows by consistency (equation 4.70). Hence the method is
convergent of order p by equation 4.52.
Definition 4.9 (Local Error). The local error
I n = ỹ(t n ) − y n (4.75)
is the amount by which the numerical solution y n differs from the exact solution of
the Local IVP
ỹ ′ (t) = f(t, ỹ(t)) (4.76)
ỹ(t n−1 ) = y n−1 (4.77)
Theorem 4.5. In general for the IVP methods we will consider,
h|N ỹ(t n )| = |I n |(1 + O(h)) (4.78)
|d n | = |N ỹ(t n )| + O(h p+1 ) (4.79)
Ideally we want a method to be as accurate as possible, in the sense that the absolute
error is minimized. In general, reducing the step size of an O(h p ) method will
reduce the error by a factor of h p . The question naturally arises as to what is a sufficiently
small step size. For example, consider the test equation y ′ = y, y(0) = 1. We
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007