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

CHAPTER 4. IMPROVING ON EULER’S METHOD 75
Theorem 4.2. Euler’s method is zero-stable.
Proof.
‖N u n − N u ∗ n‖ = ∥ u n − u ∗ n
− u n−1 − u ∗ n−1
− ( f(u n−1 ) − f(u ∗
h
h
n−1) )∥ ∥ (4.55)
≥ ∥ u n − u ∗ n
∥ − ∥ u n−1 − u ∗ n−1
+ ( f(u n−1 ) − f(u ∗
h
h
n−1) )∥ ∥ (4.56)
By the triangle inequality and the Lipshitz condition
∥ u n − u ∗ n
∥ ≤ ‖N u n − N u ∗ h
n‖ + ∥ u n−1 − u ∗ n−1
∥ + K ∥ ∥u n−1 − u ∗ ∥
n−1 (4.57)
∥
h
∥u n − u ∗ ∥
n ≤ h‖N u n − N u ∗ n‖ + ∥ ∥u n−1 − u ∗ ∥
n−1 + hK ∥ ∥u n−1 − u ∗ ∥
n−1 (4.58)
= h‖N u n − N u ∗ n‖ + (1 + hK) ∥ ∥u n−1 − u ∗ ∥
n−1
(4.59)
Apply the result recursively n − 1 additional times to give
∥
∥u n − u ∗ ∥
n ≤ h‖N u n − N u ∗ n‖ + (1 + hK) ∥ ∥u n−1 − u ∗ ∥
n−1
(4.60)
≤ h‖N u n − N u ∗ n‖+ (4.61)
(1 + hk) [ h‖N u n − N u ∗ n‖ + (1 + hk) ∥ ∥u n−2 − u ∗ ∥ ] n−2 (4.62)
≤ .
≤ h‖N u n − N u ∗ n‖
n∑
(1 + hK) n−i + (1 + hk) n∥ ∥ u0 − u ∗ ∥
0 (4.63)
i=1
∥∥
≤ k{
u 0 − u ∗ }
∥
0 + ‖N u n − N u ∗ n‖
(4.64)
where
{
k = max h
n∑
(1 + hK) n−i , (1 + hk) n} (4.65)
i=1
Therefore the method is 0-stable.
A zero-stable method depends continuously on the initial data. If a method is
not zero stable that a small perturbation in the method could potentially lead to an
infinite change in the results. Suppose that
y n = y n + 1 + hφ(t n − 1, y n−1 , ...) (4.66)
is a numerical method. Then we define a perturbation δ of the method as
ỹ n = y n + 1 + hφ(t n − 1, y n−1 , ...) + δ n (4.67)
The following theorem is normally taken as the definition of zero-stability.
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge