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

74 CHAPTER 4. IMPROVING ON EULER’S METHOD
Definition 4.5. [ Local Truncation Error] Let u(t n ) be any function defined on
a mesh, and define the difference operator N by
N u(t n ) = u(t n) − u(t n−1 )
h n
− φ(t n−1 , u(t n−1 )) (4.45)
The Local Truncation Error d n is given by N y(t n ), where y(t n ) is the exact
solution evaluated on the mesh:
d n = N y(t n ) = y(t n) − y(t n−1 )
h n
− φ(t n−1 , y(t n−1 )) (4.46)
The local truncation error gives an estimate of the error in discretizing the
differential equation at y n assuming that there are no errors at y n1 - the local error
in the calculation of the derivative. For Euler’s Method,
y n+1 = y n + h n f(t n , y n ) (4.47)
we can derive the local truncation error using a Taylor Series approximation about
y(t n−1 ),
d n = N (t n ) = y(t n) − y(t n−1 )
h
= y(t n−1) + hy ′ (t n−1 ) + 1 2 h2 y ′′ (t n−1 ) + · · · − y(t n−1 )
h
Using the fact that y ′ (t n−1 ) = f(t n−1 , y(t n−1 ),
− f(t n−1 , y(t n−1 )) (4.48)
− f(t n−1 , y(t n−1 ) (4.49)
d n = h 2 y′′ (t n−1 ) + O(h 2 ) (4.50)
Thus the local truncation error for Euler’s method is proportional to h. We write
this as d n = O(h), and say that Euler’s method is a first order method.
Definition 4.6 (Convergence). A one step method is said to converge on an
interval [a, b] if y n → y on [a, b] as n → ∞ for any IVP
y ′ = f(t, y), y(t 0 ) = y 0 (4.51)
with f(t, y) Lipshitz in y. A method is said to be convergent of order p if for
some positive integer p the global error e n = |y(t n ) − y n | satisfies
e n = |y(t n ) − y n | = O(h p ) (4.52)
Definition 4.7 (Stabilty, Zero-Stability, 0-Stability). A one-step method is
said to be stable if for each IVP (4.44) with f Lipshitz in y there exists K, h 0 > 0
such that the difference between two different mesh functions (not necessarily solutions,
just functions that are defined on the mesh) y n and ŷ n satisfies
[
]
|y n − ŷ n | ≤ K |y 0 − ŷ 0 | + ‖N y n − N ŷ n ‖
(4.53)
for all h ∈ [0, h 0 ], where ‖ · ‖ denotes the sup-norm,
‖u n ‖ = max
1≤j≤n |u j| (4.54)
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007