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

86 CHAPTER 4. IMPROVING ON EULER’S METHOD
4.4 Other Euler Methods
The Trapezoidal Method has an iteration formula
y n = y n−1 + h n
2 (f(t n, y n ) + f(t n−1 , y n−1 )) (4.113)
The local truncation error for the Trapezoidal Method is O(h 2 ):
d n = y(t n) − y(t n−1 )
h
Use Taylor series expansions about t = t n ,
− 1 2 [f(t ,y(t n )) + f(t n−1 , y(t n−1 ))] (4.114)
y(t n−1 ) = y(t n ) − hy ′ (t n ) + h2
2 y′′ (t n ) − h3
6 y′′′ (t n ) + · · · ⇐= (4.115)
f(t n−1 , y(t n−1 )) = y ′ (t n−1 ) = y ′ (t n ) − hy ′′ (t n ) + h2
2 y′′′ (t n ) · · · (4.116)
Hence the local truncation error is
d n = y ′ (t n ) − h 2 y′′ (t n ) + h2
8 y′′′ (t n ) + · · · (4.117)
· · · + 1 ]
[y ′ (t n ) + y ′ (t n ) − hy ′′ (t n ) + h2
2
2 y′′′ (t n ) + · · · (4.118)
= 3h2
8 y′′′ (t n ) (4.119)
= O(h 2 ) (4.120)
For the test equation, the trapezoidal method gives
y n = 2 + hλ
2 − hλ y n−1 (4.121)
Substituting hλ = x + iy we find that the region of absolute stability is precisely
the left-half plane x ≤ 0.
The theta method is given by
y n = y n−1 + h [θf(t n−1 , y n−1 ) + (1 − θ)f(t n , y n )] (4.122)
The theta method is implicit except when θ = 1, where it reduces to Euler’s method.
For θ = 1/2 it becomes the trapezoidal method. The method is first order except
when θ = 1/2. The theta method can be used to eliminate error in specific terms in
the Taylor expansion besides the lowest order term. For example, setting θ = 2/3
gets rid of the O(h 3 ) term in the error even though the O(h 2 ) term remains. This
could be of use in cases where the higher order term has a sufficiently high coefficient
that for larger step sizes it overwhelms the lower order term. The theta-method is
A-stable if and only if 0 ≤ θ ≤ 1/2.
Math 582B, Spring 2007
California State University Northridge
c○2007, B.E.Shapiro
Last revised: May 23, 2007