Lecture Notes in Differential Equations - Bruce E. Shapiro

369
Expanding the final term in a Taylor series,
f(t n , y n ) = y ′ (t n ) (33.58)
= y ′ (t n−1 ) + hy ′′ (t n−1 ) + h2
2 y′′′ (t n−1 ) + · · · (33.59)
= f(t n−1 , y n−1 ) + hy ′′ (t n−1 ) + h2
2 y′′′ (t n−1 ) + · · · (33.60)
Therefore the Trapezoidal method is a second order method:
LTE(Trapezoidal = 1 2 f n−1 + h 2 y′′ n−1 + h2
6 y′′′ n−1 + · · ·
The theta method is given by
− 1 2 f n−1 − 1 2 hy′′ n−1 − 1 4 h2 y ′′′
n−1 + · · · (33.61)
= − 1
12 h2 y n−1 ′′′ + · · · (33.62)
= O(h 2 ) (33.63)
y n = y n−1 + h [θf(t n−1 , y n−1 ) + (1 − θ)f(t n , y n )] (33.64)
The theta method is implicit except when θ = 1, where it reduces to Euler’s
method, and is first order unless θ = 1/2. For θ = 1/2 it becomes the trapezoidal
method. The usefulness of the comes from the ability to remove the
error for specific high order terms. For example, when θ = 2/3, there is no
h 3 term even though there is still an h 2 term. This can help if the coefficient
of the h 3 is so larger that it overwhelms the the h 2 term for some values of h.
The second-order midpoint method is given by
(
y n = y n−1 + h n f t n−1/2 , 1 )
2 [y n + y n−1 ]
(33.65)
The modified Euler Method, which is also second order, is
y n = y n−1 + h n
2 [f(t n−1, y n−1 ) + f(t n , y n−1 + hf(t n−1 , y n−1 ))] (33.66)