Lecture Notes in Differential Equations - Bruce E. Shapiro

368 LESSON 33. NUMERICAL METHODS
If we expand y in a Taylor series about t n−1 ,
Thus
y(t n ) = y(t n−1 ) + hy ′ (t n−1 ) + h2
2 y′′ (t n−1 ) + · · · (33.50)
= y(t n−1 ) + hf(t n−1 , y n−1 ) + h2
2 y′′ (t n−1 ) + · · · (33.51)
LTE(Euler) = h 2 y′′ (t n−1 ) + c 2 h 2 + c 3 h 3 + · · · (33.52)
for some constants c 1 , c 2 , ... Because the lowest order term in powers of h
is proportional to h, we say that
LTE(Euler) = O(h) (33.53)
and say that Euler’s method is a First Order Method. In general, to
improve accuracy for a given step size, we look for higher order methods,
which are O(h n ); the larger the value of n, the better the method in general.
The Trapezoidal Method averages the values of f at the two end points.
It has an iteration formula given by
y n = y n−1 + h n
2 (f(t n, y n ) + f(t n−1 , y n−1 )) (33.54)
We can find the LTE as follows by expanding the Taylor Series,
LTE(Trapezoidal) = y(t n) − y(t n−1 )
− f(t n , y n ) (33.55)
h
= 1 )
(y(t n−1 ) + hy ′ (t n−1 ) + h2
h
2 y′′ (t n−1 ) + h3
3! y′′′ (t n−1 ) + · · · − y(t n−1 )
− 1 2 (f(t n, y n ) + f(t n−1 , y n−1 )) (33.56)
Therefore using y ′ (t n−1 ) = f(t n−1 , y n−1 ),
LTE(Trapezoidal) = 1 2 f(t n−1, y n−1 ) + h 2 y′′ (t n−1 ) + h2
6 y′′′ (t n−1 ) · · · − 1 2 f(t n, y n )
(33.57)