Lecture Notes in Differential Equations - Bruce E. Shapiro

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result with a smaller step size. However, in order to reduce computation
time, it is desirable to find methods that will give better results withou
a significant decrease in step size. We can do this by making φ depend
on values of the solution at multiple time points. For example, a Linear
Multistep Method has the form
y n+1 + a 0 y n + a 1 y n−1 + · · · = h(b 0 f n+1 + b 1 f n + b 2 f n−1 + · · · ) (33.42)
For some numbers a 0 , a 1 , . . . and b 0 , b 1 , . . . . Euler’s method has a 0 =
−1, a 1 = a 2 = · · · = 0 and b 1 = 1, b 0 = b 2 = b 3 = · · · = 0
Here we introduce the Local Truncation Error, one measure of the
“goodness” of a numerical method. The Local truncation error tells us
the error in the calculation of y, in units of h, at each step t n assuming
that there we know y n−1 precisely correctly. Suppose we have a numerical
estimate y n of the correct solution at y(t n ). Then the Local Truncation
Error is defined as
LTE = 1 h (y(t n) − y n ) (33.43)
= 1 h (y(t n) − y(t n−1 ) + y(t n−1 ) − y n ) (33.44)
Assuming we know the answer precisely correctly at t n−1 then we have
so that
For Euler’s method,
hence
LTE = y(t n) − y(t n−1 )
h
= y (t n ) − y(t n−1 )
h
y n−1 = y(t n−1 ) (33.45)
+ y n−1 − y n
h
(33.46)
− 1 h φ(t n, y n , . . . ) (33.47)
φ = hf(t, y) (33.48)
LTE(Euler) = y(t n) − y(t n−1 )
h
− f(t n , y n ) (33.49)