Several methods exists for finding better numerical solutions to first order differential equations of the form
than Euler's simple formula
One method uses forwards and backwards Euler formulae to derive a 'central' formula.
Subtracting these gives
The
term means that the error at each iteration is of the order of
The error for the simple Euler forwards or backwards formulae are of the order of
so the central formula is more accurate.
Another method uses Euler's simple formula to find an estimate for
using
at
then uses this estimate to find a second estimate for
at the point
An average of these two values ofis used to provide an improved estimate for
The process is
Use this value of
to find
Then find
Example. If
with
estimate
with
The problem can be solved exactly by separation of variables.
hence
This is very accurate.