A method for solving ordinary differential equations using the formula
 |
which advances a solution from 
to 
. Note that the method increments a solution through
an interval 
while using derivative information from only the beginning of the interval. As a
result, the step's error is 
. This method is called simply "the Euler method"
by Press et al. (1992), although it is actually the forward version of the
analogous Euler backward method.
While Press et al. (1992) describe the method as neither very accurate nor very stable when compared to other methods using the same step size, the accuracy is actually not too bad and the stability turns out to be reasonable as long as the so-called Courant-Friedrichs-Lewy condition is fulfilled. This condition states that, given a space discretization, a time step bigger than some computable quantity should not be taken. In situations where this limitation is acceptable, Euler's forward method becomes quite attractive because of its simplicity of implementation.
