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.