It is easy to see that Euler's method converges for the special case of the equation with solution . For this example,

(5) |

Error estimates show

- The global error at
*T*is*O*(*h*) (*first order accuracy*). - The error grows exponentially in time.
- The error increases with
*M*. This suggests that one should take smaller steps where the solution is changing more rapidly. We will return to this below.

The error analysis above ignores round-off error. If one assumes that
a fixed error is added at each time step, then the error estimate
of is modified to . That is,
taking more steps reduces the discretization error, but increases
the round-off error. Therefore, there is a point of diminishing
returns where the total error increases as *h* decreases.
Better results require not more effort, but more efficiency. The
key is to take more terms of the Taylor series and reduce the
discretization error to *O*(*h*^{p}), with *p*>1.