The midpoint method is a second-order Runge-Kutta method for numerically solving an initial
value problem 
.
With step size 
,
it computes
 |  |  |
(1)
|
 |  | ![y_n+hf[t_n+1/2h,y_n+1/2hk_1].](/images/equations/MidpointMethod/Inline8.svg) |
(2)
|
Thus the slope used for the step is evaluated at an estimated midpoint of the solution curve. The method has local truncation error of order 
and global error of order 
under the usual smoothness hypotheses. It is distinct from
the midpoint rule, a form of the rectangle rule
for numerical integration.
