Editing Runge–Kutta methods (section)

===Second-order methods with two stages===
An example of a second-order method with two stages is provided by the explicit [[midpoint method]]:
:<math> y_{n+1} = y_n + hf\left(t_n+\frac{1}{2}h, y_n+\frac{1}{2}hf(t_n,\ y_n)\right). </math>
The corresponding tableau is

:{| style="text-align: center" cellspacing="0" cellpadding="3"
| style="border-right:1px solid;" | 0
|-
| style="border-right:1px solid; border-bottom:1px solid;" | 1/2 || style="border-bottom:1px solid;" | 1/2 || style="border-bottom:1px solid;" |
|-
| style="border-right:1px solid;" | || 0 || 1
|}

The midpoint method is not the only second-order Runge–Kutta method with two stages; there is a family of such methods, parameterized by α and given by the formula<ref>{{harvnb|Süli|Mayers|2003|p=327}}</ref>

:<math> y_{n+1} = y_n + h\bigl( (1-\tfrac1{2\alpha}) f(t_n, y_n) + \tfrac1{2\alpha} f(t_n + \alpha h, y_n + \alpha h f(t_n, y_n)) \bigr). </math>

Its Butcher tableau is

:{| style="text-align: center" cellspacing="0" cellpadding="8"
| style="border-right:1px solid;" | 0
|-
| style="border-right:1px solid; border-bottom:1px solid;" | <math>\alpha</math> || style="border-bottom:1px solid; text-align: center;" | <math>\alpha</math> || style="border-bottom:1px solid;" |
|-
| style="border-right:1px solid;" | || <math>(1-\tfrac1{2\alpha})</math> || <math>\tfrac1{2\alpha}</math>
|}

In this family, <math>\alpha=\tfrac12</math> gives the [[midpoint method]], <math>\alpha=1</math> is [[Heun's method]],<ref name="Süli 2003 328"/> and <math>\alpha=\tfrac23</math> is Ralston's method.