Editing Runge–Kutta methods (section)

===Examples===
The RK4 method falls in this framework. Its tableau is<ref name="Süli 2003 352">{{harvnb|Süli|Mayers|2003|p=352}}</ref>

:{| style="text-align: center" cellspacing="0" cellpadding="3"
| style="border-right:1px solid;" | 0
|-
| style="border-right:1px solid;" | 1/2 || 1/2
|-
| style="border-right:1px solid;" | 1/2 || 0 || 1/2
|-
| style="border-right:1px solid; border-bottom:1px solid;" | 1 || style="border-bottom:1px solid;" | 0
| style="border-bottom:1px solid;" | 0 || style="border-bottom:1px solid;" | 1
| style="border-bottom:1px solid;" |
|-
| style="border-right:1px solid;" | || 1/6 || 1/3 || 1/3 || 1/6
|}

A slight variation of "the" Runge–Kutta method is also due to Kutta in 1901 and is called the 3/8-rule.<ref>{{harvtxt|Hairer|Nørsett|Wanner|1993|p=138}} refer to {{harvtxt|Kutta|1901}}.</ref> The primary advantage this method has is that almost all of the error coefficients are smaller than in the popular method, but it requires slightly more FLOPs (floating-point operations) per time step. Its Butcher tableau is

:{| style="text-align: center" cellspacing="0" cellpadding="3"
| style="border-right:1px solid;" | 0
|-
| style="border-right:1px solid;" | 1/3 || 1/3
|-
| style="border-right:1px solid;" | 2/3 || −1/3 || 1
|-
| style="border-right:1px solid; border-bottom:1px solid;" | 1 || style="border-bottom:1px solid;" | 1
| style="border-bottom:1px solid;" | −1 || style="border-bottom:1px solid;" | 1
| style="border-bottom:1px solid;" |
|-
| style="border-right:1px solid;" | || 1/8 || 3/8 || 3/8 || 1/8
|}

However, the simplest Runge–Kutta method is the (forward) [[Euler method]], given by the formula <math> y_{n+1} = y_n + hf(t_n, y_n) </math>. This is the only consistent explicit Runge–Kutta method with one stage. The corresponding tableau is

:{| style="text-align: center" cellspacing="0" cellpadding="3"
| width="10" style="border-right:1px solid; border-bottom:1px solid;" | 0
| width="10" style="border-bottom:1px solid;" |
|-
| style="border-right:1px solid;" | || 1
|}