Editing Runge–Kutta methods (section)

===Examples===
The simplest example of an implicit Runge–Kutta method is the [[backward Euler method]]:

:<math>y_{n + 1} = y_n + h f(t_n + h,\ y_{n + 1}). \,</math>

The Butcher tableau for this is simply:

:<math>
\begin{array}{c|c}
1 & 1 \\
\hline
  & 1 \\
\end{array}
</math>

This Butcher tableau corresponds to the formulae

:<math> k_1 = f(t_n + h,\ y_n + h k_1) \quad\text{and}\quad y_{n+1} = y_n + h k_1, </math>

which can be re-arranged to get the formula for the backward Euler method listed above.

Another example for an implicit Runge–Kutta method is the [[trapezoidal rule (differential equations)|trapezoidal rule]]. Its Butcher tableau is:

:<math>
\begin{array}{c|cc}
0 & 0& 0\\
1 & \frac{1}{2}& \frac{1}{2}\\
\hline
  &  \frac{1}{2}&\frac{1}{2}\\
  & 1 & 0 \\
\end{array}
</math>

The trapezoidal rule is a [[collocation method]] (as discussed in that article). All collocation methods are implicit Runge–Kutta methods, but not all implicit Runge–Kutta methods are collocation methods.<ref>{{harvnb|Iserles|1996|pp=43–44}}</ref>

The [[Gauss–Legendre method]]s form a family of collocation methods based on [[Gauss quadrature]]. A Gauss–Legendre method with ''s'' stages has order 2''s'' (thus, methods with arbitrarily high order can be constructed).<ref>{{harvnb|Iserles|1996|p=47}}</ref> The method with two stages (and thus order four) has Butcher tableau:

:<math>
\begin{array}{c|cc}
\frac12 - \frac16 \sqrt3 & \frac14                  & \frac14 - \frac16 \sqrt3 \\
\frac12 + \frac16 \sqrt3 & \frac14 + \frac16 \sqrt3 & \frac14 \\
\hline
                         & \frac12                  & \frac12 \\
                         & \frac12+\frac12 \sqrt3   & \frac12-\frac12 \sqrt3
\end{array}
</math> <ref name="Süli 2003 353"/>