linguistic:paraphrasing
Examples

The simplest example of an implicit Runge–Kutta method is the backward Euler method:

The Butcher tableau for this is simply:

This Butcher tableau corresponds to the formulae

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. Its Butcher tableau is:

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.

The Gauss–Legendre methods 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). The method with two stages (and thus order four) has Butcher tableau: