Next we present a few examples, in increasing order of accuracy. The simplest one is a forward finite-difference scheme [cf. Eq. (7.70)].
Example 42. The Euler method (first-order, one-stage):
Example 43. Second-order, two-stages RK :
Example 44. Third-order, four-stages RK:
Example 45. Third-order, three-stages Heun–RK method
Example 46. The standard fourth-order, four-stages RK method:
Table 2 shows representations of Examples 42, 43, 44, 45, while Table 3 shows the classical fourth-order Runge–Kutta method of Example 46.
The above examples explicitly show that up to, and including, fourth-order accuracy there are Runge–Kutta methods of order and stages with . It is interesting that even though the first RK methods date back to the end of the 19h century, the question of whether there are higher-order (than four) RK methods remained open until the following result was shown by Butcher in 1963 : cannot be achieved anymore starting with fifth-order accurate schemes, and there are a number of barriers.
However, there are fifth and sixth-order RK methods with six and seven stages, respectively. Butcher in 1965  and 1985  respectively showed the following barriers.
Seventh and eighth-order methods with and stages, respectively, have been constructed, as well as a tenth-order one with stages.
In practice many approaches in numerical relativity use an adaptive timestep method. One way of doing so is to evolve the system of equations two steps with timestep and one with . The difference in both solutions at can be used, along with Richardson extrapolation, to estimate the new timestep needed to achieve any given tolerance error.
In more detail: if we call the solution at evolved from time in two steps of size , and the solution at the same time advanced from time in one step of size , then the following holds
Embedded methods also compute two solutions and use their difference to estimate the error and adapt the timestep. However, this is done by reusing the stages. Two Runge–Kutta methods, of order and (in most cases – but not always – or ) are constructed, which share the intermediate function values, so that there is no overhead cost. Therefore, their Butcher table looks as follows:
Embedded methods are denoted by , where is the order of the scheme, which advances the solution. For example, a method would be of fifth order, with a fourth-order scheme, which shares its function calls used to estimate the error.
Table 5 shows the coefficients for a popular embedded method, the seven stages Dormand–Prince . Dormand–Prince methods are embedded methods, which minimize a quantification of the truncation error for the highest-order component, which is the one used for the evolution.
Living Rev. Relativity 15, (2012), 9
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