This is the third post in a series on Runge-Kutta methods. The first post in the series introduces Runge-Kutta methods and Butcher tableau. The next post looked at Fehlberg’s adaptive Runge-Kutta method, first published in 1969. This post looks at a similar method from Dormand and Prince in 1980.

Like Fehlberg’s method, the method of Dormand and Prince can be summarized in a big, intimidating tableau, which we will display below. However we will discuss three differences between the methods:

- Order 4(5) vs 5(4)
- Derivative reuse
- Precision / computation ratio

## Dormand Prince tableau

Here’s the Butcher tableau for the Dormand-Prince method in all it’s glory:

The only detail of the table that will be important below is that 7th and 8th rows are identical.

## Order 4(5) vs order 5(4)

Fehlberg’s method, a.k.a. RKF45, computes each update to the solution using a 4th order Runge-Kutta method, and uses a 5th order Runge-Kutta method to estimate the error.

The method of Dormand and Prince also uses 4th and 5th order Runge-Kutta methods, but in the opposite way. The fifth order method is used to advance the solution, and the 4th order method is used for comparison to estimate error.

## Derivative reuse

The work in solving

by a Runge-Kutta method is roughly proportional to the number of stages. Dormand-Prince is a 7-stage method while Fehlberg is a 6-stage method, so it would seem that the latter is more efficient. However, if you look back at the Dormand-Prince tableau, the last row above the horizontal line equals the first row below the line. That means that the last evaluation of *f* at one step can be reused at the first evaluation of *f* at the next step.

## Precision per unit work

In their book Solving Differential Equations, vol. 1, Hairer et al compare several adaptive Runge-Kutta methods, including Fehlberg (RKF45) and Dormand-Prince, and conclude that the latter produces more precision per unit work.

We again see that the [Fehlberg] method

underestimatesthe local error. Further, with the use of local extrapolation, the advantage of RKF4(5) melts away to a large extent. The best method of all these is without a doubt the coefficient set of Dormand and Prince.

Nice post, but did you drop negative signs from your Butcher tableau?