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.
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 underestimates the 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.