Error estimates for Runge-Kutta Methods:

We have seen that the bounds for the local truncation error, as discussed earlier, are very complicated and of little use in practice for deciding the appropriate step size control policy. What is needed, in place of a bound, is a readily computable estimate of the local truncation error. The most commonly used estimate arises from an application of the Richardson extrapolation. Under the usual localizing assumption that no previous errors have been made, we can write

(3.44)

where p is the order of the Runge-Kutta method. Now let us compute

, a second approximation to ), obtained by applying the same method at but with step size 2h. With the same localizing assumption, it follows that

(3.45)

on expanding about . Subtracting (3.44) from (3.45),

We get

Thus, the principal local truncation error, which is taken as an estimate for the local truncation error, may be written as

(3.46)

This estimate is quite adequate for step size control policy.