Error control

Runge-Kutta methods can be written in the general form

,

,

where the are approximations for the values which satisfy the initial value problem

, in , and .

We would like to ensure that for a given small value , a number of steps can be chosen such that the global error is bounded by , i.e.,

for all .

Since the global error is not easy to calculate, we will focus on the local truncation error (which assumes the previous calculated approximation can be treated as exact) . It can be shown that under suitable conditions, a connection exists between the local truncation error and the global error, in the sense that a bound on the local truncation error produces a corresponding bound on the global error.

Therefore, we will try to estimate the local trunction error made in a calculation.

First, consider a method with a local truncation error of order ,

,

Second, consider a method with local truncation error of order

,

with the equivalent of for the second method.

The first method produces estimates and the second method yields estimates . Then assume that

.

Then

,

or

Similarly,

Then,

or,

.

But while , so that

.

Hence, the local truncation error can be estimated in terms of the calculated values and . This requires repeating the calculation of the estimate using a method of a higher order. This extra calculation, however, will provide an estimate of the accuracy of the first estimate (obtained by the lower order method). This is often worth the extra effort.

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Can we now choose the step-size such that the truncation error is kept below a given bound ?

Consider the formula above and the fact that , or,

, for some constant .

Then,

.

Now consider a constant, positive factor , then the truncation error for a step-size is given by

,

so that, to keep the truncation error bounded by a value , we must have

,

and the required step-size is found using a value of which satisfies the inequality

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This formula allows you to estimate at every step in the calculation the step size needed to keep the local truncation error for that step as small as you want.

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