Let us assume that the solution of the initial value problem has a continuous second derivative in the interval of interest. To assure this, we can assume that , and are continuous in the region of interest. If f has these properties and if is a solution of the initial value problem, then
and by the chain rule
Since the right side of this equation is continuous, is also continuous.
Then, making use of a Taylor polynomial with a remainder to expand about , we obtain
where is some point in the interval . Subtracting Eq. (1) from this equation, and noting that and , we find that
To compute the local truncation error we apply Eq. (5) to the true solution , that is, we take to be . Then we immediately obtain from Eq. (5) that the local truncation error is
Thus the local truncation error for the Euler method is proportional to the square of the step size h and the proportionality factor depends on the second derivative of the solution . The expression given by Eq. (6) depends on n and, in general, is different for each step. A uniform bound, valid on an interval [a, b], is given by
where M is the maximum of on the interval . Since the equation given above is based on a consideration of the worst possible case, that is, the largest possible value of , it may well be a considerable overestimate of the actual local truncation error in some parts of the interval . One use of Eq. (7) is to choose a step size that will result in a local truncation error no greater than some given tolerance level. For example, if the local truncation error must be no greater than , then from Eq. (7) we have
The primary difficulty in using any of Eqs. (6), (7), or (8) lies in estimating or M. However, the central fact expressed by these equations is that the local truncation error is proportional to . Thus, if h is reduced by a factor of , then the error is reduced by , and so forth. In Golub/Ortega's book, it is mentioned that the local truncation error is as opposed to . It is because they implicitly divide it by h. Their derivation of local trunctation error is based on the formula
where is the local truncation error.
More important than the local truncation error is the global truncation error . The analysis for estimating is more difficult than that for . However, knowing the local truncation error we can make an intuitive estimate of the global truncation error at a fixed as follows. Suppose that we take n steps in going from to . In each step the error is at most ; thus the error in n steps is at most . Noting that , we find that the global truncation error for the Euler method in going from to is bounded by
This argument is not complete since it does not take into account the effect that an error at one step will have in succeeding steps. Nevertheless, it can be shown that the global truncation error in using the Euler method on a finite interval is no greater than a constant times h. The Euler method is called a first order method because its global truncation error is proportional to the first power of the step size.
Because it is more accessible, we will hereafter use the local truncation error as our principal measure of the accuracy of a numerical method, and for comparing different methods. As an example of how we can use the result (6) if we have a priori information about the solution of the given initial value problem, consider the illustrative example.
on the interval . Let be the solution of the initial value problem. Then, as noted previously,
Equation (6) then states that
The appearance of the factor 19 and the rapid growth of explain why the results in the preceding section with were not very accurate.
For example, the error in the first step is
It is clear that is positive and, since , we have
Note also that ; hence . The actual error is 0.1090418. It follows from Eq. (10) that the error becomes progressively worse with increasing t; Similar computations for bounds for the local truncation error give
in going from 0.4 to 0.5 and
in going from 0.9 to 1.0.
These results indicate that for this problem the local truncation error is about 40 or 50 times larger near t = 1 than near t = 0 . Thus, to reduce the local truncation error to an acceptable level throughout , one must choose a step size h based on an analysis near t = 1. Of course, this step size will be smaller than necessary near t = 0 . This results in more calculations than necessary, more time consumed, and possibly more danger of unacceptable round-off errors. Another approach is to keep the local truncation error approximately constant throughout the interval by gradually reducing the step size as t increases. In the example problem we would need to reduce h by a factor of about seven in going from t = 0 to t = 1 . A method that provides for variations in the step size is called adaptive. All modern codes for solving differential equations have the capability of adjusting the step size as needed. This includes the two routines ode23 and ode45 in Matlab.