Chapter 8.2, Problem 25P

In this section we stated that the global truncation errorfor the Euler method applied to an initial value problem over a fixed interval is no more than a constant times the step size $h$. In this problem, we show you how to obtain some experimental evidence in support of thisstatement. Consider theinitial value problem in Example 1 for which some numerical approximations are given in Table 8.2.1. Observe that, foreach step size, the maximum error $E$ occurs at the endpoint $t=2$. Now let us assume that $E=C{h}^p$, where the constants $C$

and $p$ are to be determined. By taking the logarithm of eachside of this equation, we obtain

$\ln E=\ln C+p\ln h,$

Which is the equation of a straight line in the $(\ln h)(\ln E)-$ plane. The slope of this line is the value of the exponent $p$ and the intercept on the $(\ln E)$-axis determines the value of $C$.

a) Using the data in Table 8.2.1, calculate the maximum error $E$ for each of the given values of $h$.

b) Plot $\ln E$ versus $\ln h$ for the four data points that you obtained in part (a).

c) Do the points in part (b) lie approximately on a singlestraight line? If so, then this is evidence that the assumed expression for $E$ is correct.

d) Estimate the slope of the line in part (c). If the statementin the text about the magnitude of the global truncation erroris correct, then the slope should be no greater than 1.

Note: Your estimate of the slope p depends on how youchoose the straight line. If you have a curve-fitting routinein your software, you can use it to determine the straight linethat best fits the data. Otherwise, you may wish to resort toless precise methods. For example, you could calculate theslopes of the line segments joining (one or more) pairs ofdata points, and then average your results.