In this section we stated that the global truncation err
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In this section we stated that the global truncation error for the Euler method applied to an initial value problem over
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In this section we stated that the global truncation error for 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 this statement. Consider the initial value problem in Example 1 for which some numerical approximations are given in Table 8.2.1. Observe that, for each step size, the maximum error E occurs at the endpoint t = 2. Now let us assume that E = Chp, where the constants C and p are to be determined. By taking the logarithm of each side 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 single straight 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 statement in the text about the magnitude of the global truncation error is correct, then the slope should be no greater than 1. Note: Your estimate of the slope p depends on how you choose the straight line. If you have a curve-fitting routine in your software, you can use it to determine the straight line that best fits the data. Otherwise, you may wish to resort to less precise methods. For example, you could calculate the slopes of the line segments joining (one or more) pairs of data points, and then average your results.
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