The Chemistry Maths Book, Second Edition

588 Chapter 20Numerical methods

Euler’s method is called a first-order method because only the first power of his

retained in the Taylor expansion. The truncation error at each step is therefore of

order h

2

, so that halving the step size hreduces the error by a factor of 4. However, the

cumulative (global) error for a series of steps is of order h.

EXAMPLE 20.18Given the initial value problem

y′(x) 1 = 1 y 1 + 1 1, y(0) 1 = 11

use Euler’s method to obtain an approximate value ofy(1)using step sizesh 1 = 1 0.2

andh 1 = 1 0.1.

The results of applying the recursion

y

n+ 1

1 = 1 y

n

1 + 1 h(y

n

1 + 1 1)

are summarized in Table 20.10, and compared with the exact values obtained from

the solutiony(x) 1 = 12 e

x

1 − 1 1.

Table 20.10 Example of Euler’s method

hnx

n

y

n

exact error hnx

n

y

n

exact error

0.2 0 0.0 1.0000 1.0000 0.0000 0.1 0 0.0 1.0000 1.0000 0.0000

1 0.1 1.2000 1.2103 0.0103

1 0.2 1.4000 1.4428 0.0428 2 0.2 1.4200 1.4428 0.0228

3 0.3 1.6620 1.6997 0.0377

2 0.4 1.8800 1.9836 0.1036 4 0.4 1.9282 1.9836 0.0554

5 0.5 2.2210 2.2974 0.0764

3 0.6 2.4560 2.6442 0.1882 6 0.6 2.5431 2.6442 0.1011

7 0.7 2.8974 3.0275 0.1301

4 0.8 3.1472 3.4511 0.3039 8 0.8 3.2872 3.4511 0.1639

9 0.9 3.7159 3.9192 0.2033

5 1.0 3.9766 4.4366 0.4599 10 1.0 4.1875 4.4366 0.2491

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Figure 20.11