# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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2.0 8934.0696 5435.7294 4172.7228 3597.4478 3540.2001

Errors in Numerical Approximations. The use of a numerical procedure, such as the Euler formula, to solve an initial value problem raises a number of questions that must be answered before the approximate numerical solution can be accepted as satisfactory. One of these is the question of convergence. That is, as the step size h tends to zero, do the values of the numerical solution y1, y2,, yn,... approach the corresponding values of the actual solution? If we assume that the answer is affirmative, there remains the important practical question of how rapidly the numerical approximation converges to the solution. In other words, how small a step size is needed in order to guarantee a given level of accuracy? We want to use a step size that is small enough to assure the required accuracy, but not too small. An unnecessarily small step size slows down the calculations, makes them more expensive, and in some cases may even cause a loss of accuracy.

There are two fundamental sources of error in solving an initial value problem numerically. Let us first assume that our computer is such that we can carry out all computations with complete accuracy; that is, we can retain an infinite number of decimal places. The difference En between the solution y = ?(t) of the initial value problem (1), (2) and its numerical approximation is given by

En = <P(tn) - yn, (14)

and is known as the global truncation error. It arises from two causes: First, at each

step we use an approximate formula to determine yn+1; second, the input data at each

step are only approximately correct since in general $( tn) is not equal to yn. If we assume that yn = $(tn), then the only error in going one step is due to the use of an approximate formula. This error is known as the local truncation error en.

The second fundamental source of error is that we carry out the computations in arithmetic with only a finite number of digits. This leads to a round-off error Rn defined by

Rn = yn - Yn, (15)

where Yn is the value actually computed from the given numerical method.

The absolute value of the total error in computing $( tn) is given by

¹(tn) - Yn1 = \4>(tn) - yn + yn - Yn|-

(16)

8.1 The Euler or Tangent Line Method

425

Making use of the triangle inequality, |a + b|<|a| + |b|, we obtain from Eq. (16)

¹(tn) - Yn| < \0(tn) - yn| + y - Yn|

<En | + R I (17)

Thus the total error is bounded by the sum of the absolute values of the truncation and round-off errors. For the numerical procedures discussed in this book it is possible to obtain useful estimates of the truncation error. However, we limit our discussion primarily to the local truncation error, which is somewhat simpler. The round-off error is clearly more random in nature. It depends on the type of computer used, the sequence in which the computations are carried out, the method of rounding off, and so forth. While an analysis of round-off error is beyond the scope of this book, it is possible to say more about it than one might at first expect (see, for example, Henrici). Some of the dangers from round-off error are discussed in Problems 25 through 27 and in Section 8.5.

Local Truncation Error for the Euler Method. Let us assume that the solution y = 0(t) of the initial value problem (1), (2) has a continuous second derivative in the interval of interest. To assure this, we can assume that f, ft, and f' are continuous. Observe that if f has these properties and if 0 is a solution of the initial value problem (1), (2), then

0\t) = f[t ,0(t)],

and by the chain rule

0"(t) = ft [t ,0(t)] + fy [t ,4(t)W(t)

= ft[t,0(t)] + fy[t,0(t)] f[t,0(t)]. (18)

Since the right side of this equation is continuous, 0" is also continuous.

Then, making use of a Taylor polynomial with a remainder to expand 0 about tn, we obtain

0(tn + h) = 0(tn) + 0'(tn )h + 2 0" (ln )h2, (19)

where ~tn is some point in the interval tn < \ < tn + h. Subtracting Eq. (4) from Eq. (19), and noting that 0(tn + h) = 0(t11+1) and 0'(tn) = f [tn, 0(tn)], we find that

0(tn+1) - yn+1 = [0(tn) - yn] + h{f [tn, 0(tn)] - f(tn, yn)} + 20"(-tn)!?. (20)

To compute the local truncation error we apply Eq. (20) to the true solution 0(t), that is, we take yn to be 0(tn). Then we immediately see from Eq. (20) that the local truncation error en 1 is

en+1 = 0(tn+1) - yn+1 = 20"( Jn)h2. (21)

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 0. The expression given by Eq. (21) depends on n and, in general, is different for each step. A uniform bound, valid on an interval [a, b], is given by

|en| < Mh2/2, (22)

where M is the maximum of |0"(t)| on the interval [a, b]. Since Eq. (22) is based on

a consideration of the worst possible case, that is, the largest possible value of |0/;(t )|,

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Chapter 8. Numerical Methods

it may well be a considerable overestimate of the actual local truncation error in some parts of the interval [a, b]. One use of Eq. (22) 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 e, then from Eq. (22) we have

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