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a very small coefficient. Nevertheless, since e~^nt tends to zero and e710^t grows very rapidly, the latter eventually dominates, and the calculated solution is simply a multiple of e71^1 = 03(t).
To be specific, suppose that we use the Runge-Kutta method to calculate the solution y = 04( t) = e710n t of the initial value problem
/ 10n2 y = 0, y (0) = 1, y'(0) = VlOn.
(The Runge-Kutta method for second order systems is described in Section 8.6.) Using single-precision (eight-digit) arithmetic with a step size h = 0.01, we obtain the results in Table 8.5.4. It is clearly evident from these results that the numerical solution begins to deviate significantly from the exact solution for t > 0.5, and soon differs from it by many orders of magnitude. The reason is the presence in the numerical solution of a small component of the exponentially growing solution 03(t) = t. With eightdigit arithmetic we can expect a round-off error of the order of 108 at each step. Since e-ZTont grows by a factor of 3.7 x 1021 from t = 0 to t = 5, an error of order 108 near t = 0 can produce an error of order 1013 at t = 5 even if no further errors are introduced in the intervening calculations. The results given in Table 8.5.4 demonstrate that this is exactly what happens.
Equation (18) is highly unstable and the behavior shown in this example is typical of unstable problems. One can track a solution accurately for a while, and the interval can be extended by using smaller step sizes or more accurate methods, but eventually the instability in the problem itself takes over and leads to large errors.
8.5 More on Errors; Stability
TABLE 8.5.4 Exact Solution of y" 10n2y = 0, y(0) = 1, y'(0) = VTOn and Numerical Solution Using the Runge-Kutta Method with h = 0.01
t Numerical Exact
0.0 1.0 1.0
0.25 8.3439 X 102 8.3438 X 10 2
0.5 6.9631 X 103 6.9620 X 10 3
0.75 5.9390 X 104 5.8089 X 10 4
1.0 2.0437 X 104 4.8469 X 10 -5
1.5 2.2394 X 102 3.3744 X 10 -7
2.0 3.2166 2.3492 X 10 9
2.5 4.6202 X 102 1 .6356 X 10 11
3.0 6.6363 X 104 1.1386 X 10 13
3.5 9.5322 X 106 7.9272 X 10 16
4.0 1.3692 X 109 5.5189 X 10 -18
4.5 1.9667 X 1011 3.8422 X 10 -20
5.0 2.8249 X 1013 2.6750 X 10 22
Some Comments on Numerical Methods. In this chapter we have introduced several numerical methods for approximating the solution of an initial value problem. We have tried to emphasize some important ideas while maintaining a reasonable level of complexity. For one thing, we have always used a uniform step size, whereas production codes currently in use provide for varying the step size as the calculation proceeds.
There are several considerations that must be taken into account in choosing step sizes. Of course, one is accuracy; too large a step size leads to an inaccurate result. Normally, an error tolerance is prescribed in advance and the step size at each step must be consistent with this requirement. As we have seen, the step size must also be chosen so that the method is stable. Otherwise, small errors will grow and soon render the results worthless. Finally, for implicit methods an equation must be solved at each step and the method used to solve the equation may impose additional restrictions on the step size.
In choosing a method one must also balance the considerations of accuracy and stability against the amount of time required to execute each step. An implicit method, such as the Adams-Moulton method, requires more calculations for each step, but if its accuracy and stability permit a larger step size (and consequently fewer steps), then this may more than compensate for the additional calculations. The backward differentiation formulas of moderate order, say, four, are highly stable and are therefore indicated for stiff problems, for which stability is the controlling factor.
Some current production codes also permit the order of the method to be varied, as well as the step size, as the calculation proceeds. The error is estimated at each step and the order and step size are chosen to satisfy the prescribed error tolerance. In practice, Adams methods up to order twelve and backward differentiation formulas up to order five are in use. Higher order backward differentiation formulas are unsuitable due to a lack of stability.
Finally, we note that the smoothness of the function f, that is, the number of continuous derivatives that it possesses, is a factor in choosing the order of the method
Chapter 8. Numerical Methods
to be used. High order methods lose some of their accuracy if f is not smooth to a corresponding order.
1. To obtain some idea of the possible dangers of small errors in the initial conditions, such as those due to round-off, consider the initial value problem
y = t + y 3, y(0) = 2.
(a) Show that the solution is y = 01 (t) = 2 t.
(b) Suppose that in the initial condition a mistake is made and 2.001 is used instead of 2. Determine the solution y = 02(t) in this case, and compare the difference 02(t) 01 (t) at t = 1 and as t ^to .