Unformatted text preview: and subtract the above di erence equations to obtain
n = n; 1 + ( h f tn;1 zn;1 ); ( n
ft 1 ; yn;1 )] : Using the Lipschitz condition (1.2.4) j nj (1 + )j n 1j
hL ; : Iterating the inequality and using (2.1.10) leads us to j nj (1 + )n j 0j
hL hLn e j 0j LT e j 0j k 8 nh T where = LT and j 0j .
Thus, Euler's method is stable. So what's wrong with the results of Table 2.1.3? This
di culty points out additional shortcomings of De nition 2.1.5. First, it is applicable in
the limit of small step size. Computations are performed with a prescribed step size and
it is often di cult to determine if this step size is \small enough." Second, De nition
2.1.5 allows some growth of the solution for bounded times. In Example 2.1.5,
If the solution of the IVP is stable or asymptotically stable, we cannot tolerate any
growth in the computed solution unless either is small or computations are performed
for very short time periods . If the solution of the IVP is unstable, some growth of
the perturbation is acceptable. The following concept of absolute stability will provide
a more useful tool than that of De nition 2.1.5 when IVP solutions are not growing in
k e k> L T De nition 2.1.6. A numerical method is absolutely stable for a step size and a given
h ODE if a change in y0 of size is no larger than for all subsequent time steps. 12 Remark 4. In contrast to De nition 2.1.5, absolute stability is applied at a speci c
value of h rather than in the limit as h ! 0.
De nition 2.1.6, like De nition 2.1.5, still depends too heavily on the di erential
equation. In order to reduce this dependence, it is common to apply absolute stability
to the \test equation"
y 0 = y y (0) = (2.1.15) y0 : In Chapter 1, we saw that (2.1.15) was useful in deciding the stability of the nonlinear
di erential equation (2.1.1). We now seek to use it to infer the stability of a di erence
equation. De nition 2.1.7. The region of absolute stability of a di erence equation is the set of
all real non-negative values of h and complex values of for which it is absolutely stable
when applied to the test equation (2.1.15).
Example 2.1.6. Consider Euler's method (2.1.2) applied to (2.1.15)
yn zn = (1 + )
h yn;1 = (1 + )
h zn;1 z0 = y0 + 0 where j 0j . Subtracting the two di erence equations and, once again, letting
n ; n yields
n = (1 +
z n = y h ; : Since the di erence equation is linear, the perturbed problem satis es the original di erence equation with the perturbation as its initial condition.
Iterating, we nd the solution of the perturbed problem to be
n = (1 + )n
h 0: Thus, the initial perturbation will not grow beyond j 0j if j1 + j 1
h 13 : (2.1.16) Im(h λ)
1 -2 -1 1 Re(h λ) -1 Figure 2.1.2: Region of absolute stability for Euler's method.
As shown in Figure 2.1.2, the region of absolute stability is a unit circle centered at
(;1 0) in the complex plane.
Example 2.1.7. Let us solve the IVP
h y 0 = y y (0) = 1 0 t 1 by Euler's method. The exact solution is, of course, ( ) = t . The interval 0
is divided into = 2k , = 0 1 , uniform subintervals of width = 1 . In order
to enhance roundo e ects, a...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
- Spring '14