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Unformatted text preview: at because they are open ended, nite di erence or nite element approximations in time can be sensitive to small errors, e.g., introduced by round o . Let us
illustrate the phenomena for the weighted average scheme (9.2.12c) M + tK]cn+1 = M ; (1 ; ) tK]cn + t (1 ; )ln + ln+1]: (9.4.1) We have assumed, for simplicity, that K and M are independent of time.
Sensitivity to small perturbations implies a lack of stability as expressed by the following de nition. De nition 9.4.1. A nite di erence scheme is stable if a perturbation of size k k introduced at time tn remains bounded for subsequent times t
t t0 . T and all time steps We may assume, without loss of generality, that the perturbation is introduced at
time t = 0. Indeed, it is common to neglect perturbations in the coe cients and con ne
the analysis to perturbations in the initial data. Thus, in using De nition 9.4.1, we
consider the solution of the related problem M + tK]~n+1 = M ; (1 ; ) tK]~n + t (1 ; )ln + ln+1]
~0 = c0 + :
Subtracting (9.4.1) from the perturbed system M + tK] n+1 = M ; (1 ; ) tK] n 0 = (9.4.2a) where
n = ~n ; cn:
c (9.4.2b) Thus, for linear problems, it su ces to apply De nition 9.4.1 to a homogeneous version
of the di erence scheme having the perturbation as its initial condition. With these
restrictions, we may de ne stability in a more explicit form. De nition 9.4.2. A linear di erence scheme is stable if there exists a constant C > 0
which is independent of t and such that k nk < C k 0k
as n ! 1, t ! 0, t T . (9.4.3) 20 Parabolic Problems Both De nitions 9.4.1 and 9.4.2 permit the initial perturbation to grow, but only
by a bounded amount. Restricting the growth to nite times t < T ensures that the
de nitions apply when the solution of the di erence scheme cn ! 1 as n ! 1. When
applying De nition 9.4.2, we may visualize a series of computations performed to time
T with an increasing number of time steps M of shorter-and-shorter duration t such
that T = M t. As t is decreased, the perturbations n, n = 1 2 : : : M , should settle
down and eventually not grow to more than C times the initial perturbation.
Solutions of continuous systems are often stable in the sense that c(t) is bounded for
all t 0. In this case, we need a stronger de nition of stability for the discrete system. De nition 9.4.3. The linear di erence scheme (9.4.1) is absolutely stable if
k nk < k 0 k: (9.4.4) Thus, perturbations are not permitted to grow at all.
Stability analyses of linear constant coe cient di erence equations such as (9.4.2)
involve assuming a perturbation of the form
n = ( )nr: (9.4.5) Substituting into (9.4.2a) yields M + tK]( )n+1r = M ; (1 ; ) tK]( )nr:
Assuming that 6= 0 and M + tK is not singular, we see that is an eigenvalue and
r is an eigenvector of
M + tK];1 M ; (1 ; ) tK]rk = k rk k = 1 2 : : : N: (9.4.6) Thus, n will have the form (9.4.5) with = k and r = rk when the initial perturbation
= rk . More generally, the solution of (9.4.2a) is the linear combin...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
- Spring '14
- The Land