1549301742-The_Theory_of_Difference_Schemes__Samarskii

Stability of a difference scheme^93

expresses the stability of the problem (10) solution.

The condition maxk I qk I < 1 is fulfilled if -1 < qk = 1 - TAk < 1,

that is, T Ak < 2 for all k = 1, 2, ... , n. For the latter to hold, it is sufficient

that

(13)

2

T< -

Ci'

where Ci = max1c Ak = An.

By a rather similar argument we take y 0 = cen, leaving us with ak = 0

for k = 1, 2, ... , n - 1, an= c and the relations

since I qn I = T An - 1 = T Ci - 1 > 1 if T Ci > 2. For instance, when

T Ci = 2 + p, p > 0, we draw the conclusions that I qn I = 1 + p and

I q~ I = (1 + p)j ____, = as j ____, =· In that case scheme (10) becomes

unstable and makes the method so inefficient as to be unusable.

In mastering the difficulties involved, we impose the initial condition

Yo = c ei, there by providing

where q 1 = 1-TA 1.

It may happen that T > 2/6, but TA 1 < 2 and I q 1 I< 1, which assures

us of the validity of the estimate

(14)

Because of rounding errors, y 0 is determined with son1e error e still subject
to the approved clec0111position
n
e = 2= e" ek,
k=l
which is not surprising. No matter how the value e 11 is chosen, there always
exists j 0 such that I en 11 qn jJo > M 00 , where M 00 is computer infinity,
meaning that for j = j 0 abnormal termination occurs because I qn I > 1.
Thus, scheme (10) turns out to be unstable with respect to rounding errors
under any initial condition in the case where T Ci > 2.
For the solution Yj of problem (10), the requirement of having an
estimate similar to inequality (8) necessitates imposing one more condition
l - TA 1 > T >.,, - 1 or, what amounts to the same,