Stability of a difference scheme 95Table 2Variant (^1) Variant 2
J Yj zj J Yj Z· J
8 -5.52 .10-^3 5.52. 10-^3 8 0.227 0.227
9 2.76 .10-^3 -2.76 .10-^3 8 0.193 -0.193
10 -1.38 .10-^3 1.38. 10-^3 10 0.164 -0.164
17 1.08-10-^5 -1.08 .10-^5 13 0.101 -0.101
18 -5.46. 10-^6 5.33. 10-^6 14 8.49. io-^2 -8.61-lo-^2
19 2.82 .10-^6 -2.57 .10-^6 15 7.56 .10-^2 -6.96 -10-^2
20 -1.60-10-^6 1.09. 10-^6 16 4.69 .10-^2 -7.65 .10-^2
25 8.18 .10-^6 8.10 .10-^6 17 0.127 2.16. 10-^2
26 -1.63 .10-^5 -1.62 .10-^5 18 -0.326 -0.415
27) 3.26 .10-^6 19 1.89 1.81
28 -6.51 .10-^5 20 -9.23 -9.30
30 -2.60 .10-^4 21 ) 4.63.^101
50 -2.73 .10^2 22 -2.32 -10^2
68 -7.16-10^7 38 -:3.53 -10^13
69 1.43 -10^8 39 1.77-10^14
70 -2.86 -10^8 40 -8.83 -10^14
105 CX)^46 CX)
*) Once started with this number j we have zj = Yj.
Observe that in variant 1 the accuracy in specifying the first eigenvec-
tor ei is of significant importance. For the values
(17) Yo= 1, z 0 = -1,
abnormal termination occurred at the 197th time step. At the beginning
the variables Yj and zj continue down to some quantities of order 10-^19 for
j ~ 65-70 and then begin to grow. Unlike this tendency, the initial data
( 17) did not change essentially the results of computations for variant 2:
abnorn1al termination occurred for j = 4 7.
If for /j = 1 and .6. = 2 we could take the value T = 2.1/ .6., for which
the condition T < ~ is slightly violated (r is exceeded by 5%), the same
still holds at the 760-790th time steps.
The preceding examples provide enough reason to conclude that the
concept of stability with respect to the input data is identical with the con-
cept of continuous dependence of the solution of a difference problen1 upon