1549301742-The_Theory_of_Difference_Schemes__Samarskii

94 Basic Concepts of the Theory of Difference Schemes

Assuming this to be the case, we deduce in light of the obvious relations

1 - T .. \ < exp{-,\ T} and I q 1 I > I qk I for k > 1 that

11 Y j 11 < m:x I q k I j 11 Yo 11 < ( 1 - T ) f 11 Yo 11 < exp { -\ t j } 11 Yo 11 ·

We have here one of many examples reinforcing the view that the

general ideas of stability are sensible. Consider a system of two equations

du

-+au+ bv = 0,

clt

dv

di +bu+ av= 0

with the matrix A= ( ~! ) , where a= ~ (.6. + b) and b = ~ (.6. - b). It

is straightforward to verify that its eigenvalues and eigenvectors are equal

to

With these entries, scheme (10) reduces to

(15)

y 1 +i - y· 1

T + a Yj + b zj =^0 ,

ZJ+l - Zj

~--~+by·+az T J J =0.

j = 0,1,2, ... '

Observe that the vector {Yo' Zo} coincides with the first eigenvector el if we

agree to consider

(16) Yo= )2,

After scrutinising the available information we initiate the review of final

results of calculations for problem (15)-(16) with the following values of

parameters:

1) b - 1 .6.-2

T= -·^3

• ' - , .6. ,

2) b = 10, .6. = 400,

6

T = -.6..

In both cases I l - Tb I < 1 and, therefore, for a solution of problem
(15)-(16) estimate (14) is valid, but T > 2/ .6.. Because of rounding errors,
the computational process is unstable: for large j the growth of its solution
causes abnormal termination in the computer realization of the algorithm.
The final results of calculations are tabulated in Table 2 with the
values of the functions Yj and zj for several j. For variants 1 and 2 abnormal
terminations have occurred at the 105th and 46th time steps, respectively.