1549301742-The_Theory_of_Difference_Schemes__Samarskii

14 Preliminaries

node i = i 0 , 0 < i 0 < N, and to compute the coefficients a; and /]; for

0 < i < i 0 + 1 according to formulae (10)-(15):

B;

i = 1, 2, ... , z 0 ,

C; - a; A; '

A z /]· ?, + F i

C; - a; Ai '

i = 1, 2 ,... , ·1 0 ,

and for i 0 < z < N the coefficients ~i and l/i can be found by fornmlae

(17)-(20):

A;

~i C; - ~;+

1 B; '

B; l/i+1 + F;

l/i =

C; - ~i+1 B; '

i = N - 1, N - 2,... , i 0 ,

i = N - 1, N - 2,... , z 0 ,

The next step is to put the first solution of the form (10) together with the

second one of the form (9) the node i = i 0. The outcorne of this is

yielding
B; 0 +1 + a; 0 +1 17; 0 +1
1 - O'.i 0 +1 l/i 0 +1
As far as 1 - a; 0 + 1 l/io+i > 0, this forrnula is rneaningful because under
conditions (16) at least one of the modules: either I a; 0 +1 I or ll/io+i I is less
than 1. With knowledge of y; 0 the re111aining values Yi arise in the process
of parallel calculations: for i < i 0 by formula (10) and for i > i 0 by formula
(19).
Of course, the counter elimination rnethod could be especially effective
in an at tern pt to deterrnine y; merely at only one node i = i 0.

8. Maximum principle. To make our exposition more transparent, the case
   of interest is related to the first kind boundary-value problem with x 1 = 0
   and x 2 = 0:

[, [y;] = A; Yi-1 - C; Yi+ B; Y;+1 = -F;,
(21)
i = 1, 2, ... , N - 1,