1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternating direction method^549

and the difference boundary conditions, for example, of the special type

(12)

(13)

where

(14)

The tneaning of the boundary condition (12) is known to us. On the other

hand, condition (13), which assigns the boundary value i), needs certain

clarification. In this way, the difference boundary-value problem (9)~(14)

can be put in correspondence with proble1n (7). The tnethod for solving

this difference problem is mostly based on alternative forms of equations

(9)~(10):

2

• y -Ai y = F, F=~ y +A~ y + 'P,
  T T
  (15)
  2 -


• y -A2y = F, F=

2

y + A1 f) + 'P.

T T

To make our exposition more transparent, it is more convenient to introduce

the new members

Y = Yi1i2
and approve the following rule: when one of the subscripts is kept fixed, we
omit it for a while in relevant expressions. This should cause no confusion
and guides a. proper choice of alternative forms of equation ( 15) for later
use:

(16)

i1 = 1, 2, ... , S 1 - 1 .. fJ =fl for i1 = 0, Ni,

(17)

i2 = 1, 2, ... , N2 - 1, y = p for i2 = 0, N2.