1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference equations 13

7. The left elimination method. The counter elimination method. Still
   using the framework of the right elimination method (formulae (10)-(15))
   in reverse order, we obtain through such an analysis the computational
   formulae of the left elimination rnethod:

(17)

(18)

(19)

and

(20)

(~) Ai

~ i = C; - ~i+1 B; '

(~) 1 Bi 7/i+r + F;

7i = -----

C; - ~i+1 B; '

i = N - 1, N - 2, ... , 2, 1, ~N = X2;

i = N - 1, N - 2, ... , 2, 1, 7/N = μ2;

(--+)

Y i+l = ~i+1 Yi+ 17;+1, i = 0, 1, ... , N - 1,

Indeed, with the relation Yi+1 = ~i+l Yi + 7];+1 granted we eliminate suc-

cessively Yi+1, y; = ~i Yi-I + T}i frorn equation (9). The outcome of this

IS

-Fi= A; Y1-1 + (B; ~i+1 - C;) Yi+ B; 7/i+r

= [A; - (Ci - B; ~i+1) ~;] Yi-1 + B; 7/i+1 - (C; - B; ~i+1) 7];.

Equation (9) is valid if we agree to consider

A; - ( C; - B; ~i+1) ~i = 0 ,

from which formulae (17) and (18) immediately follow. The value y 0 can

be found from the condition Yo = x 1 y 1 + μ 1 and the formula y 1 = ~ 1 y 1 +7] 0.

The presented inequalities

confirm that the correctness and stability of the left elimination method
are ensured by conditions (16), since l~il < 1 for all i = 1, 2, ... , N.
Joint use of the left and right elimination n1ethocls refers to the counter
elimination 111ethod. The essence of this method is to consider a fixed inner