1549301742-The_Theory_of_Difference_Schemes__Samarskii

10 Preliminaries

under agreement (10).

By going through the matter chronologically the forward elimination

path for finding o:i, f3i and the backward path for finding Yi do arise during

the course of the elimination method. Let us clarify the situation. With

knowledge of o:; and f3i one can determine, according to (10), all the values

Yi moving from i + 1 to i. To find o:i and f3i, we succeed the reverse order

(from i to i + 1) according to formulae (11)-(12).

Two relevant aspects are worth noting in this context:

1) equations (11)-(12) for finding O:i and f3i, being nonlinear, reproduce the

relationships between the values at two adjacent points;

2) for each unknown o:, (3 or y, it is necessary to solve the corresponding

Cauchy problem.

These requirements necessitate imposing auxiliary values which can

serve as the initial conditions. The assignment of boundary conditions is

aimed at specifying these or those values. Having substituted i = 0 into

( 10), we get Yo = o: 1 y 1 + (3 1. On the other hand, Yo = x 1 y 1 + ~L 1 , giving

( 13)

(14)

Thus, we set up for o: and (3 the Cauchy problems described by (11), (13)

and (12), (14), respectively, thereby completing the forward path of the

elimination.

At the second stage, knowing O:i and f3i, the boundary value yN is

recovered from the system of the equations

=X2YN-1+μ2,

=o:NyN+f3N

under the constraint 1 - o:N x 2 -::/-0.

Hence, the initial condition in question becornes for (10)

(15)

The cornputational formulae (10) and (15) constitute what is called the
backward elimination path.
The algorithm presented below as the sequence of applied formulae is
called the right elimination method and is showing the gateway for the