1549301742-The_Theory_of_Difference_Schemes__Samarskii

36 Preliminaries

We note in passing that formulae (71 )-(72) just established are very flexible

and more convenient for computational procedures in the case a; > > 1.

The requirement ai < < 1 necessitates making some modifications for

stability of this or that difference scheme. As a final result of minor changes,

the recurrence formulae have the representations

(71*)

(72*)

a;+1 ( o:; + d;)

ai+1 + (o:i +di) '

Under conditions (63) forn1ulae (71) and (71 *) imply that o:; > 0. This
provides support for the view that the coefficient o:; / ( O:i + di) in (70) is
always less than 1, providing stability in the process of calculations of the
flow w;.
For determination of Yi we rely on the appropriate formulae: for ai > >
1

(7 4)

andfora;<<l

(75)

Formulae (7 4)-(75) show that the eli1nination method is stable. The
values w"' and y"' should be known before proceeding to the applications
of (70), (7 4) and (75). For this reason we involve here the second boundary
condition (67) and relation (69) for i = N:

IN a"' ( l - X2) - o:"' a"' v 2

a N ( l - X2 ) + X2 Cl'. N

Observe that under conditions (63) the denominators in relevant expressions
are always positive.
To distinguish two essentially different approaches, the final algorithm
is of rather complicated structure and consists of the following steps: