Econmnical factorized schernes 575Knowing the values y1 -l and y1 we do follow the same algorithm for
determination of the value yJ+l, whose use permits us to find thatB1=E+20" 1 r A1, B2 = E + 20" 1 r A2.
A similar procedure works for the boundary conditions imposed for w(lJ as
was done for a two-layer factorized scheme and for this reason it is omitted
here.
Stability of the factorized scheme (35) can be established on account
of the general theorems from Chapter 6, Section 3, clue to which it follows
from the foregoing that the conditions
0" 1 > 0" 2 , 0" 1 + 0" 2 > 0.5, Aa =A:> 0, Ai A2 = A2 Ai
are sufficient for the stability of the scheme concerned. If 0" 1 > 0" 2 , 0" 1 +0" 2 >
0.5, Aa = A~ > 0, then the primary scheme is stable, since B > E and
4R > A. As far as the operators ~ Ai ~ and A 2 are commuting, we deduce
that A 1 A 2 > 0, meaning B > B and R > R. Due to this fact the stability
of the primary scheme implies that of the factorized schen1e (35).
A particular case where R = O" A, O" = 0. 5 ( 0" 1+0"2), is showing the gate-
way to the future research, whose aims and scope are connected with the
general method for constructing three-layer economical factorized schemes
by means of the regularization principle of difference schemes. A simple
example
(36) y 1 +r^2 RYrt+Ay=r.p. O<t=jr<t 0 , y(O)=u 0 , y(r)=u 0 ,
can add interest and help in understanding. Later we will elaborate on
this for rather complicated cases. Here the value y( T) = u 0 for t = T is so
taken as to provide a second-order approximation in T. Also, the stability
property guides a proper choice of the operator R.
That is why a reasonable form of the primary scheme is
(37) (E+2rR)yt=-F, F=(2rR-E)y 1 -2Ay+2r.p.
Let R be a sum of "economical" operators such that R = Ri + R 2 + · · · + Rp.
By replacing in ( 37) the operator
p
E + 2 TR = E + 2 T L Ra
a=i