568 Economical Difference Schemes for Multidimensional Problemsand the boundary conditions are imposed to be( 17)that is, w 11 J = 0 and w 1 ~ 1 = U on the boundary ih if pis independent. oft.
Observe that in giving scheme ( 1) in inatrix (operator) form it is pos-
sible to take into consideration the hon1ogeneous boundary conditions by
rearranging the right-hand side cp at the near-boundary nodes. The design
of the factorized scheme also involves the homogeneous boundary conditions
(Y(i) = yj = 0, w(l) = w( 2 ) = 0 for x E 11 .), but the retention of the approx-
i1nation order necessitates imposing the extra member -0'^2 r^2 h;^2 A 2 μt on
the right-hand side of this scheme at the near-boundary nodes for i 1 = 1
and i 1 = N 1 - 1.- Constructions of economical factorized schemes. Using the regulariza-
tion method behind, we try to develop the general method for constructing
stable economical difference schemes on the basis of the primary stable
scheme
(18)yn+l _ y"
B + Ay" = cp"
T
with an operator of the structure(19) B=E+rR.
The relation B > 0.5 r A is ensured by the stability property of this sche1ne.
In such a setting it is preassumed that R is a sum of a finite number
of "econon1ical" operators Rex, Cl' = 1, 2, ... , p:
(20) R = Ri +···+RP.
The operator B can be factorized by replacing B = E + T ( R 1 + · · · + Rp)
by the factorized operator
(21) Bex = E + T Rex ,
making it possible to ignore the primary sche1ne (1) in favour of the factor-
ized scheme
(22) B1 · · ·BP Yt + A y = (jJ.