Economical factorized schemes 567- The boundary conditions. As one might expect, stability and approx-
i1nation take place for the factorized scheme (1). In this view, it see1ns
reasonable to adopt equations ( 6) or ( 10)-( 11) as a perfect computational
algorithm in designing the factorized sche1ne ( l ). But this equivalence can
be established only with consistent boundary conditions and needs certain
clarification.
In the forthcoming exa1nple the first boundary-value problem is posed
for the heat conduction equation in the rectangle Go = {O < x 1 < 11 , 0 <
X 2 < 12 } with the boundary f:
(12)OU
8t = (L1 + L2) u + .f(x, t), uir = μ(:e, t), u(x, 0) = u 0 (a:),where Lau= 82 u/8x~, Cl'= 1, 2.
'vVe begin by placing the factorized sche1ne (1) with Acxy = Yxnxn on a
rectangular grid wh = { ( i 1 h 1 , i 2 h 2 )} with steps h 1 and h 2 in the specified
domain Go:(13) B1 B2 Yt = Ay + 'P, y^1 !,,Ii = μ^1 , y^0 = 1t 0 (:r).
Here and in all that follows Bex = E - (J"T Acx, A = Ai + A2 and /h is the
boundary of the grid wh. In passing from one layer to another algorithn1
(6) is perforn1ed for problem (13):( 14)which is put together with the boundary conditions yJ+i I l'h = 1-t1+^1.
As far as the operator B 1 B 2 on wh is concerned (including the bound-
ary x 1 = 0, x 1 = 11 ), the equation B2 y1 +i = Y(i) should be valid not only
for 0 < x 1 < 11 , but also on the boundary for x 1 = 0 and x 1 = 11 • Since
the values yJ+l I = pJ+I are already known, it follows from the foregoing
l'h
that
(15) Y(i) = (E - (]" r A2) /lJ+I = μ^1 +^1 - (]" r A 2 μJ+l for x 1 = 0, 11 •
When specifying Y(i 1 for x 1 = 0, 11 by the preceding formula, problen1s
(13) and (14)-(15) become equivalent. This fact can easily be verified
during the course of the elimination of Y(I) from (14). In the fran1ework of
the second algorith1n we accept
( 16)