Econmnical factorized sche1nes 569Sometimes such a passage pennits one to justify the approximation being
rp instead of cp near the boundary of the grid domain,If the primary scheme (18) is stable, then so is the fac-
torized schen1e (22) in the case, where the operalors R 1 ,
R2, ... , Rp are self-adjoint (Re, = R~), non-negalive
(Rex > 0) and pairwise conunulative (Rex Rr1 = R 13 Rex,
Cl', /3 = 1,2, .,. ,p).By virtue of the indicated properties of the operators Rex their prod-
ucts Rex Rf3, Rex R(3 R 1 , etc., will be self-adjoint non-negative linear opera-
tors. This provides enough reason to conclude that= B + r^2 Ri R 2 > B for p = 2 ,
where Q; = Qp > 0.
Thus, B > B > 0.5 TA, meaning the stability of the factorized scheme
(22). The operators Rex are so chosen as to satisfy the condition of approx-
imation, too. The forthcoming example helps clarify what is done.
Example 1 VVe are looking for a solution of the first boundary-value
problem for the heat conduction equation with variable coefficients
(23)
011
at=L1u+L2u+f(x,t), xEG, t>O,itlr = p(x, t), u(J:, 0) = tt 0 (x),
a=l,2,
G = {O < x°' < l°', Cl'= 1, 2}.