566 Economical Difference Schernes for Multidiinensional ProblemsEspecial attention is being paid to the factorized schemes with the members(8)p
R2Y = - L
a= 1where R 1 and R 2 are "triangle" operators with associated triangle matrices.
These fall within the category of explicit alternating direction sche111es. Let
us stress here that the operators R1 and R 2 111ay be, generally speaking,
non-self-adjoint, but they are alwayt> n1l1tually adjoint to each other. In
that case a solution to equation ( 3) can be found by "through execution"
formulas.
Undoubtedly, the reader con1es across difference operators Ba of the
structure Ba = E - (}TAa, where Aa approxin1ates the differential operator
La with partial derivatives of one argument a:a. For exan1ple, if Lcrv· =
EP·u/ox~, then Aay = Yx 0 "' 0 is a three-point operator, whose use pern1its
us to solve equation (3) by the eli1nination 1nei.hocl. It is worth 111entioning
here that any difference schen1e can be reduced to a sequence of sin1pler
schemes in a number of different ways. This is certainly so with scheme
(1), implying thatwhere w1 is a solution to the equation(SJ) <PJ = r.pi - A yi.
The value wi can be recovered from the governing syste111 of p equations
(10) B1w(l)=<P^1 , Baw(a)=w(a-l)> Cl'=2,3, ... ,p,
with further reference to( 11) "Uj ' -- W Ip)·It is worth recalling here that the first econon1ical sche111es were intended
for the elin1ination of intermediate values with no problems. The main idea
behind this approach is to involve factorized schemes "in integer steps",
a key role of which is to relate the values yi and yi +^1 in some or other
convenient ways.