636 Econmnical Difference Sche111es for M ulticli1nensional Proble111s- Additive schemes for a syste111 of equations. Later in this section we
will survey s01ne devices that can be used in trying to produce additive
schemes for systems of parabolic equations. With this aim, problem (50)
we have completely posed in Section 2 will serve as a basis for the up-to-
date presentation of tools and techniques, their theory and applications. In
this connection we n1ay attempt the operator L in the fonn L = L- + L +
with "triangle" operators L-, L +, the associated matrices kaa of which
arrange themselves as smns k aa = k-era + k+ aa' where k-aa = (k-srn) aa and
k+ aa = (k+sm) aa are triangle matrices with entries
Observe that the matrices k-O'O' and k+ O'C\.' are svn1n1etric " each to other, since
k-sm CYC-¥ = k+ms aa ' whence it follows that.By introducing a few auxiliary operators with !.he properties
et-1 Ct
L; u = L;a ll + ~ Lap u = ~ L~(3 ll, L;;(3 '1l = Ln(3 ll for {3 <CY,
(3:=1 (3=1
p p
Lt u = Lta u + ~ Laμ u = ~ Ltμ it, for f3>CY,
f3=n+1 /3 =et
and representing the operator L by a sun1(8:3) L+ Ct ·u J
n=l
the solution of the system of equations (50) or( 84)
where p
2= (r; + (t) = r,
<->=1