620 Economical Difference Sche1nes for Multidimensional Proble1nsIf (Co:μ) = c- is a lower triangle matrix and all the operators Cao:
are invertible, then the procedure of solving equations ( 42') can be reduced
to successive solution of the equationsc~ -1
C 0: 0: "j+o:/m iJ ~ ~ D o:O yi ~ L ~ C 0: ;3 ' J 1J+/3/m + J'i 0: , ex = 2, 3, ,.. , 1n.
/3=1
Such a triangle additive scheme will be economical once we involve econmni-
cal diagonal operators C o:o:, ex = 1, 2, ... , m. Economical schemes arising in
practical implementations of multidimensional 1nathematical-physics prob-
len1s turn out to be triangle additive schemes (usually lower, but sometimes
upper), whose 1natrices are of a special structure. As a rule, nonzero ele-
ments of the matrix (Co:μ) stand only on one or two diagonals adjacent to
the main diagonal. With this in mind, the schemeC aa Yj+o:/m + C aa-l , Y "j+(o:-1)/m -~ D aO Yj t + Jj a , ex= 1,2,. .. ,m,
may be of assistance in achieving the final aims. In particular, when Dao =
0 for all the values ex= 1, 2, ... , m, the preceding reduces to
C O'O'c yj+n/m + C ~'aa-l:J "j+(o:-1)/rn ~ - Jj Ll'lone special case of which is the weighted schen1e
, j+cx/m ~, j+(o:-1)/m
Y Y + Acx (a-o: yj+o:/m + (1 ~ a-o:) y1+(o:-l)fm) = 'P~.
T
Such locally one-dimensional schemes were investigated before, all the triks
and turns remain here unchanged. The following issues are yet to be an-
swered in the possible theory:
1) the estimation of stability and accuracy of an additive schen1e;
2) the design of an economical additive sche1ne for a multidimensional
problem in mathematical physics.- Methods for the convergence rates of additive schemes. So far we have
established many times that approximation and stability of a difference
scheme provide its convergence. For additive schemes we shall need stability
with respect to the right-hand side so that it follows from the condition of
summarized approximation