548 Economical Difference Schernes for Multidimensional Problernsboundary-value problem of the form(8) A;y;_ 1 - C,yi + Bn.J;+i = -F;, z = 1, 2, ... , N - 1,
Yo = ll 1 , YN = ll 2 , Ai > 0, B; > 0, Ci > A; +Bi.
This problem can be solved by the standard elimination method requiring
0( 1 / h) = 0( N) operations, the amount of which is proportional to the
total number ,'V of nodal points of the grid wh = {x; = ih, 0 < i < N}.
With regard to problem (7) posed in the rectangle, it is worth noting
several things. The grid wh at hand may be treat.eel either as a collection
of the nodes along the rows i 2 = 0. l, ... , 1V'J or as a collection of t.he
nodes along the columns i 1 = 0, 1, ... , N 1 , thereby providing for subsequent.
compositions the availability of N 1 + 1 nodal points along every row as well
as N 2 + 1 nodal points along every column.
In trying to solve a typical problem like (8) by the elimination method
for fixed i 2 (or i 1 ), exactly 0( N1 N2) arithmetic operations are needed in
giving a solution at all the nodes of the grid. Their amount is proportional
to the total number of the grid nodes in the plane. The main idea behind
economical methods lies in successive solution of one-dimensional proble1ns
of the type (8) along rows and along colun1ns in passing from one layer to
another.
The scheme ascribed to Peaceman and Rachford provides s01ne
realization of this idea and refers to implicit alternating direction schemes.
The present values y = yn and fJ = yn+l of this difference scheme are put
together with the intermediate value iJ = yn+^1!^2 , a formal treatment of
vvhich is the value of y at moment t = i 71 + 1 ; 2 = t 11 + T/2. The passage
fron1 the nth layer to the ( n + 1 )th layer can be done in two steps with the
appropriate spacings 0.5 T:
(9)(10)These equations are written at all inner nodes x = X; of the grid wh
and for all t = tn > 0. Let us stress here that the first scheme is i1nplicit
along the direction x 1 and it is explicit along the direction x 2 , while the
second one is explicit in the direction x 1 and it is implicit in the direction
x 2. Equations (9)-( 10) are supplemented with the initial conditions
( 11) y( X 1 0) = Uo ( X) 1