596 Economical Difference Sche1nes for Multidhnensional ProblemsAn alternative form of writing includes the intermediate value yj+l/^2 and
a smaller step T in two times:yj+l - yj+l/2.
-----=(}A 11+1.
T
Here the passage fro1n the jth layer to the (j + 1 )th layer is canied out in
the following two steps: the first one involves the explicit schenrn and the
second one - the implicit scheme as suggested above.
Of course, this example is not of global character and can serve mainly
as an illustration of such theory. It should be noted that in the general case
the elimination of intermediate values with fm·ther reduction to a scheme
including the values of y only at integer steps inay be impossible and even
meaningless in the theoretical research.- Reduction of a nmltidin1ensional problen1 to a chain of one-din1ensional
problems. The multiple equation we 111ust solve is
(5)OU
m=Lu+f(x,t), O<t<t 0 ,where L is a linear differential operator acting on u(x, t) as a function of
x, x = (x 1 , x 2 , ... , xp) is a point in the p-dimensional do1nain G with the
boundary f, on which proper boundary conditions are i1nposed in one or
another convenient way. That does not matter for us in subsequent discus-
sions. An effective tool in designing economical methods is the accepted
decomposition
L = Li + L2 + · · · + Lp
with si1nplifiecl operators. For exa1nple, if Lu = t..-u and f,o:U = 82 u/ C);r^2 '
then Lo: is the operator of the second derivative with respect to the argu-
ment xo: (operator of one variable).
Furthermore, to problem ( 5) there corresponds the first chain of "one-
di1nensional" equations by reducing either (5) or
Pu=m-Lu-Jx,t OU '( ) =0
to p
L Po: 1l = 0'
o:=l1 OU.
Po: u. = p at - Lo: u - j o: ,