544 Economical Difference Schemes for Multidimensional Problemsimal execution time in g1vmg an approximate solution with a prescribed
accuracy E > 0. The total number of the necessary arithmetic operations
Q( E) for doing so is the main characteristic of the algorithms in question,
since other characteristics such as the quality of the related software and
the availability of advanced-architecture computers are beyond our control.
In view of this, the economy requirement becmnes rather urgent and
extremely important in nmnerical solution of tnultidin1ensional problems
arising time and again in mathematical physics.
To understand the nature of this a little better, we focus our attention
on the simplest examples serving to motivate what is done with economi-
cal difference schen1es and regarding to some prelin1inaries. The object of
investigation is the heat conduction equation in the space RP:(1)OU
ot =Lu'p
Lu= L La U,
Cf= l82 u
Lau= vxa J'l 2 'xEG, tE(O,t 0 ], ulr=O, u(x,O)=u 0 (x).
Let G =Gap= {O < xa < 1, C\'. = 1, 2, ... ,p} be a cube of the dimension
p; wh = {(i 1 h, ... , iph) E G} be a cubic grid with step h in all directions
xa, CY= 1, 2, ... ,p, and w 7 be the grid with step T = t 0 /n 0 on the segment
0 < t < t 0. At the next stage the operator32u
Lau = J:i 2
vxa
is approxin1atecl by the difference operator Any = lJ.c 0 ,,. 0 , so that A
L~=l AL,, leaving us with the weighted two-L1yer schemeYt=A(O"y+(l-O")y), xEwh, O<t=nT<t 0 ,
(2)
Yl1'h = 0, y(x, 0) = y 0 (x).We recall from Chapter 5, Section 3 that scheme (2) is stable with respect
to the initial data under the constraint
By merely setting O" = 0 we have at our disposal the explicit scheme
(3) Yt = A lJ or fJ = y + T A y ,