Heat conduction equation with several spatial variables 347The maximum principle can be applied to any such scheme with
weights under the constraint T < To, where1 ( p 1 )-!
To = 2 ( 1 - O") ~ h; ,by means of which it is not hard to show that for O" = 1 there is no limitation
on T. Because of this, a priori estimate (11) for the problem solution is
certainly true. Indeed, it is possible to reduce (14) to (6), by means of
which
p
A(P) = 1+2 ()" L ~ ,
a=I a
T
B(P, Q) = ()" h2 ,T
(1 - ()") h2 ,
°' °'
The restriction T < T 0 follows from the initial assumption concerning the
nonnegativity of the coefficients B(P, Q). It is clear that D(P) = 0.
Instead of (14) it is reasonable to deal with the scheme with different
weights O" °' related to the directions x °':
p
Yt = L Aa ( O" °' Y + ( 1 - O" °') y) ,
a=lfor which
- The scheme with increased accuracy. For the stationary problem
6.u=-f(x),
in a parallelepiped the scheme of accuracy 0( I h 14 ) has been constructed
in Chapter 4. In the two-dimensional case p = 2 this scheme reduces to
A' y = -<p,where
h2 h2
'P = f + 1 ~ Ar f + 1 ~ A2 f.