1549301742-The_Theory_of_Difference_Schemes__Samarskii

The maximum principle 259

_The grid w is taken to be connected, that is, for fixed points P E w

and P E w there always exists - _a - sequence of neighborhoods {Patt'( P)} such

that the passage from P to P can be done using only the nodes of those

neighborhoods or, in other words, one can select nodes P1, P2, ... , Pm of

the grid w such that

P1 E Patt'(F), P2 E Patt'(Pi), ... , Pm E Patt'(Pm-1), PE Patt'(Pm)

with

B(P;, P;+i) # 0, i = 1, 2, ... , in - 1,

(4)

B(F,P1)#0, B(Pm,F)#-0.

In the case of the difference scheme for the Dirichlet problem (24 )-(26) of
Section 1 the definition ( 4) of connectedness coincides with anothe_r defi-
nition from Section 1. The very definition implies that the point P may
be boundary and, hence, the connectedness is to be understood that every
point of the boundary belongs to the neighborhood Patt'(P) of at least one
inner node.
Within the notation

(5) £ y(P) = A(P) y(P) - B(P, Q) y(Q),
QEPatt'(P)

we may attempt equation (1) in the form

(6) £ y(P) = F(P).

An alternative forn1 of£ y(P) may be useful in the further development:

(7) £ y(P) = D(P) y(P) + B(P, Q) (y(P) - y(Q)).
QEPatt'(P)

In the preceding section the Dirichlet difference problem was set up
in the form (1), (3). Consider as one possible example the so-called scheme
with weights for the heat conduction equation

8u 82 u

8t = ox 2 + f(x, t), 0 < x < 1, t > 0,