416 Stability Theory of Difference Schemes- Stability of the weighted scheme. As an example of applying the
theorems just established the weighted scheme comes first
( 46) Yt + A ( O"Y + ( 1 - O") y) = 'P, y(O) =Yo.
Recall that in Section 1 scheme ( 46) has been already reduced to the
canonical fonn
( 47) ( E + () T A) Yt + Ay = 'P J y(O) =Yo ·
By comparing (46) with (1) we see that B
exists an inverse operator A-^1. Applying A-^1
canonical form of the weighted scheme:
E + O"TA. Suppose there
to ( 4 7) reveals the second
- By 1 + Ay = <p, y(O) = Yo ,
(48)
B=A-^1 +0"rE,
In the sequel the form ( 47) is more convenient for the case of self-adjoint
operators A and the form ( 48) - for the case of non-self-adjoint positive
definite operators A = A( t). We will elaborate on this later.
The first analysis is connected with the case when A is a constant
self-adjoint positive operator A= A* > 0. As we have shown in Section 2,
a necessary and sufficient condition for the stability of the weighted scheme
(47) with respect to the initial data is
1 1
O"o = - - ---
2 rll All.
Under this con di ti on estimate ( 15) holds true for a solution of problem
(47). In particular, for an explicit scheme (for O" = 0) the condition O" > 0" 0
implies T < 2/llAll, that is, an explicit scheme is stable in the space HA
for T < 2/llAll· A sche1ne with T >~is unconditionally stable, that is,
for any T. In Section 2.4 a model exan1ple with Ay = -Ay = -Yxx for
0
y E Dh = Hh has been considered in full details, in which II A II< 4/h^2 and
the appropriate explicit scheme is stable for T < ~h^2.
For the heat conduction equation with a variable coefficient k( x) we
might have
- By 1 + Ay = <p, y(O) = Yo ,
o ( ou)
Lu = ox k( x) ox 'A y = (a( x) Yx) x ,
and 0" 0 < ~ - h^2 /(4c 2 r), while an explicit scheme is stable for T < ~h^2 /c 2.