1549301742-The_Theory_of_Difference_Schemes__Samarskii

402 Stability Theory of Difference Schemes

With the relations A< II A llE and E >A/II A II in view, we obtain

B-0.5rA> (1/llAll+(cr-0.5)r)A.

The preceding inequality implies that condition (14) is equivalent to the

constraint

This condition is necessary and sufficient for the stability of the weighted

scheme, which interests us.

In the case of a model heat conduction scheme

Yt + A (cry + ( 1 - er) y) = 0 , A y = Yxx , 0 < x = ih < 1 , hN = 1 ,

y(O, tn) = 0, y(l, tn) = 0, y(x, 0) = u 0 (x), x = ih E [O, 1],

corresponding to the first boundary-value problem for the heat conduction

equation

O<x<l, t>O,

u(O, t) = u(l, t) = 0, t > 0 1 U ( X 1 0) = u 0 ( X) 1

we deduce that (see Chapter 5, Section 1)

0

A-- -A ' Ay = Ay for y ED h = Hh,

4 27rh 4 1

II A II= h2 cos 2 < h2' CTo = (^2 4) T cos (^2) "h ·
2
The condition er> iJ" 0 , iJ" 0 = h^2 /( 4r) > cr 0 , has been found in Chapter
5 by the method of separation of variables.
Suppose now the operator A > 0 not to be self-adjoint. Then scheme
(18) does not belong to the primary family. However, it can be replaced by
an equivalent scheme from the primary family. Since A > 0, there exists an
inverse operator A-^1 > 0, whose use with regard to equation (18) permits
us to confine ourselves to

• 
  
  
  
  • B Yt + Ay = 0, B=A-^1 +crrE, A=E.