1549301742-The_Theory_of_Difference_Schemes__Samarskii

434 Stability Theory of Difference Schemes

where II Y II is defined in accordance with rule (24).

Indeed, for B > 0 identity (23) implies that

II Y(t + r) 112 <II Y(t) 112, II Y(t + r) II< II Y(t) II<···< II Y(r) II·

Remark 1 Theorem 1 holds true under the constraints

A> 0.

However, in that case II Y II may fail to be a norm, but it is always a

seminorm.

Remark 2 The three-layer scheme (la) can be reduced to the two-layer

scheme

(29) BYi+AY=O, Yli E H^2 ,

where Y = Yn E H^2 , Yi = (Y;,+1 - Yn)/r, A and Bare operators in the
space H^2. This can be done using the vector

and treating the operators A and B as operator matrices with elements

being operators in the space H:

(

B+0.5rA

B=

-r(R-:1)A)

If A = A, R = R, then the operator A : H^2 ~ H^2 is self-adjoint:
A = A*, while the operator B is non-self-adjoint. All this enables us to
conclude that

In addition, we have A > 0, provided that A > 0 and R > t A.
It is easy to verify that the stability condition in the space H~ of the
two-layer scheme (29)

(30)

T

B>~A - 2