1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable three-layer schemes 441

6. Weighted schemes. In practice the reader frequently encountered the
   weighted schemes

T < t = n T < t 0 ,

(63)

y(O) = Yo , y( r) = Y1 ,

where y(a-1,0-2) = (J' 1 y + (1 - (]' 1 - (J' 2 )y + (J' 2 y. The accepted view is that

stability and accuracy of the scheme are governed by selection rules for real

numbers (]' 1 and (]' 2. In Section 1 scheme (63) was written in the canonical

form (1), making it possible to recover the operators

(64) R = (]'1 + 2 (]'2 A.

Assumming that there exists an inverse operator A-^1 and applying it,

on the same grounds, to both sides of (1) with operators (64), we obtain

(65)

where

~ 2 ~ ~

B yo t + T R Ytt + A Y = <p ,

y(O) = Yo ,

T < t = n T < t 0 ,

Y( r) = Y1 ,

A-E - )
~ ~
This implies that constant operators A and R are self-adjoint.

<p= - A-1 <p.

On account of Theorem 1 with regard to ( 65) the operator inequalities

hold:

(66) R--A= ~ l~ ((]'1+(]'2 --1) E>O

4 2 4

for

(67) B = A-^1 + ((]' 1 - (]' 2 ) TE> 0 for (]' 1 > (]' 2 and any A(t) > 0.

Theorem 7 If A(t) is a variable positive operator and the conditions

(68)

are fulfilled, then scheme ( 63) is stable and the estimate holds:

t'=r