1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable three--layer schemes 433

In our basic account A and R are taken to be constant self-adjoint

positive operators and B refers to a non-self-adjoint nonnegative operator:

(22) A= A> 0, R = R > 0, B > 0.

These restrictions permit us to specify and extract a primary family of

schemes.

With the regard to problem (la) identity (18) takes the form

(23) 2r (Byo, t yo)+ t II Y(t + r) 112 =II Y(t) 112 , t = nr '

where

(24) 11 Y(t + r) 112 = % (A(y(t + r) + y(t)), y(t + r) + y(t))

• T^2 ( ( R - i-A) Yt , Yt) ,

(25) II Y(t) 112 = % (A(y(t) + y(t - r)), y(t) + y(t - r))

To avoid cumbersome calculations, we will also use index denotations by
setting Y(t + r) = Yn+l and

II Yn+l 112 = i- (A(Yn+l + Yn), Yn+l + Yn) + T

2

( (R - % A) Yt,n 1 Yt,n)

= t II Yn+l + Yn II~ + llYn+l - Yn 11~_ lA 4 ·

Observe that relation (24) i1nplies that II Y(t+r) 112 > 0 for any y(t) #
0, y(t + r) # 0, provided that the operators A and R - A/4 are positive,
A> 0 and R > A/4.

Theorem 1 Let A = A > 0 and R = R > 0 be positive operators. Then
the conditions

(26)

(27)

B = B(t) > 0 for all t E w 7 ,

R>lA 4

are sufficient for the stability of scheme ( 1) with respect to the initial data.
Under conditions (26) and (27) for problem (la) the estimate holds:

(28) llY(t+r)ll < llY(r)ll,