1549301742-The_Theory_of_Difference_Schemes__Samarskii

428 Stability Theory of Difference Schemes

6.3 CLASSES OF STABLE THREE-LAYER SCHEMES

1. The problem statement. In this section we establish sufficient stability
   conditions and a priori estimates for three-layer schemes on the basis of
   their canonical form

(1)

B Yo t + T^2 R Ytt + A Y = ip( t) , y(O) = Yo , Y( r) = Y1 ,

0 < t = nr < t 0 , n = 1, 2,... , n 0 - 1 ,
Here and below, y 0 and y 1 are arbitrary given vectors of a finite-
dimensional real space H, ip(t) is a given arbitrary abstract H-valued func-
tion of the variable t E w 7 ; A, B and R are linear operators in the space
H. The dependence of y(t) = Yhr(t), ip(t) = 'Phr(t), A(t) =Ahr, B, R, Yo
and y 1 on h and T is not explicitly indicated. We proceed as usual. This
amounts to introducing more compact notations:

Y = y(tn + r) = Yn+l, y = y(tn - r) = Yn-1,

Yt = ( y - y) IT , Yt = ( y - i)) IT ' yo t = (Y - i))/(2 r),

Ytt = (y - 2y + y)/r^2 ,
allowing a simpler writing of the ensuing formulae.
All the tricks and turns remain unchanged by analogy with Section 2:
first, a solution of problem (1) can be arranged as a sum y = i) + y, where
y is a solution of the homogeneous equation:

(la)

B Yo t + T^2 R Ytt + A Y = 0 , y(O) = Yo ,

0 < t = nr < t 0 ,

Y( r) = Y1 ,

and y is a solution of the nonhomogeneous equation with the zero initial

data:

( 1 b)

B yo t + T^2 R Ytt + A Y = ip( t) ' y(O) = y(r) = 0,

0 < t = nr < t 0 ,

Second, an alternative form of writing appears useful:

(2)

(B + 2 rR) Yn+l = n'