1549301742-The_Theory_of_Difference_Schemes__Samarskii

362 Difference Schen1es with Constant Coefficients

The next step is to multiply the resulting equation by 2r Yxt = 2 C!Jx - Yx)

and take into account that

with v = Yt. The outcome of this is

With the aid of the relations

N N

L (Yi)x, i h = L [(Yt)I - (Yt)T_ 1 ] = (Yt )jy - (Yt)Z

i= 1 i = 1

and Yt,o = Yt(O,t) = 0, since y(O,t) = 0, we get

(13) r(yt)Jv + 2 ( (O" - !) T + !h) II Yxt 112 +II Yx 112 =II Yx 112

as a final result of multiplying once again by h and summing over all grid

nodes x = ih, i = 1, 2, ... , N. It follows from the foregoing that

if (O" - !) T +! h > 0, giving O" > 0" 0. This scherne is stable in the energy

norm II Y 11(1) = II Yx II·

In Chapter 6 we will deal with the two-layer scheme of general form

(14) B Yt + Ay = 0,

where both operators A and B really act in a prescribed Euclidean space
H, A= A* > 0 and B > 0. A necessary and sufficient stability condition
will be established in the forrn

(15)

Moreover, II yj+i llA < 11 y^0 llA in the norm 11 y llA = J(Ay, y).

In studying problem (12) we refer to the space Hof all grid ~mctions

defined on the grid wh and vanishing for i = 0 and the operator Ay = Yx,
for which the estimate

(Ay, y) =! YJv +! h II Yx l/^2 > 0