1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable two-layer schemes^407

we find upon substituting (28) into (la) that

Putting this together with the orthogonality of the system { ~k} we arrive

at

N N

y(t + r) = L ck(t + r)~k = L (1-T ,\k) cdt)~k.

k=l

The norm II y(t+r) II~ = (Ay(t+r), y(t+r)) can be most readily evaluated
by observing that

All this enables us to deduce that

yielding

Whence it follows that

(29) II y(t + r) llA < II y(t) llA < II y(O) llA

if

k=l,2, ... N.

This condition provides support for the view that -1 < 1 - T A1; < 1 or

(30) k=l,2, ... N.