1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference schemes for the equation of vibrations of a string 375

Indeed, from the first Green formula (see Chapter 2, Section 3) it follows

that

• ( Ay, Yo t ) = ( V, Vo t ]

with v = Yx. Having involved in the further development

V · Vo = -1 ( ( V + V). 2 ) - -T2 ( 2)

8

t 8 t (Vt) t ,

we arrive at identity (36).

Substituting (35) and (36) into (34) yields the energy identity

or [j+i = Ej, where

Let us find the values of O", for which the quantity E j is nonnegative when

y j and yj -^1 are arbitrarily taken, by simple observations (see Chapter 2,

Section 3) that

I I Ytx J I^2 < h^4 2 I I Yt 11 2

and

With this in mind, we conclude that the right-hand side of (38) is nonneg-
ative if

(39) ()" >

1

4

1

4,2 ,

T

1=

h

Here the expression ( [J ·)1/2 = 11yJ11.. can be viewed as a norn1 (or, more
exactly, as a seminorm), permitting us to write down

(40) Ej =II Yj II;= II Yf 112 + ( O" - ~) r^2 II Yfx ]1^2 + i II Yk + Yi-^1 ll^2 ·

Nate that such combined norms depending on values of y on several layers
are typical for nrnltilayer schernes. This is especially true for three-layer
schemes.