1549301742-The_Theory_of_Difference_Schemes__Samarskii

318 Difference Schemes with Constant Coefficients

and taking into consideration the trivial identities

y 2 1 ( y A + y ) - 2 1 T Yt'

(T y + ( 1 - (T) y = ( (T - ~) T Yt + ~ (ii + y).

Putting these together with (16b) we recast the probletn concerned as

Yt - (er - ~) T A Yt - ~ A ( ii + Y ) = zp ,

( 42)

y(x,0)=0, y(O, t) = y(l, t) = 0,

which reduces by multiplying equation (42) by 2 TYt h = 2 (ii - y) h and

summing the resulting equality over the inner nodes x = ih of the grid wh

to

By appeal to the Green difference formula derived in Chapter 2, Section

3.1

(Av, W) = ( V;;;x , W) = -( Vx, Wx],

with v = Yt, w = Yt and v = y + y, w = y - y incorporated, we find in view
of Yo = YN = 0 that

Substitution of these expressions into ( 43) leads to the energy identity

which is valid for any er. Let er> cr 0 • The accepted view is that a reasonable
form is

By virtue of the estimate emerged in Chapter 2, Section 3

( 45)