1549301742-The_Theory_of_Difference_Schemes__Samarskii

30 Preliminaries

with the well-established notations of the right and left differences: 6. Yi =

Yi+i - Yi and VYi =Yi -Yi-1, so that \ly;+1 = 6.yi.

vVith this relation established, we find in a step-by-step fashion that

Yi 6.vi + Vi+16.Yi =Yi (vi+l - vi)+ Vi+l (Yi+l - Yi)

= Yi+l Vi+l - Yi Vi = 6. (Yi v;).

Likewise, one can check the second formula based on another relation

v (Yi v;) = 6. (Yi-I Vi_i) ·

An important role in the theory of difference schemes is played by the

identities serving on this basis as grid analogs of integration by parts:

b b

j 1/ v dx uul~ - f ilv^1 clx

a a

For any functions Yi = y( i) and v; = v( i) defined on the grid w = { i =

0, 1, 2, ... , N}, it will be sensible to introduce analogs of the integral

b

(u, v) 0 :=fa uv dx:

N-l

(y, v) := L y;v;,

i=:l

N

(y, v] := L Yi Vi ,

i=:l

N-l

[y, v) := L Yi Vi.

i=:O

vVith these, the summation by parts formula hholds:

( 48) (y, 6.v) =-(v, \ly]+yNvN -y 0 v 1.

With this ain1, we replace Yi 6. Vi by its expression ( 4 7):

Yi 6. v; = 6. (y; vi) - v1+1 v Yi+1

in the smn

N-1

(y, 6. v) = L Yi 6. Vi

i=:l

N-1 N-l

L 6. (y; vi) - L Vi+l v Yi+l

i=:l

N

= YN vN - y 1 v 1 - L Vi, v y;1,

i'-::::2