1549301742-The_Theory_of_Difference_Schemes__Samarskii

Mathe1natical apparatus in the theory of difference sche1nes 99

As in the previous case simila1· formulae of two types

(3)

(4)

are valid for grid functions keeping the notations

N-1 N N-1

(5) (u, v) = L u; V; h, (u, v] = L 1l; v; h,

i = 1

[u,v)= L uivih.

i = 1 i = 0

Let us prove, for instance, (3). First, applying formula (1) yields

N-1 N-1 N-1

(6) L (uvx)i h = L (uv)x,i h - L (ux vC+l)); h

i=l i = 1 i= 1

N

= (uv)N - (uv) 1 - L (ux v); h

i = 2

N

= (uv)N - u 1 v 1 - L ·ux,i V; h + (ux v) 1 h.

i=l

Second, it is obvious that h( ux v) 1 = u 1 v 1 - u 0 v 1. Su bstit u ti on of the
preceding into (6) along with (5) yields (3).
In what follows we will frequently employ a non-equidistant grid de-
noting it by w h in contrast to an equidistant in trod need in Section 1. On
any such grid the formulae of the inner product and the difference .summa-
tion by parts formulae are somewhat different:

(7)

N-1

(u, v). = L 1l; V; n;,

i=l

N

(u,v] = L uiv;h;,

i=l

N-1

( u, v) = L 1li V; hi+1 ,

i=l

where n; = ~(h; + h;+ 1 ). The notations for the difference derivative on a
non-equidistant grid