1549301742-The_Theory_of_Difference_Schemes__Samarskii

100 Basic Concepts of the Theory of Difference Sche1nes

and the inner product on a non-equidistant grid (, ). allow a simpler writing

of the ensuing formulae.

To prove (7), a simple observation that

may be useful. Indeed, substituting this expression into the inner product

N-1

where (u,w)= L 1l;W;h;+ 1 ,

i=l

and following the proof of identity (3), we arrive at (7) as desired.

3) The first Green formula. The equality

1 1

j 1l (kv^1 )1 dx = - j kH^1 v^1 dx + kuv^116

0 0

is usually called the first Green formula.

For grid functions an analog of the Green formula can be obtained by

the summation by parts formulae. This can be done by substituting

1l = z,

into (3). The outcome of this is the first difference Green formula

(8)

If z 0 = zN = 0, the last two terms in (8) vanish and the first Green formula
appears in simplified form:

(z, Ay) = -(ayx, zx],

In particular, for z = y the preceding becomes

(8//) Yo= YN = 0.

In the case of a non-equidistant grid we obtain a similar result:

(z, (ayx),J. = -(ctyx, zxl + azyx IN - al Yx,o Zo,

(z, (ayx)i:). = -(ay,", zxl for z 0 = zN = 0.