1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference schemes as operator equations. General formulations 139

structure [y, v] = L~~l Yi vi h + ~ h (Yo v 0 + yN vN ). Then problem (9)

reduces to an operator equation of the form

Ay=<p,

where y and <p are the vectors of the dimension N + 1:

and the operator A acts in accordance with the rule

i = 0,

(54) ( Ay ); =

i = N.

The negative norm of operator (54) is expressed by the collection of formu-
lae

f!., L_, h 8 1 2 + ~ (J' (^8) N+l^2
i=l 2
i-1
Si = ~ h 'Po + L h 'Pk , i = 2, 3, ... , N,
k=I
N-1
SN+1=~h(<po+<f'N)+ L h<pk
k=l
and admits the estimate
(57)
In establishing these relations the grid should be enlarged by recording
two "artifical" points x 1 = -h and x N+i = 1 + h and assigning the values
y( x 1 ) = y 1 = 0 and y( x N+i ) = YN+i = 0. All this enables us to impose
the boundary condition in (9) for i = 0 as follows:
( Y1 - Yo) - ao( Yo - Y.1)
h2 = 'Po '

• 1
  'Po = 2 'Po ·