1549301742-The_Theory_of_Difference_Schemes__Samarskii

Other problems^195

The standard condition

(54)

is imposed for r = R. As a final result we get the difference equation (51)
subject to the boundary conditions (53) and (54).
Let ~7 = y(f) be a solution of this problern. We begin by placing the
following problem for the error z = ~7 - u;

(55) r; = (i - ~) h, i = 2, 3, ... , N - 1,

r 1 a z 1 r, 1 _ d + ,/, _ O

r- I^1 z^1 <r1 - ,

1 i

where 1jJ is the approximation error equal to

1

1/J; = 1/;(f;) = -=-- (ri_ 1 a,:_ 1 ur,i),. i - d; ui +'Pi, i = 2, 3, ... , N - 1,

ri ,

with
u; = u(r;) = u;_ 112 •

Combination of the resulting expressions and the balance equation on the
interval r; _ 1 < r < ri gives

i = 2, 3, ... , N,

1'i

1/J,* = tp,· - 'Pi^0 - (d - d)^0 u + -^1 J

z z ' f. l h rq(r) (u(r)-u(f;)) dr = 0,

where
r· z r· z

(^0 1)
J
(^0 1)
J
'P· = f.h r f(r) dr, d
f; h
z z rq(r)dr.
z 1·.
z-1 r1:-1