1549301742-The_Theory_of_Difference_Schemes__Samarskii

198 Homogeneous Difference Schemes

After replacing the variable by r = r; + sh we find that

r;+1/2 1/2

h ~ 2 j Jr^2 dr = : 2 j J(r, +sh) (r, + sh)^2 ds

z ''i-1/2 ' -1/2

1/2 1/2 1/2

J

2h J h

2

J(r; +sh) ds + ~ s J(ri +sh) ds + r[ J J(ri +sh) s^2 ds

-1/2 -1/2 -1/2

2 h2 I h2 2

= Ji+ O(h ) + ~ Ji+ 12 r2 Ji+ O(h )

z z

= (1+ l~~T) J; + 6h:i Jl +O(h2).

Also, a similar expression can be derived for the second integral by merely
setting qu in place of J. Frorn here and formula (63) it is plain to show
that
1/J; * = h 2 ( )' ( 2)
6 r qu - J i +^0 h ,
'
if tpi and d; are specified by the formulae

(

h2 ) ( h2 )

(64) tp;= 1+12r2 J;, di= 1+12r2 q;,

z '

i = 1, 2, ... , N - 1,

or by other formulae differing from (64) only by the quantity O(h^2 ).
Arguing as in the preceding section, we verify that the residual in the
boundary condition reduces to
ai u,. o
(65) v= h,' -q 0 u 0 +J 0 =0(h).

To develop those ideas, we contrived to do it with further reference to the
problem for the enor z = y - u

(66) Az=-1/J, rEwh,

A solution of this problem can be estimated in a similar way as was done
in the preceding section, but with b; = rz_ 112 a;. In concluding this sec-

tion we establish through such an analysis that scheme (59)-(61) converges
uniformly at the rate O(h^2 ):

llzllc = lly-ullc = O(h^2 ).