1549301742-The_Theory_of_Difference_Schemes__Samarskii

194 Homogeneous Difference Schemes

By inserting in (50) the estirnates

v = O(h),

0 2

I 77i I= O(h ),

we conclude that II v lie= O(h^2 ) and, hence,

II z lie < 2 II v lie < M h

2

·

This means that scheme (38) is of second-order accuracy in the space C.

Let us turn to a scheme of the second type known as a "scheme on

the current grid". We split up the segment [O, R] into JV parts by the nodes

(current points)

1' 0 - - (^0) , ... ' 1'; = (i - ~) h, ... '
i'N-1 = (N - ~) h, fN = (N - ~) h = R.
Denote by Yi = y( fi) the values of a grid function at those nodes.
The balance equation for (26), which is an analog of (30), is aimed at
designing the difference scheme, making it possible to write on the interval
1'; 1 = f; 112 < r < fi+i; 2 = r; the difference scheme
(51) i = 2, 3, ... , N - 1,
where ri = ih, fi = (i - ~)/h and a, d, tp are chosen by analogy with
(32)-(33), so that in the simplest case
(52) l.p; = f(f;),
The balance equation written on the interval 0 < r < r 1 = h
h
w^1 -w^0 + - (^1) J (f(r) - q(r) u(r)) r dr = 0,
1'1 h 1'1 h
w(r) = r k(r) u'(r),
0
and the condition w 0 = 0 imply the difference equation
(53)
1'1 a1 Y 1
--_-hr, - d1 Y1 + l.f1 = o
1'1
for r = f 1 = ~ h, where a 1 , d 1 , t_p 1 are specified by formulae (52).