1549301742-The_Theory_of_Difference_Schemes__Samarskii

188 Homogeneous Difference Schemes

vVhen r = Rone of the usual boundary conditions applies here, for instance,

(28) u(R) = μ 2.

Let u 1 (r) and u 2 (r) be linearly independent solutions to equation (26),

u 1 ( r) being bounded for r E [O, R]. A brief survey of their properties 1s

presented below.

1) If q(O) and f(O) are finite, then u 1 (0) #- 0 and u~ (0) = 0.

2) If q(r), f(r) E C(^2 l[o, R] and k(r) E C(^3 l[o, R], then the derivatives

u' 1' u^11 J' u( 1 3 ) and t/ 1 4 ) are bounded for 0 < - r < - R.

3) The second solution u 2 (r) of equation (26), which is linearly inde-

pendent of u 1 ( r), has a logarithmic singularity at the point r = 0.

Conditions (27) and (28) together provide the existence of a unique

solution to equation (26). By virtue of property 1) condition (27) can be

replaced by

(29) u'(O) = 0.

We proceed as usual and introduce on the segment 0 :::; r :::; R the equidis-

tant grid wh = {r; = ih, i = 0, 1, ... , N, hN = R}.

Still using the framework of Section 2, a difference scheme for equation

(26) can be obtained by the balance method:

(30)

where

Wi+l/2 - Wi-1/2

7'; h

7'; = ih,

R

h= -,

N'

1

1·;1+112 1 1·;1+:12

qurdr=-- J(r)rdr,

r z h 7' z h

"i-1/2 "i-1/2

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

1

7';±1;2 = 7'; ± 2 h,

du

w = r k(r)

dr

Approximating the flow w by the expression

and substituting d; ui ri hand 'Pi ri h, respectively, for the integrals in the
balance equation (30), we arrive at the difference equation

(31) i = 1, 2, ... , N - 1,