1549301742-The_Theory_of_Difference_Schemes__Samarskii

Other problems 189

where

Yi - Yi-1

Yr, i = h

Yi+1 - Yi

Y,., i = h

the coefficients ai, di and 'Pi are chosen so that

(32) 'Pi= f; + O(h^2 ).

In the simplest case we accept

(33) 'Pi = Ji.

Let us approximate the boundary condition at r = 0 that can be

declared to be the condition of the zero flow at r = 0: w(O) = 0. We are

going to show that the difference boundary condition

(34)

h

a 1 y,.(O) = "4 (q(O)y(O) - J(O))

has the approximation error O(h^2 ) on a solution to equation (26) satisfying

the boundary condition (27).

Indeed, the residual for (34) is equal to

(35) v - = a 1 1t,.(O) - "4 h (q(O) y(O) - f(O)).

The forthcoming substitutions

u,.(O) = 1t'(O) +! h^11 u(O) + O(h^2 )

yield

(36) v = (ku')o + ~ h (ku')~ - ~ (q 0 u 0 - f 0 ) + O(h^2 ).

From equation (26) we deduce that

(ku ')' =qu-f--. k u'

Since u' --+ 0 as r --+ 0, we have

---+ ku' (k u ')' 0

r

r

as r --+ 0