1549301742-The_Theory_of_Difference_Schemes__Samarskii

82 Basic Concepts of the Theory of Difference Schemes

Example 1. The third boundary-value problem. for an ordinary

second-order differential equation:

(39)

d^2 u

--qu=-f(x),

dx^2

du(O)

dx = uu(O)-μ 1 ,

q = const , 0 < x < 1 ,

u(l) = μ 2.

On the equidistant grid wh = {xi = ih, 0 < i < N} we may attempt the

difference equation in the forrn

( 40) Yxx - q Y = -<p

with '-Pi = J (xi), where J ( x) is a continuous function.

At the point x = 1 the boundary condition is satisfied exactly:

( 41) y(l)=YN=μ2.

Furthermore, we replace the first derivative u'(O) by the first difference

derivative Yx ' ' 0 = (y 1 - Yo)/ h and impose the boundary condition at the

point x = 0 such as

( 42) Yx O =^17 Yo - μ1 or
'
where an operator lh is defined on the two-point pattern (0, h). Substituting
here y = z + u, where u is a solution of problem (29), we establish for the
error z the condition

where the error of approxirnation v 1 for the boundary condition on a solu-
tion is equal to v 1 = μ 1 + u,~,o - uu 0.
Developing u(x) about the node x = 0 in Taylor's series

we find that

(43) ux ,o = uo I +^1 2 1lo // + O(h2) '

v 1 =[μ 1 +1/(0) -uu(O)] + ~ hu^11 (0) + O(h^2 )

= ~hu^11 (0)+0(h^2 ),