1549301742-The_Theory_of_Difference_Schemes__Samarskii

182 Homogeneous Difference Schemes

which are continuous at the point x = 0 (x = 1), so that k(l - 0) =

k(O + 0) = k(O), etc. Let

(14) k(x) 2: c 1 > 0, q(x) > c 1 > 0.

Assuming this to be the case, it is required to find a solution to equation

(12) subject to the periodicity condition u(x+l) = u(x), which is equivalent

to the requirements

(15) u(O + 0) = u(l - 0), k ·u'lx=O+O = k u'l.r=l-ll ·

The maximum principle implies that problem (12)-(15) is uniquely solvable.

With this in mind, we start from the scheme for 0 < x = ih < 1:

(ayrJ x - dy = -<p(x), x -- ih ' i=l,2, ... ,N-l,

keeping y 0 = YN. The coefficients a, d, <p can be recovered from the condi-

tions of second-order approximation in Section 2.4.

With the aid of the equalities

the condition ku'lx=O+o = ku'lx=l-O is approximated to second order by

the difference relation

a 1 Yx,o -1 h(q(O) Yo - f(O + 0)) = aN Yx,N + 1 h(q(l - 0) YN - f(l - 0)).

Under the agreements YN+l = y 1 and aN+i = a 1 the preceding reduces to

(ayx)x - dy = -<p(x), x=xN=l,

with

d = dN ::: 1(q(O+0) + q(l - 0)), <p = <f'N =! (f(O + 0) + f(l - 0)),

leaving us with the periodic difference scheme

x = i h, i=l,2,. .. ,N,
(16)