1549301742-The_Theory_of_Difference_Schemes__Samarskii

148 Homogeneous Difference Schemes

Reducing scheme (2) to the form (4) from Section 2.1 we find that

b = k + k i+1 - k i-1

! i 4 )

k - k

(3) a· i = k· i - i+1 4 z-1 di ='Pi = 0)

whence it follows that scheme (2) belongs to family ( 4) from the preceding

section.

Conditions (5) and (6) from Section 2.1 hold true, since on segments,

where the function k(x) is smooth enough, the relations occur:

a·= z k i - l 2 h k' z + O(h^2 ) l b i = k z + 2 l h k' z + O(h^2 ) )

so that ai > 0 and bi > 0 for sufficiently small h.

vVe are going to show that scheme (2) is divergent even in the class of

step coefficients

(4)

with ~ being an irrational number such that ~ = xn +eh, xn = nh, 0 <

e < 1.

The exact solution of problem (1), (4) subject to the continuity con-

ditions is of the form

(5) {

1 - cv 0 x, 0 < x < ~,

u(x) =

/3 0 ( 1 - x) , ~ < x < 1 ,

cv 0 = (x+ (1-x)~)-^1 ,

f3o = x CVa ' x = kl/ k2.

We proceed to solve the difference problem (2), ( 4) in the usual way. Since

ai = bi = k 1 for 0 < i < n and ai = bi = k 2 for n + 1 < i < N, equation (2)

reduces to Yi-i - 2 Yi+ Yi+l = 0 for i f:- n and i f:- n + 1. Whence it follows

that

(6)

0 < x < xn,

Xn+l < X < 1.

The coefficients cv and f3 can be most readily recovered from the appropriate
equations with i = n and i = n + 1 incorporated:

(7)

bn [/3(1-Xn+ 1 ) - (1-cvxn)] + ancvh = 0,

bn+l f3 h + an+l [f3 (1 - Xn+ 1 ) - (1 - CV xn)] = 0.