1549301742-The_Theory_of_Difference_Schemes__Samarskii

Other proble1ns 183

subject to the conditions

(17) Yo= YN'

For determination of Yi, i = 1, 2, ... N, we obtain the system of equations

i= 1,2,. .. ,N,

supplied by the conditions of periodicity: y 0 = YN and YN+l = y 1. This

system can be solved by the cyclic elimination method (for more detail see

Chapter 1, Section 1.2).

Since a > c 1 > 0 and d 2 c 1 > 0, the maximurn principle is still valid

for problern ( 16)-( 17), due to which

The inequality obtained permits us to derive for the error z

estimate

II z lie= O(h^2 ),

y - u the

since 1/J; = 0( h^2 ) for i = 1, 2, ... , N. Thus, scheme (16)-(17) is of second-
order accuracy in the space C when k(x) E cC^3 ) and q(x), f(x) E C(^2 l.

3. Monotone schemes for an equation of general form. The object of
   investigation is the boundary-value problem

Lu= (kit')'+ r(x)u' -q(x)u = -f(x), O<x<l,

(18)

u(O) = 11. 1 , u(l) = u 2 , k(x) 2 c 1 > 0, I r(x) I< c 2 , q > 0.

The main idea here is connected with the design of a new difference schen1e
of second-order approximation for which the maximum principle would be
in full force for any step h. The meaning of this property is that we should
have (see Chapter 1, Section 1)

(19) Ai Yi-1 - Ci Yi+ B; Yi+1 =-Fi, i = 1, 2, ... , N - 1,

where A; > 0, B; > 0 and C; - Ai - Bi = D; 2 0.
Any such scheme is said to be monotone. As before, the operator
Lu = ( ku')' - qu is approximated to second order by the homogeneous
three-point scheme Ay = ( ayx) x - dy.