1549301742-The_Theory_of_Difference_Schemes__Samarskii

Conservative schemes 155

the inner product (y, v) = Lf:~^1 Yi vi h. Being elements of the space rh,

0

any functions y, v E Dh are subject to the identity:

N-1 N-1

L (ai Yi-1 - ci Yi+ bi Yi+1) V; = Z (bi-1 vi-1 - ci vi+ ai+1 vi+1) Yi·

i=l i=l

This provides enough reason to conclude that the condition

(Ay, v) = (y, Av)
0
holds for arbitrary y, v E Dh if and only if bi = ai+l > i = 1, ... , N - 1
(see Section 2, Chapter 1). The condition b; = ai+l for scheme ( 4) of
Section 1 n1eans that we should have B [ k( x +sh)] = A [ k ( x + (s + 1 )h)] or
B[k(s)] = A[k(s + 1)] for any k(s) E QC^0 l[-1, l]. Evidently, it is possible
only in the case when the functional A[k( s )] is independent of the values
of k( s) on the segment 0 < s < 1. The same applies equally well to the
functional B[k(s)] and the function k(s) on the seg111ent -1 < s < 0, so
that
a(x) = A[k(x +sh)] for -l<s<O. - -
Conditions ( 5) of Section 1 relating to the second-order local approximation
for the conservative scheme (17) acquire the form

( 18)

a(x + h) - a(x) = k'(x) + O(h^2 ),

h

a(x + h) + a(x) = k(x) + O(h^2 ),

2

d(x) = q(x) + O(h^2 ), <p(x) = f(x) + O(h^2 ),

implying that a( x) = k( x) - ~ h k' ( x) + O(h^2 ) or a( x) = k( x - ~ h) + 0( h^2 )
In Section 3.2. the integro-interpolational method was aimed at con-
structing the homogeneous conservative scheme (16) with the coefficients
a, d and <p of the special form (15), namely with pattern functionals such
that
1/2
( 151 ) F [f(s)] j f(s) ds.
-1/2

In this case the coefficients a, d and <p are calculated by integrating
the functions k(x), q(x) and f(x) (see (15)).