1549301742-The_Theory_of_Difference_Schemes__Samarskii

Homogeneous schemes for second-order equations 147

with step h. Let k( s) be a vector function defined for -m 1 < s < m 2 and

called the coefficient pattern. In the sequel we are dealing with pattern

functionals

AJ^11 [k(s)], Eh [k(s)],

which usually depend on the parameter h and are defined for the vector

functions k(s), s E [ -niJJ m 2 ]. By a linear with respect to a grid function

yh homogeneous difference sche1ne is meant (L~k)yh)i = 0, where

Omitting the subscript i one can rewrite the preceding as

m2

L A~[k(x +sh)] yh(x + mh) + Bh[k(x +sh)].

The principal question in the theory of homogeneous difference sche-

mes is connected with further design of admissible schemes within a primary

family for solving a class of typical problems as wide as possible and choos-

ing the most efficient ones (in accuracy, volume of computations, etc.).

3.2 CONSERVATIVE SCHEMES

1. An example of the sche1ne which is divergent in the case of discontinuous
   coefficients. We now consider problen1 (1) of Section 2.1 with q _ 0 and
   f - 0 incorporated:

(1) (ku')' = 0, O<x<l, u(O)=l, u(l)=O.
As one might expect, the derivative ( ku')' should be replaced by ku" + k' u'.
As a first step towards the construction of a second-order approxirnation,
it will be sensible t'o carry out the forthcmning substitutions

k' ~ ka = k z+1 - k z-1

x 2 h

'U. I ,...._, u 0 = -----1li+1 - l!i-1

x 2 h

u II ,...._, Uxx '

Within these notations, a reasonable form of the difference scheme is

(2) k· Yi+1 -^2 Yi + Yi-1 +
ki+1 - ki-1 Yi+1 - Yi-1
! h2 =0
2h 2h )
0 < i < N, Yo = 1, YN = 0.