214 Homogeneous Difference Schemes
at some isolated points on each segment [xi-I, xi+il· Such schemes permit
one to retain the order of accuracy in the case of discontinuous coefficients
k, q, f on the grids wh(J() when discontinuity points fall into the nodes of
the grid wh(J().
Exact and truncated schemes are quite applicable in the estimation
of the accuracy of schemes (16)-(17) in Section 2. This approach allows to
weaken or get rid of the srnoothness of the functions k, q, f involved in the
estimation of the accuracy order of schemes (16)-(17) in Section 2.
3.8 METHODS FOR DESIGNING DIFFERENCE SCHEMES
- General remarks. Frorn such reasoning it seams clear that in the space
of grid functions difference schemes should retain the basic properties of dif-
ferential equations such as self-adjointness, the validity of certain a priori
estimates (for instance, the maximum principle), etc. Moreover, a scheme
which interests us must satisfy, first at all, the requirements of solvability,
stability, approximation and, hence, accuracy of a certain order; at last,
an algorithm must be expedient in computer resources. The expediency
depends not only on a scheme, but also on making a substantiated choice
of the method for solving difference equations and a grid which, gener-
ally speaking, may be non-equidistant and depends on the behaviour of a
solution.
The construction of schemes with the indicated properties and a de-
sired quality is one of the outlines of the possible theory.
As a result, a considerable amount of effort has been expended in
designing various methods for providing difference approximations of dif-
ferential equations. The simplest and, in a certain sense, natural method
is connected with selecting a suitable pattern and imposing on this pattern
a difference equation with undetermined coefficients which may depend on
nodal points and step. Requirements of solvability and approxirnation of
a certain order ca\lse some limitations on a proper choice of coefficients.
However, those constraints are rather mild and we get an infinite set (for
instance, a multi-parameter family) of schemes. There is some consensus of
opinion that this is acceptable if we wish to get more and more properties of
schemes such as homogeneity, conservatism, etc., leaving us with narrower
classes of admissible schemes.
Actually this way has been demonstrated in Sections 1-3. Several
methods find a wide range of applications in designing difference schemes
of a desirable quality, among them
1) the integro-interpolational method (see Section 2);
