146 Homogeneous Difference Schemesof the governing differential equation, a class of boundary and initial con-
ditions as well as of forming a functional space comprising the coefficients
of the differential equation. Naturally, such universal difference schemes
should possess the convergence and stability properties on any sequence of
grids and for any original problern from the given class. The requirement
of universality of computational algorithms for solving a class of problems
necessitates imposing the notion of homogeneous difference scheme. By a
homogeneous difference scheme we mean one whose forrn is indepen-
dent of a concrete problem from this class and the choice of a grid. At
all grid nodes the difference equations take the san1e form for any problem
from this class. The coefficients of a homogeneous difference scheme are
treated as functionals of the coefficients of the differential equation.
For instance, of great interest are "through" or "continuous" execu-
tion schemes available for solving the diffusion equation with discontinuous
diffusion coefficients by means of the same formulae (software). No se-
lection of points or lines of discontinuities of the coefficients applies here.
This means that the scheme remains unchanged in a neighborhood of dis-
continuities and the computations at all grid nodes can be carried out by
the sanre forn1ulae without concern of discontinuity or continuity of the
diffusion coefficient.
Homogeneous "through" execution sche1nes are quite applicable in the
cases where the diffusion coefficient is found as an approxin1ate solution
of other equations. For instance, such schen1es are aimed at solving the
equations of gas dynamics in a heat conducting gas when the diffusion
coefficient depends on the density and has discontinuities on the shock
waves.
In the theory of difference schemes with a pri1nary family of sche1nes
the coefficients of a homogeneous difference scheme are expressed through
the coefficients of the initial differential equation by means of the so-called
pat tern functionals; the arbitrariness in the choice of these functionals
is limited by the require1nents of approximation, solvability, etc. There
are various ways of taking care of these restrictions. The availability of a
primary family of homogeneous difference schemes is ensured by a family
of admissible pattern functionals known in advance.
Let us clarify the subject of investigation in a more si1npler situation
through the use of difference operators acting on functions of only one
variable xi = i h, i = 0, ±1, .... One way of proceeding is connected with
the following two steps. A difference operator is defined beforehand on
an integral pattern, that is, on a set of the type 9J1 0 = { -ni 1 , -ni 1 +
1, .. ., -1, 0, 1, .. ., m 2 }, where m 1 , in 2 are positive integers. The next
step is the transition to the real grid w h = {xi = i h, i = 0, ± 1, ... }