Vlll Prefacechoice depends on the type of problems being solved) the set of admissi-
ble methods is gradually reduced by successive introduction of necessary
constraints and requirements regarding approxin1ation, stability, economy,
convergence, etc. The following general principle plays a crucial role in such
matters: a difference scheme (a discrete schenw) inust reproduce properties
of the initial differential equation as well as possible.
In practical imple1nentations, in order to construct difference schemes
of a desired quality one should formulate the general approaches, heuristic
tricks, turns, and rules for later use. The design of any method for solv-
ing a problem is stipulated by the h01nogeneity and conservatism of the
appropriate difference scheme. The 1neaning of conservatism is that the
difference scheme reproduces some conservation law (a balance equation)
on a grid. Conservatis1n of homogeneous sche1nes is a necessary condition
for convergence in the class of discontinuous coefficients for stationary and
nonstationary problems in mathematical physics.
The property of the difference scheme conservatism for linear equa-
tions is generally equivalent to the self-adjointness of the relevant difference
operator (see Chapters 3-4).
The basic notions of the theory of difference sche1nes are the error of
approxi1nation, stability, convergence, and accuracy of difference sche1ne.
A more detailed exposition of these concepts will appear in Chapter 2. They
are illustrated by considering a nm11ber of difference schemes for ordinary
differential equations. In the same chapter we also outline the approach
to the general formulations without regard to the particular forn1 of the
difference operator.
The question of the accuracy of the schen1e, being of principal impor-
tance in the theory, a1nounts to studying the error of approxi1nation and
stability of the sche1ne. Stability analysis necessitates i1nposing a priori es-
timates for the difference problen1 solution in light of available input data.
This is a problem in itself and needs investigation.
On the other hand, neither is the estimation of the approximation er-
ror a trivial issue. Even the simplest example of the scheme on a nonequidis-
tant grid for a second-order equation shows that it is desirable to evaluate
the approxin1ation error not in the C'-nonn, but in a weaker nonn of spe-
cial type (in one of the negative non11s), thus i1nposing the need for a priori
estirriates for a solution of a difference problem from the right-hand side
in a weaker norm. Various estimates of this sort arising in Chapter 3 ap-
ply equally well to establishing the convergence of homogeneous difference
schemes in the case of discontinuous coefficients.
This book includes many remarkable examples illustrating different
approaches to the stability analysis of difference sche1ne. Thus, in Chapter