750 Concluding Remarksshould cause no confusion. This n1ethod was independently proposed by
1nany authors in the early fifties. The bibliographical sources of Godunov
and Ryabenkil (1964) and Richt111yer ( 1957) help clarify its appearance a lit-
tle bit. Faddeev and Faddeeva (1963) have treated the elimination method
as the well-established Gaussian elimination for systems of equations with
a tridiagonal matrix.
Most of the propositions in Chapters 2-4 are of independent value
although, for the present book they are used only as part of the auxiliary
mathematical apparatus. Some of the111 were known earlier and the rest
were discovered and proven in recent years in connection with the rapid
development of the theory of difference schemes.
Relevant results frorn functional analysis, operator theory and the
well-developed numerical methods are given at the very beginning of Chap-
ter 2. All proofs af the 1nain statements as well as a detailed exposition of
the foundations of functional analysis, operator theory and the theory of
difference schemes are outlined in the textbooks and 1nonographs by Ames
(1977), Mitchell and Griffits (1980), Morton and Mayers (1994), Ortega and
Poole (1981), Samarski1 (1987), Samarskil and Gulin (1989). The basic con-
cepts, notions and mathematical apparatus of the contemporary theory of
difference methods for solving mathematical physics problems completed in
late fiftieth owe a debt to Forsythe and Wasow (1960), Richtmyer (1957),
Richtmyer and l!Iorton (1967), Ryabenkil and Filippov ( 1956). The the-
orern on connection between approximation and stability of a difference
schen1e and its convergence to a solution of the" original problem appeared
in the works of Filippov (1955), Lax and Richt1nyer (1956).
The theory of homogeneous difference schen1es for ordinary differential
equations of second order with variable coefficients and, in particular, with
discontinuous ones, was developed by A.N. Tikhonov and A.A. Samarskil
in the 111iddle of 50th - in the early 60th. The first results in this area
were obtained by Tikhonov and San1arskil ( 1956). Later a. complete and
syste111atic study was carried out by Tikhonov and Sa111a.rskil ( 1961) along
these lines. Special investigations of convergence of difference schemes on
non-equidistant grids have been done by Tikhonov and Samarskil (196lab,
1962). The monograph of Samarskil, Lazarov and Makarov (1987) includes
n1ore a detailed exploration of the theory of explicit difference schemes and
schemes of any accuracy order for ordinary differential equations, while the
original works of Alekseevskil ( 1984), Bagmu t ( 1969), Chao Show ( 1963)
and Prikazchikov (l 965) concentrate on special cases in tackling singular-
perturbecl proble1ns, equations with singularity, smne proble1ns associated
with a four-order equation and the Sturm-Liouville problem, respectively.
The notion of the conservatism of a difference scheme played a crucial