Concluding Ren1arks 751role in the further recognition and establish1nent of convergence of homo-
geneous difference schemes with discontinuous coefficients (for more detail
see Tikhonov and Samarski! ( 1961 ab)). It was shown therein that conser-
vatism is a necessary property for convergence of a homogeneous scheme in
the class of discontinuous coefficients. Further development of the conser-
vatisrn principle provided by Sa111arski! and Popov (1969, 1992) gave rise
to the notion of full conservatism.
Being concerned with homogeneous conservative difference schemes,
A.N. Tikhonov and A.A. Samarskil recommended the design of the ba-
lance method being an integro-interpolation one and providing the neces-
sary framework for constructing difference schemes ~n arbitrary irregular
grids. Sa1narskil 1 Koldoba, Poveshchenko, Tishkin and Favorski! (1966)
followed these procedures in solving the global problems of 1nathen1atical
physics under such an approach. vVe note in passing that. the 1nethods based
on Marchuk's identity and well-established by Babuska et al. ( 1966) and
Marchuk ( 1975) are conceptually close to the balance 111ethod mentioned
above. A greater gain in accuracy of difference schemes on a sequence of
grids were the main idea behind this approach justified in full details by
Marchuk and Shajdurov (1979).
Some modification of the describing monotone difference scheme for
"divergent" second-order equations was made by Golant (1978) and Ka.-
retkina. ( 1980). In Andreev and Savin (1995) this sche1ne applies equally
well to some singular-perturbed problen1s. Various classes of rnonotone
difference schemes for elliptic equations of second order were composed by
Sa111arskil and Vabishchevich (199.5), Vabishchevich (1994) by means of the
regularization principle with concern of difference schemes.