592 Economical Difference Sche1nes for Multidimensional ProblemsThe quasilinear heat conduction equation reproduces the case when ko: =
ko:(x, t, u) and f = J(x, t, u).
Of course, the words "arbitrary domain" cannot be understood in a
literal sense. Before giving further motivations, it is preassumed that the
boundary r is smooth enough to ensure the existence of a smooth solution
u = ·u( x, t) of the original problem (1 )-(2). In the accurate account of
the approximation enor and accuracy we al ways take for granted that the
solution of the original proble111 associated with the governing differential
equation exists and possesses all necessary derivatives which do arise in the
further development.
A co111mon algorithmic idea behind available economical methods is
connected with further reduction of numerical solution of a multidimen-
sional proble111 to the process of solving a few simpler proble111s. In order
to understand the nature of this a little better, we focus the reader's atten-
tion on second-order equations of hyperbolic and parabolic types for which
the "basic algebraic problem" is related to a three-point difference problem
(a second-order difference equation). The three-point difference proble111
obtained through such an approximation can be solved by the elimination
method and it can be treated, as a rule, as a difference approximation to
the one-dimensional (in xo:) differential equation. Some consensus of opin-
ion is to create on this basis a chain of si111pler algorithms which constitute
what is called an economical algorithm for con1plex proble111s. This idea
lies in the origin and tenninology of 111any econo111ical 111ethods available for
solving multidimensional problems. Among them, the alternating direction
111ethod permits us to solve at every stage a one-dimensional problem along
a fixed direction x°', the method of "fractional steps" necessitates placing
in the storage intennediate (non-integer) values at every stage of a complex
con1putational procedure, the method of separation of variables in a com-
n10n setting of the problem reduces to a nmnber of particular si111pler tasks,
etc. All these terms reflect one of the real advantages and the essence of
econo111ical methods.
However, throughout this book, the classification of difference 111eth-
ocls is mostly based on the origin of difference schemes rather than on a
possible way of constructing them and a perfect tool for solving this or that
difference scheme (equation).- The notion of su1nmarized approximation. In the preceding sections and
chapters the basic fundamental property of difference sche111es is to generate
approximations on a solution to the governing differential equation. In what
follows we get rid of the classical notion by introducing a more weaker
condition of smnmarized approximation, expanding our possibilities and