Preface Vllshould be closely linked with further developn1ent of numerical a.lgorih1ns.
l\IIa.thematica.1-physics problen1S a.re fairly con1plica.ted and a.lgoritl11ns for
solving then1 a.re extre111ely cumbers01ne. In n1a.stering the difficulties in-
volved, one might reasonably try to split them into blocks or "1nodules".
Fortunately, quite often processes having different physical backgrounds
will be described by the sa1ne equations (for instance, the processes of
diffusion, heat conduction, and magnetization). One and the sa.rne 1na.the-
ma.tica.l model could be suitable for various physical problems even if ea.ch
one has its own physical particulars. On the other hand, a mathematical
model may change substantially a number of times during the course of
numerical experiments and this obstacle necessitates improving the algo-
rithm constructed and the corresponding program. Therefore, it becomes
extremely important to create programs (a progra.n1 package) in conformity
with the tnodulus principle, whose use pern1its us efficiently to conduct nu-
1nerica.l experiments and solve various types of problems of 1niscellaneous
physical genesis. This is one of the 111odern trends in progra.nuning and
solving major n1a.thema.tica.l-physics proble1ns lea.cling to the clevelopn1ent.
of new nmnerica.l methods.
Usually the finite difference method or the grid method is aimed at
nmnerical solution of various problems in 111athe1na.tical physics. Under
such an approach the solution of partial differential equations a.n10u11ts to
solving systems of algebraic e4ua.tions.
This book is devoted to the theory of difference methods (schen1es)
applied to typical problems in mathematical physics. There seem to be at
lea.st two widespread approaches within the theory:
( 1) Composition of discrete (difference) approximations to equations of
mathematical physics and verifying a priori quality characteristics of these
a.pproxi1na.tions, ma.inly the error of a.pproxi1na.tion, stability, convergence,
and accuracy of the difference sche1nes obtained;
(2) Solution of difference equations by direct or iterative methods
selected on the basis of the economy criteria. for the corresponding compu-
tational algorithms.
Because of the enormous range of difference approximations to an
equation having similar a.sy1nptotic properties with respect to a grid step
(the sa.111e order of accuracy or the number of necessary operations), their
11un1erica.l realizations resulted in the appearance of different schen1es for
solving the basic proble111s in nrni.henrntical physics.
Of course, one strives to develop the best possible method, whose
use permits us to obtain the desired solution in 1ninima.l c01nputing time.
Indeed, the search for such nmnerica.l procedures among admissible meth-
ods is the ma.in goal of such theory. In designing an optimal method (its