1549301742-The_Theory_of_Difference_Schemes__Samarskii

Preface Xl

This condition can easily be verified in the case of discrete schen1es for
equations of mathe1natical physics. It allows one to extract from the pri-
1nary family of schemes the set of stable sche1nes, within which one should
look for sche1nes with a prescribed accuracy, volume of computations, and
other desirable properties and para1neters.
One of the important corollaries to stability theory is the general
method of regularization in the class of stable schen1es (by changing the
operators A and B) for the design of schemes of a desired quality.
Chapter 6 includes a priori estimates expressing stability of two-layer
and three-layer schemes in terms of the initial data and the right-hand side
of the corresponding equations. It is worth noting here that relevant ele-
ments of functional analysis and linear algebra, such as the operator norm,
self-adjoint operator, operator inequality, and others are much involved in
the theory of difference schemes. For the reader's convc~nience the necessary
prerequisities for reading the book are available in Chapters 1-2.
The book includes many good examples illustrating the practical use of
general stability theory with regard to particular schemes to assist the users
in subsequent i1nplen1entations. Stability is probably the most pressing
problem in any algorithn1, since it is a necessary rather than a sufficient
condition for accuracy.
Despite: the great generality of the research present.eel in this book, it
is of a constructive nature and gives the reader an understanding of relevant
special cases as well as providing one with insight into the generctl theory.
I hope it proves to be useful and sti1nulat.ing.
One of the popular branches of modern n1athematics is the theory of
difference schemes for the nmnerical solution of the differential equations
of rnathen1atical physics. Difference schemes are also widely used in the
general theory of differential equations as an apparatus available for proving
existence theore1ns and investigating the differential properties of solutions.
The theory of difference schemes has a number of special problen1s.
In the final analysis, of greatest importance fr01n the viewpoint of
nu1nerical analysis is the design of algorithms permitting one to obtain a
solution of a differential equation on a computer with a prescribed accuracy
in a finite number of operations. The user can encounter in this connection
the question of the quality of an algorithm, that is, the manner in which
the accuracy of the algorithm depends on

1. the available information on the original problen1,


2. the amount of calculation (viz. the machine Lime spent in solving
   the problem with a prescribed accuracy).

Experience with computers has sti1nulated the formulation of a nmnber of