1549301742-The_Theory_of_Difference_Schemes__Samarskii

XU Preface

special (for the theory of difference tnethods) problems:

1. the determination of the attainable order of accuracy of difference
   schemes for various classes of problems,


2. the design of sche1nes for the solution of a wide class of problems
   with a certain guaranteed accuracy,


3. the construction of schemes with increased accuracy in narrower
   classes of problems,


4. the develop1nent of n1ethods for investigating stability and conver-
   gence of difference schemes,


5. the formulation of the general principles for constructing stable dif-
   ference schen1es and econon1izing the a1nount of calculation ( eco-
   nomical sche1nes)
   and others.
   The main purpose of the final chapters is to show how the results
   of the general theory of difference schemes are aimed at stating principles
   for constructing difference schemes of a prescribed quality. This approach
   requires forsaking a more detailed description of the structure of difference
   operators for concrete classes of differential equations and presenting the
   theory in the language of functional analysis. The difference schen1es re-
   lating to analogs of nonstationary equations of mathe1natical physics are
   treated in this connection as difference (with respect to the variable t) equa-
   tions with operator coefficients in an abstract space of any dimension. The
   difference schemes for elliptic equations are viewed as operator equations of
   the first kind. It should be emphasized, ho\vever, that the indicated notions
   of sche1nes have a much 1nore general meaning.
   Stability theory is quite applicable to fonnulnte a general principle
   for regularizing difference schernes in order to obtain stable schernes of a
   prescribed quality.
   The theory of iterative methods for solving the equation Au = f,
   where A E (Hr--+ H) is a linear operator in a Hilbert space H, is treated as
   a contemporary part of the general stability theory of operator-difference
   schemes. Our main concern is with obtaining effective estimates for the
   rate o{ convergence of the iterations and with choosing an optimal set of
   iteration parameters. Special attention is being paid to a class of im-
   plicit schemes with a factorized operator B on the upper level of the form
   B = (E + wRi) (E + wR 2 ), where E is the identity operator, w > 0 is
   a nmnerical parameter, and Ri and R2 = Ri are adjoint or "triangular"
   (with a triangular matrix) linear operators. A formula for the parameter w
   is obtained through such an analysis from the condition that the number
   of iterations be n1inimized.