1549301742-The_Theory_of_Difference_Schemes__Samarskii

Preface IX

5 which is devoted to difference schemes for nonstationary equations with

constant coefficients we discover asyn1ptotic stability of difference schernes

for the heat conduction equation, a property intrinsic to the differential

equation.

Chapters 2-5 are concerned with concrete difference schemes for equa-

tions of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on ho-

1nogeneous difference sche111es for ordinary differential equations, by means

of which we try to solve the canonical problen1 of the theory of difference

schen1es in which a primary fainily of difference schemes is specified (in such

a case the availability of the family is provided by pattern functionals) and

sche1nes of a desired quality should be selected within the primary family.

This problem is solved in Chapter ;3 using a particular forrr1 of the sche1ne

and its solution leads us to conservative h01nogeneous schen1es.

Chapter 4 provides the general theory of difference sche1nes in which

it see1ns reasonable to eliminate constraints on the structure and implicit

fonns of difference operators. Such a theory treats difference sche1nes as

operator equations (an analog of grid approxi1nations for elliptic and in-

tegral equations) and operator-difference equations (difference equations

in t with operator coefficients), which are analogs of difference schen1es

for time-dependent equations of mathe1natical physics (for instance, equa-

tions of parabolic and hyperbolic types). The operators of schemes are

viewed in such a setting as linear operators in an abstract normed vector

space H h depending on the vector parameter h (an analog of a grid step

in x = (x 1 , x 2 , ••• , xp)) equipped with the norm lhl > 0. Thus, in the

sequel two types of schen1es will be given special investigation.

The operator sche1ne is associated with an operator equation of the

first kind

Ay = f,

which can always be parainetrized by a real variable h, where A = Ah,

A1z: Hh r-+ H h (the operator Ah depending on h really acts from H1z into

Hh), f = fh E Hh is a known vector and y = Yk E Hh is the vector of

unknowns.

The two-layer opera tor-difference scheme we are interested in acquires

the canonical form

j==0,1., ... ,

where T is the step in t: tj = j T, j = 0, 1, ... ; A, B: Hh r-+ Hh are

operators depending on h and T an cl, generally, on t j; yj = Yh. 7 ( tj) E H h
and zpj = 'Ph, 7 (tj) E H1i are, respectively, the sought and given functions