1549301742-The_Theory_of_Difference_Schemes__Samarskii

x Preface

of the discrete argmnent tj :::::: j T with the values in the space H h (below

we omit the subscripts h and T).

Multilayer schemes, that is, the schemes containing the values y(t) at

some moments t = tj, tj+i> tj+ 2 , .•• may be reduced to two-layer schemes,

where A and B are operator matrices.

Stability theory is the central part of the theory of difference schemes.

Recent years have seen a great number of papers dedicated to investigat-

ing stability of such schernes. Many works are based on applications of

spectral methods and include ineffective results given certain restrictions

on the structure of difference operators. For schemes with non-self-adjoint

operators the spectral theory guides only the choice of necessary stability

conditions, but sufficient conditions and a priori estimates are of no less

importance. An energy approach connected with the above definitions of

the scheme permits one to carry out an exhaustive stability analysis for

operators in a prescribed Hilbert space H1i.

Clearly, stability is an intrinsic property of schen1es regardless of ap-

proximations and interrelations between the resulting schemes and relevant

differential equations. Because of this, any stability condition should be

imposed as the relationship between the operators A and B. l\llore specifi-

cally, let a farnily of schemes specified by the restrictions on the operators

A and B be given: A= A* > 0 or (Ay, v) = (y, Av) and (Ay, y) > 0 for

any y, v E H, where ( , ) is an inner product in Ii, B > 0 and B -::f. B*

( B is non-self-adjoint). The problem statement consists of extracting from

that family a set of schemes that are stable with respect to the initial data,

having the form

J=0,1,.. .,

and satisfying the inequality

II y llA = J(Ay, Y).

The rneaning of stability here is the validity of the preceding estimate.

A necessary and sufficient condition for a two-layer scheme to be stable

can be written as the operator inequality

B>0.5rA

or
(By, y) > 0.5 T (Ay, y) for any y EH.