1549301742-The_Theory_of_Difference_Schemes__Samarskii

384 Stability Theory of Difference Schemes

the variable t (time) plays a key role and, following established practice,

should be marked out throughout the entire chapter. Here Lis a differential

operator acting on u(x, t) as a function of a point x = (x 1 , x 2 , ... , xp) in

some p-dimensional domain G. For any fixed t, the function u(x, t) is an

element of the Banach space B. Therefore, instead of u(x, t) we obtain an

abstract function u(t) of the variable t, 0 < t < t 0 , with the values in the

space B; meaning u(t) E B for all t E [O, t 0 ]. The operator L acting on

u( x, t) as a function of the variable x is replaced by an operator A acting

in the space B. The operator A generally acts from a space B 1 into a

space B2 (its domain D(A) C B1 is everywhere dense in B 1 , while its range

R(A) C B2). In this regard, we take for granted that B 1 = B 2 = B, making

it possible to set up the abstract Cauchy problem

du

dt +Au= f(t), u(O) = u 0 ,

where u 0 is a given elernent from the domain D(A).

The above reasoning is of a heuristic nature and is aimed at carrying

out some analogy between the methods of the general theory of differential

equations and those of the theory of difference schemes, the framework and

methodology of which are outlined in this chapter.

A Cauchy problem is said to be stable with respect to the initial

data and right-hand side if

t

II u(t) II< M1 II u 0 II+ M2 j II f(t') II dt',

0

where M 1 = canst > 0 and M2 = canst > 0.
In conformity with the superposition principle (A is a linear operator),
the stability of the Cauchy problem with respect to the right-hand side
follows from the uniform stability with respect to the initial data

II u(t) II < M1 II u(t') 11, t > t' > 0'

where u(t) is a solution of the homogeneous equation.

2. Operator-difference schemes. 'vVe now consider a linear system Bh de-
   pending on a pararneter h as a vector of some normed space equipped with
   the norm I h I· With regard to the linear system Bh, it is reasonable to
   introduce a collection of norrns 11 · llh' 11 · ll(lh), 11 · 11(21<)' ... , thus causing

linear normed spaces Bh , B~^1 ) , B~^2 ),.... For the sake of simplicity, we