Operator-difference schemes

In this chapter we study the stability with respect to the initial data and the right-hand side of two-layer and three-layer difference schemes that are treated as operator-difference schemes with operators in Hilbert space. Necessary and sufficient stability conditions are discovered and then the corresponding a priori estimates are obtained through such an analysis by means of the energy inequality method. A regularization method for the further development of various difference schemes of a desired quality (in accuracy and economy) in the class of stability schemes is well-established. Numerous concrete schemes for equations of parabolic and hyperbolic types are available as possible applications, bring out the indisputable merit of these methods and unveil their potential. 

Operator-difference schemes. We now consider a linear system 8h depending on a parameter h as a vector of some normed space equipped with the norm h. With regard to the linear system 8h, it is reasonable to introduce a collection of norms li 111,. II II/, m II ll m. .., thus causing... 

Approximation and convergence. The notions of approximation, convergence and accuracy for operator-difference schemes are introduced by analogy with the corresponding notions for the operator schemes A y — arising earlier in Chapter 2, Section 4. Only a few editorial changes will... 

In this view, it seems reasonable to turn to operator-difference schemes... 

The two-layer operator-difference scheme we are interested in acquires the canonical form... 

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