Stability Theory of
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 schernes of a desired quality (in
accuracy and econorny) in the class of stability schernes 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.
6.1 OPERATOR-DIFFERENCE SCHEMES- Introduction. In Section 4 of Chapter 2 the boundary-value problems
for the differential equations Lu= -f(x) have been treated as the operator
equations Au = f, where A is a linear operator in a Banach space B.
In the study of nonstationary processes described by partial differen-
tial equations of parabolic and hyperbolic types
au 82 u
8t = Ltt+f(x,t), ot 2 = Lu+f(x,t), 0<t<t 0 ,
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