Classes of stable two-layer schemes

The problem statement. We pursue the stability analysis of two-layer schemes by having recourse to their canonical form 

Let Bfi = -ff/j be a finite-dimensional real space, (, ) be an inner product and x-1 = / x, x) be the associated norm in the space The operators A and B of scheme (1) depend, in general, on h, t and t. Here and below, we agree not to indicate explicitly the dependence on t. 

The main goal of our studies is to find out sufficient conditions for the stability of scheme (1) and obtain a priori estimates for a solution of problem (1) expressing the stability of this scheme with respect to the right-hand side and the initial data. In preparation for this, a solution of problem (1) can be written as a sum y = y + y, where y is a solution to the homogeneous equation with the initial condition y(0) = j/(0) = y  

We will also use an alternative definition of stability of a scheme with respect to the right-hand side  

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The forthcoming energy norms will be taken as the norm addition to the approved basic norm y = / y, y)  

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