Classes of stable two-layer schemes 3976.2 CLASSES OF STABLE TWO-LAYER SCHEMES- The problem statement. We pursue the stability analysis of two-layer
schemes by having recourse to their canonical fonn
(1) But+ Ay = ip(t), t = nr E WT, y(O) =Yo.
Let Bh = Hh be a finite-dimensional real space, (, ) be an inner
product and II x II= ~be the associated norm in the space H 1 ,. 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+f), where y is a solution to the
homogeneous equation with the initial condition y(O) = y(O) = y 0 :
But + Ay = 0, t =E WT, y(O) =Yo,
and y is a solution to the nonhomogeneous equation with the zero initial
condition:
But+ Ay = ip(t), t =E WT , y(O) = 0.The meaning of the estimate
(2)of a solution of problem (la) is that scheme (1) is stable with respect to
the initial data, while the estin1ate
(3) lly(t + r)llc1J < M2 max llip(t')llc2J
O<t'<t
.
of a solution of problem (lb) expresses the stability of scheme (1) with
respect to the right-hand side.
We will also use an alternative definition of stability of a scheme with
respect to the right-hand side:
where
'Pr(t') = (ip(t') - ip(t' - r))/r.