1549301742-The_Theory_of_Difference_Schemes__Samarskii

448 Stability Theory of Difference Schemes

associated with the transfer equation

which is stable for T / h < 1.

Observe that the explicit three-layer sche111e

Ytt + Ay = 0

with a skew-symmetric operator A = -A* is absolutely unstable, while

the scheme with a self-adjoint operator A = A* > 0, as stated before, is

conditionally stable for

T2

E >-A

4

or

2

T <

(for Ay = -Yxx we obtain T < h).

9. Other a priori estimates. Together with scheme ( 1) the reader may be
   confronted with schemes written in the form

(83) (E + r^2 R) Ytt + B yo t +A y = <p, y(O) =Yo, y( r) =Yi ·

Such schemes formally can be obtained from (1) by replacing R by R = R+
E/r^2. With this substitution in mind, one can easily conclude that scheme
(83) is stable for R > A/4 and write down the appropriate estimates.
Compound norms II Y II arise naturally in connection with the energy
balance equation. Their structure seems to be rather complicated. It is
desirable to possess a priori estimates for solutions of problems (1) and
(83) in the usual energy norms of the spaces HA and HR. We proceed to
the derivation of such estimates. This amounts to setting any three-layer
scheme in the form

(84)

D Ytt + B Y1. + Ay = <p(t)' 0 < t E WT'

y(O) =Yo, Yt(O) =Yo,

where D = D(t), A= A(t) and B = B(t) are linear operators. In particu-
lar, D = r^2 R for scheme (1) and D = E + r^2 R for schen1e (83).

(84a)

(84b)

Together with (84) we consider the problen1s

D Ytt + B yo t +A y = <p(t), y(O) = y 1 (0) = 0,