1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable two-layer schem.es 427

This implies the estimate for a solution of the nonhomogeneous equation

Byt + Ay = <p:

n

II Yn+l II < P II Yn II+ T II <fn II < p"+^1 II Yo II+ L T Pn-k II <fk II.

k=O

Let now H be a complex space and B be a non-self-adjoint operator.

Then a necessary and sufficient condition for the stability in the space HA

with respect to the initial data of the scheme

B Yt + Ay = 0

is of the fonn

Bo= ReB > 0.5rA, A= A*> 0,

where Bo = ~ ( B + B *) = Bti. In particular, the scheme

i Yt + Ay = 0, A= A*> 0

is unstable in the space HA, since B = iE and Re B = 0. However,

A' = -iA is a skew-symmetric operator (A')* = -A' and, as stated before,

this scheme is conditionally stable in the space H:

0

In the case of the Schrodinger equation Ay = -Yxx, y E D h [O, 1], we have

II A 11 < 4/h^2 and the restriction on T takes the form

h4

T < - - 16 c 2 ,

which is improper for parabolic equations. However, the weighted scheme

i Yt + A y( '7) = 0 , A= A*> 0,

will be stable in both spaces HA and H for er > ~. If so,

Bo = Re B = err A > ~ TA

and II Yn llA < II Yo llA. On the other hand, in dealing with the skew-
symmetric operator A'= -iA we get the estimate II Yn II < II Yo II for er> ~·