1549301742-The_Theory_of_Difference_Schemes__Samarskii

Classes of stable three-layer schemes 447

0

Let H = SL h be the space of all complex functions defined on the grid

wh and vanishing at the points x = 0 and x = 1. Also, it will be sensible

to introduce

N-1

(y, v) = L Yk vk h,

k=l

where vk is the complex conjugate function of vk. The next step is to write

the initial equation in the form

OU. 82 u.

• = -z --+ zqu
  ot 8x^2

and introduce in the space H the operator

A y = i Yxx - i q 1l.

The operator A is skew-symmetric: (Ay, v) = -(y, Av), since

N-1 N-1

(Ay,v) =I: (iYxx,k)vkh+ I: (-iqyhvkh

k=l k=l

N-1 N-1

=-2.::: Yk(ivfa-,k)h-2.::: Yk(-iqv)kh=-(y,Av)

k=l k=l

The norm of the operator A admits the estin1ate

4

llAll < h 2 +q.

The explicit scheme (82) for the Schrodinger equation takes the fonn

0

, yo t + i Yxx. - i q y = 0 1 y E Ah.

It is stable for rll A II < 1, that is, under the constraints

or

h2

T <

4 + q h^2

As the second possible example we look at the scheme

yo t +yo x = 0