1549301742-The_Theory_of_Difference_Schemes__Samarskii

Shrodinger tin1e-dependent equation 351

The metodology of the rnethod of separation of variables guides the

choice ofy(x,t):

N-i.

'tfj(x • ·s-L_, )- ~ c^1 k Xk(x) '.s>

k=l

where Xk(x 8 ) refer to the eigenfunctions of the operator A, meaning, by

definition,

k = 1, 2, ... , N - 1,

4.? Irkh

).k = h? sin- --.

• 2

On the strength of the preceding decomposition we establish the recurrence

relation for the coefficients

k=l,2, ... ,N-1,

where

i - (1-o-) T ).k -(l - O"o) T ).k + i(l + 0" 1 T ).k)

qk= i+a-r>.k = o- 0 r>.k+i(l+o- 1 r>.k) '

and the useful relations for yj+i:

N-1

11 Yj +i 112 = L I qk I 2 I ci I 2 < mkax I qk I 2 · 11 Yj 112 •

k=l

Simple algebra gives

implying that I qk I < 1 and thereby justifying that this scheme is stable:

for

If o- 0 < ~' then I qk I > 1 and the scheme becomes unstable, thus
causing some difficulties. In particular, the explicit scheme with a- = 0 is