1549301742-The_Theory_of_Difference_Schemes__Samarskii

352 Difference Schemes with Constant Coefficients

unstable for any T / h^2 = canst. In contrast to the approved schemes for the

heat conduction equation, there are no conditionally stable schemes among

ones with r/h^2 = canst: all the schemes with 0" 0 > ~ are stable, while

others turn out to be unstable.

vVe have found earlier for the scheme of accuracy O(h^4 + r^2 ) that

1 i h^2

O"= 2-12T •

The same is still valid for the scheme at hand. From what has been said

above it is clear that for 0" 0 = ~ this scherne is unconditionally stable.

The meaning of stability II yj II < M II y^0 II with constant M > 1 is

that we should have

max I q k I < 1 + c 0 T < e co^7 ,

k

c 0 > Q.

A similar estimate for I qk I can be derived also for 0" 0 < ~ by merely setting

T = O(h^4 ) or

1

O"o = 2

Indeed, in the case of interest

4

6. = h2

< ~~~~~~~~~~ ( 1 - 20"0) T26.2

( 1 + 0"1 TAk)2 + ()"5 T2 .Af

if

T <

( 1 - 20"0) 6_2

or, what amounts to the same,

< c 0 T

c 0 h^4

for

1 c 0

()" = ----

2 T 6.^2

This provides support for the view that the explicit scheme with O" = 0 is
stable:
11 y j 11 < e co t j 11 Y^0 11

if T < c 0 6. -^2 116 c 0 h^4 , where c 0 > 0 is an arbitrary number. The
condition T < 116 c 0 h^4 is very tough and unnatural. Here T / h^2 (but not
T / h^4 ) is a dimensionless quantity. Because of this, the explicit scheme is
unacceptable for the Schrodi.nger equation.