1549301742-The_Theory_of_Difference_Schemes__Samarskii

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Stability Theory of Difference Schemes

\!Ve now consider the scheme with a skew-syn1111etric operator A:

yo + Ay = 0,

t

A= -A*.

We are going to show that this scherne is stable for rll A II < 1 and for it

the energy identity holds true:

vn+I c =GI c 1

where En+I =II Yn+I 112 + 2 r( Yn+I1 Ayn)+ II Yn 112 > 0.

All the tricks and turns remain unchanged: first, write the difference

equation in the form

y+rAy=y-rAy

and compute the squared norms on the left and right-hand sides

II Y 112 + 2 T (y, Ay) + r^2 ll Ay 112 =IIY112 - 2 T (Ay, Y) + r^2 ll Ay 112,

then add II y 112 to both sides and take into account that A is a skew-

symmetric operator, that is, (Ay, y) = -(y, Ay). As a final result we obtain

En+I =En = ... = £ 1. What is more, we claim that En+I > 0. Indeed,

En+I > II Yn+I 11

2

• 2 T II Yn+I II · II Ayn II+ II Yn 112

II Yn 112 - r^2 II Ayn 1/2 > 0

under the restriction rl I A 11 > 1. Here H is a real space. In the case of a

complex space H we have the quantity

with respect to which both the above reasoning and results are still valid.

Example Of special interest is the Schrodinger equation

q = const > 0 , 0 < x < 1 ,

u(O, t) = u(l, t) = 0, u(x, 0) = u 0 (x).

!Ve introduce, as usual, on the segrnent 0 < x < 1 a uniform grid

wh={xi=ih, i=O,l, ... ,N; hN=l}.