1549301742-The_Theory_of_Difference_Schemes__Samarskii

308 Difference Schemes with Constant Coefficients

The eigenfunctions { X (k)} constitute an orthonormal system

for which Parseval's identity takes place:

N-1

(20) 111112= 2= 1;,

k=l

Here fk are the coefficients of the expansion for an arbitrary grid function

f(x) defined on the grid wh and vanishing at the points x = 0 and x = 1:

N-1

f(x) = L fk X (k)(x),

k=l

So, problem (16a) has nontrivial solutions Y(k) = Tk X (k) '!- 0, where

Tk can be recovered fron1 the equation

(21)

or

Tk J+1 - qk yJ k - ... -_ qk J+1 yo k )

As a n1atter of fact, the constant T~ is free to be chosen in any convenient

way.

A solution to equation (16a) having the form Y(k) = Tk ~X (k) is called

a harmonic of the attached number k. Clearly, this function satisfies

problem (16a) with the initial condition u 0 (x) = T~ X (k)(x). Let us find

out the conditions under which every harmonic Y(k)> k = 1, 2, ... , N - l, is

stable. From the recurrence relations

(22) Y(k) J+1 = X (k) yJ+1 k = (^11) k X (k) Tj k,
the conclusion can be drawn that for I qk I > 1 + E, where E = const > 0
does not depend on h and T both,
as T -t 0, that is, the problem concerned becomes unstable. If I qk I < 1 and
t = jr is kept fixed, then II Y(k) II does not increase along with increasing
j(r-tO):