1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with constant coefficients 309

and the harmonic in question is stable. Under the choice I qk I < 1, we have

II Y(k) II < II Y(k) II When this is the case, the scheme is said to be stable

in every harmonic.

We now assign the values of(} such that either I qk I < 1 or -1 <

qk < 1, which guarantees the stability of the scheme in every harmonic.

It is clear from the formula qk = 1-TAk(l + (}TAk)-^1 that qk < 1 if

1 + (} T Ak > 0, that is,(}> -(r >-d-^1. The bound qk > -1 or

1

2+(2()-l)r>-k

qk + = l+(}TAk -> 0

is ensured by the restriction 2+ (2()-l)rAk > 0 or(}>~ -(r>-k)-^1 • In

that case the condition 1 + (}T Ak > 0 is automatically fulfilled. Since

4

Ak < AN-1 < h2

the following relations occur:

1

--- <

TAk -

h2

4r

Hence, the condition I qk I < 1 holds true for all k = l, 2, ... , N - 1 under

the agreement that

(23)

1

(} > -

2

h2

4r

Thus, all of the hannonics Y(k) = Tk X (k) are stable under one and the
same con di ti on (} > (} 0. We are going to show that the stability of scheme
(16a) in every harmonic known as the spectral stability implies that in the
grid L 2 -norm with respect to the initial data y(x, 0) = u 0 (x), where u 0 (x)
is a grid function defined for 0 < x < 1 and vanishing at the points x = 0
and x = 1. \tVith this aim, the general solution of problem (16a) is sought
as the sum of particular solutions having the form (22):

N-l
1111112 = 2= if,
k=!
Substituting here Tk = qk Tk and taking into account (20), we find
that
N-1
y= L qkTk ~X(k),
k=l