1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with constant coefficients 307

To decide for yourself whether scheme (16) is stable with respect to the

initial data, a first step is to evaluate the solution of problem (16a). This can

be done using the method of separation of variables and deriving estimate

(18) in the grid L 2 (wh)-norm:

N-1

II Yll(l) =II Yll, where II Yll = ~, (y, v) = L Y; v; h.

i=l

To develop those ideas, we seek the solution to equation (16a) as a product
of two functions, one of which T = T(tj) depends only on t = tj and
the other X = X(xi) only on x = X; under the approved decomposition
y(x, t) = X(x) T(t). Substituting this expression into (16a) and taking
into account that

Ay=TAX, Yt = X Tt,

we arrive at the relations

A)( - ->-

x - '

T-T

T ( () T + (1 - o-) T)

where A is a separation constant. As a final result we get

T= qT,

q - l-(l-o-)r>-

• l+o-r>-.

The difference eigenvalue problem for X can be viewed as the Stunn-

Liouville difference problem:

AX(x)+>-X(x)=O, O<x=ih<l, X(O)=X(l)=O, X(x)"!-0,

which has been under consideration in Chapter 2, Section 3.2. As stated
therein, the problem in view has nontrivial solutions identical with the
eigenfunctions

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

wich correspond to the eigenvalues

4. 2 Irkh

-sm h2 --2 '

4. 2 7rh

A 1 = - sm

h^2 2 '

k = 1, 2, ... , N - 1, 0 < Ai < ... < AN-1 '