1549301742-The_Theory_of_Difference_Schemes__Samarskii

360 Difference Schemes with Constant Coefficients

3. The boundary-value problem. We are interested in learning more about
   the boundary-value problem when the boundary value p(t) is specified at
   the point x = 0 and a solution is sought for x > 0 and t > 0:

OU au

7ft +a ox= 0' t > 0' a= canst > 0,

(10)

u(x, 0) = u 0 (x), x > 0' u(O, t) = p(t), t > 0'

under the natural premise u 0 (0) = p(O). In the case of the differentiable

members u 0 (x) and p(t) the function

{

'1l 0 ( x - at)

u(x, t) =

p(t - x/a)

for t < x /a,

for t > x /a,

is just the solution of the problem under consideration.

In working on the grid wh = {xi = ih, i = 0, 1, 2, ... } with spacing h

and the grid w 7 = { tj = jT, j = 0, 1, ... } with spacing T we construct the

implicit scheme on the pattern depicted in Fig. 17cl in the usual way:

Y

j+1 - yj j+1 j+1

k k +a Yk - Yk-1 = O.

T h

( 11)

Alternative forms of this scheme

Yt +a Yx = 0

and

Y

j+1 _ I yj+1 + 1 yj

k - 1+1 k-l 1+l k>

are more convenient for later use. With these, the computational procedure
may be carried out with the starting point k = 1, j = 0. Then

1 I 1 1 o

Yi = I + l Yo + I + l Y1

With knowledge of Yi we are able to calculate the remaining values yj up
to some j = ic· After that, we will find Yd for 0 < j < j 0 , etc. by setting
k = 2.
We now deal with the fan1ily of schemes on the four-point pattern (see
Fig. 17e)

Yt + O" Yx + (1 - O") Yx = 0,
(12)
y( X, 0) = Uo ( X) ,