1549301742-The_Theory_of_Difference_Schemes__Samarskii

The transfer equation 355

The problem we must solve is the Cauchy problem

-oo<x<oo, t > 0' u(x, 0) = 1l 0 (x),

under the natural premise a= canst f::. 0. The solution of problem (1) is a

"travelling wave":

u(x, t) = u 0 (x - at),

where a is the wave velocity and u 0 (~) is a differentiable function.

For the purposes of the present section we introduce in the plane (x, t)

the grid

Wh = { X; = ih, i = 0, ±1, ±2, ... },

W 7 = { tj = JT, J = 0, 1, 2, .. , }

with steps h and T in x and t, respectively, and begin by constructing one

of the well-known explicit schemes for the Cauchy problem

yj+1_yj j j

, , + a Yi+1 - Y; = O

T h

(2)

or Yt + a Yx = 0. The pattern of this scheme consists of the three nodes

(x;, tj), (xi+i' tj) and (xi, tj+i) (see Fig. 17a).

Obviously, scheme (2) is of first-approxirnation order with respect to

T and h, since its residual behaves as follows:

7/J = ut +a 1lx = ( u +au') + ~T ii+ ~ah u" + O(h^2 + r^2 ) = 0( T + h).

For a > 0 the sche,me at hand turns out to be absolutely unstable. Now
what we must do is to discover that some particular solution becomes un-
stable albeit with obvious modifications of equation (2):

aT

(3) 1=

h

With this aim, we seek a particular solution to this equation in harmonic
form. This amounts to

(4) i-- v-1, ~ <p f::. 0 '