1549301742-The_Theory_of_Difference_Schemes__Samarskii

The transfer equation 357

Remark Keeping I = ar/h = O(h), that is, T = O(h^2 ), we might have

I q I< 1 +c 0 T, where c 0 > 0 does not depend on T and h, and the harmonic

in question remains bounded:

for

This rneans that

with the constraint a T / h^2 < c 1 (in our exarnple I y~ I = 1), where c 1 =

canst > 0 does not depend on h and T.

We may recommend one more explicit scheme on the pattern consist-

ing of the three nodes (xi,tj), (xi,tj+i) and (xi-1>tj) (see Fig. 17b):

Y, j+1 - yj , + a yj i - yj i-1 _ 0

(5) T h - ' a > 0,

or Yt +a Yx = 0. This scheme also generates an approximation of order 1,

since 1/; = ut +a ux = 0( h + T). After scrutinising the available information

on the difference equation

we deduce for r < 1 that

II yj+i lie< (1 - 1) II Yj lie+ 1 II Yj lie= II Yj lie,

meaning the stability of the scheme in the space C:

(6) If yj+i lie < II Y^0 lie for 0 <I< 1.

By analogy with scheme (2) for a > 0, it is straightforward to verify that
scheme ( 5) is unstable for a < 0, while scheme (2) is stable under the
constraint I I I = I a I T h-^1 < 1, so that the inequality

holds true for all j = 0, 1, 2, ....