1549301742-The_Theory_of_Difference_Schemes__Samarskii

332 Difference Schemes with Constant Coefficients

with the asymptotic expansion f3 = ~ T 62 - i T^2 63 + · · · = 0( T). It

is evident that pj approximates e -o t j poorly as compared with another

scheme we have mentioned above.

Our approach is especially clear in the forthcoming example with

2

T = T 0 TD=21f.

For the value b / b. = 0.01 the quantities

0.9

Pl 17--o. 5 = Po.s = - = 1.1 0.8182,

1

Pla=l = P 1 = U = 0.8333

need to be considered with more care than usual. Their comparison with

e-ro = e-^0 2 = 0.8187 gives

Po.s = (1-0.0006)e-^70 , p 1 = (1+0.018) e-ro.

To be tnore specific, for b tj = 1 (j = 5) we have

Po.s j ~^0 ·^997 e -o t. J,

thereby clarifying that for b tj = 1 the quantity Pls differs from e-^0 tj by
0.3% and pj - by 9%.
The above example shows that although the scheme with (} = 1 is
absolutely stable and, in principle, may be used for any T, it is not accurate
enough at the stage of the regular behavior when T grows. In order to retain
a prescribed accuracy (here it is meaninful to speak only about the relative
accuracy), we should refine successively the step T = Tj with increasing t j.
So, we are n1uch disappointed by the tnain advantage of the scheme with
(} = 1 stipulated by its stability for any T > 0.
The symmetric scheme with (} = ~, which is absolutely stable in
the usual sense: II yj II < II y^0 II, is conditionally asymptotically stable for
T < T 0. Being concerned with the explicit scheme for(} = 0, we observe that
the condition of asymptotic stability T < 2 ( b + b. )-^1 practically coincides
with the usual stability condition T < 2,6.-^1 for small values of o/b..
For the heat conduction equation

4 7rh

,6. = h 2 cos2 2 ,

-sm^4 2 7rh

h^2 2 ,

4

h2