1549301742-The_Theory_of_Difference_Schemes__Samarskii

340 Difference Schemes with Constant Coefficients

For large j Lhe first harmonic yi ::::::; o: 1 piX 1 clorninates in the solution

which has been constructed before.

By means of the quantity pi = e-i log(l/p) with

it is plain to calculate

2

1--

3

log ~ = log ( 1 + ~ μ ) - log ( 1 - ~ μ^1 )

p 3 1 3 l+Jl-2μ1.

The expansion in powers of μ 1 yields log (1/ p) = μ 1 + ~ p~ + ···,so that

Thus, the three-layer scheme (13) of accuracy O(h^2 + T^2 ) possesses the

proper asymptotics as tj ____, oo under the unique restriction T 15 < 1/2,

which is not burdensome. Comparison of the final results with the two-

layer scheme (4) reveals some formal advantage of the three-layer scheme

over the symmetric two-layer scheme with O" = i, which is conditionally

asymptotically stable if we imposed the extra constraint T <To= l/../[!S..,

To = To ( h), in addition to the usual one T 15 < 1. However, this restriction

is sufficiently weak in real-life situations and, therefore, it is rneaningless

to speak about any practical advantage of the three-layer scheme. For

this reason the two-layer schen1e is quite applicable and 111ore efficient m

practical implementations.

5.3 SCHEMES FOR THE HEAT CONDUCTION EQUATION

WITH SEVERAL SPATIAL VARIABLES

1. The explicit difference scheme. The schemes considered in Section 1
   111ay be generalized to the case of the heat conduction equation with several
   spatial variables.