1549301742-The_Theory_of_Difference_Schemes__Samarskii

338 Difference Schemes with Constant Coefficients

With this relation in view, it is not difficult to derive the asymptotic

expans10n

with the members

T^2 {j3 T3^04

/3 = - 24.6 271.7 + ... ,

which serve to motivate the double inequality pJ < e-^0 t1+!lt1 < e-^0 t1

and the a priori estimate II yj 11 < e-^0 t^1 11 y^0 II for scheme (8) under the

constraint TD < 1.

4. Asymptotic stability of the three-layer scheme. The object of investi-
   gation here is the three-layer scheme

(13) 2 3 Yt - 2 I Yr + A Y A = ,^0 A-- A* , A> oE, i5 > 0,

which is unconditionally asymptotically stable for T < 1/(25). To make

sure of it, an alternative form Yj + TYft + Af; = 0, which provides an

approximation of order 2 for the heat conduction equation with constant

coefficients ( 1), will appear in the further development:

1/J = Uo t + T U[t + A U. = Uo t + T U[t - A ·u - TA Uo t - ~ - T^2 A '1lft

= U. + TU .. - L U - T L U ' - 2 I T-? L U .. + O(h? -+ T-?)

= (il-Lu)+r(u-Lii)+O(h^2 +r^2 ) = O(h^2 +r^2 ).

Here we used also the initial equation ·u = Lu and its corollary u = Lu.

In conformity with the method of separation of variables,

where Xk refers to an eigenfunction of the operator A. Recall that, by
definition,
k = 1, 2, ... , 1V,

With this in mind, we initiate the derivation of the difference equation