1549301742-The_Theory_of_Difference_Schemes__Samarskii

604 Econmnical Difference Sche111es for M ultidirnensional Problerns

possible at the saine moment only along one direction ;r 1 without concern

of others. At n1oment t = ta + f',,,t the same procedure works along one

direction J: 2 and at nloment t = ta + 2 f',,,t - along another direction x 3. As

a final result we obtain at t =ta + 3 f',,,t the sa1ne temperature distribution

as in the three-dimensional case at moment t = ta + t,,,t. As a matter of

fact, this is a way of reducing the three-dimensional process to a sequence

of one-dimensional ones of pl'olongations as well as of a real physica.l process

in three times.

Generally speaking, proper guidelines in such matters are not so obvi-

ous as the users might expect. For example, in isotropic media the operator

L 0 involved in (6^1 ) is taken to be

In spite of the fact that v(p)(x, t*) does not coincide with u(x, t*), the

asymptotics reveals itself as

v(p)(x, t) - u(x, t) = O(t*).

5. A locally one-di1nensional schen1e (LOS) for the heat conduction equation
   in an arbitrary do1nain. The 1nethod of sun1nrn.rized approximation can
   find a wide range of application in designing econo1nical additive schemes for
   parabolic equations in the domains of rather c01n plicated configurations and
   shapes. More a detailed exploration is devoted to a locally one-dimensional
   problem for the heat conduction equation in a complex domain (; = G + r
   of the dimension p. Let x = (x 1 , x 2 , ... , xp) be a point in the Euclidean
   space RP.
   The problem state111ent for the heat conduction equation in the cylin-
   der Qt 0 = G X [O < t < ta] is

( 15)

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8t = Lu+.f(x,t), (x,t)EQt 0 ,

vlr = p.(x, t), t > 0, v(.r, 0) = v 0 (x), x E G.

Here f is the boundary of the do111ain 0 and L is a second-order elliptic
operator. For the sake of simplicity we agree to consider L = t,,,, that. is,
Lau = 82 u/8x!, Cl' = 1, 2, ... , p (Laplace operator). Also, we take for
granted that.