1549301742-The_Theory_of_Difference_Schemes__Samarskii

Hornogeneous difference schemes for the heat conduction 497

as a corollary of the condition of conjugation u(O + 0, t) = u(l - 0, t). By

analogy with Chapter 3, Section 5 the second condition of conjugation is

approximated by the equation Yo t = flxx 0. By identifying the endpoints

«: = 0 and x = 1 it is supposed that ' '

In accordance with what has been said above, the difference sche1ne

in question is constructed at all the nodes i = 1, 2, ... , N of the grid wh

under the periodicity condition YN+i = YN imposed at the node i = N.

The same procedures are woTkable in constructing the appropriate

difference scheme associated with the equation with variable periodic coef-

ficients

OU

ot

ox o ( k(x, t) ou) ox + f(x, t), O<x<l,

u(x, 0) = u 0 (x), 0 < x < 1,

t > 0,

and the boundary conditions of periodicity that are known to us as the

conjugation ones:

u( 0 + 0, t) = u( 1 - 0, t) , k Otl I = ou ·1.

OX x=O+n OX x=l-0

Here all the functions k( x, t), f ( x, t) and u 0 ( x) are periodic of period

1 so that

u 0 (x + 1) = u 0 (x), f(x + 1, t) = f(x, t), k(:r + 1, t) = k(x, t).

Let us stress here that the available coefficients k(O+O, t) and k(l-0, t)
may be different: k(O + 0, t) f:. k(l - 0, t). When this is the case, the
derivatives ou/ox involved happen to be discontinuous:

ou ( 0 + 0, t) f:. Otl ( 1 - 0, t).

ox ox

By identifying the endpoints x = 0 and x = 1 on the same grounds as
before, the condition of periodicity is to be understood as the condition of
conjugation at a discontinuity point of the coefficient k(x, t). Fron1 such
reasoning it seems clear that, having stipulated the condition YN+i = y 1 ,
the design of the scheme in question includes all the nodes i = 1, 2, ... , N