1549301742-The_Theory_of_Difference_Schemes__Samarskii

Ho1nogeneous difference schemes for the heat conduction 461

The starting point is the balance equation written in the rectangle

{x;_ 112 < x < X;+ 1 ; 2 }, tj < t < tj+I with regard to the governing equation

OU

Ft= Lu+ f(x, t), Lu= ox o ( k(:r) ou) ox.

The outcome of this is

tj+ I

(5) x7°

5

[·u(x,tj+l)- u(x,tj)] dx=

;ci-0.5

j [w(x;+i/ 2 , t) - w(x;_ 112 , t)] dt

tj

tj +l Xi+IJ.5

+

./

dt .1· f(.r, t) d.i:, w(.r:.t)=k

l J. ,Vi-0.5

where the integrals and derivatives are yet to be replaced by

x7o

5

u(x, t) dx ~ h u(x;, t),

Xi-0.5

tj+I

~ J w(xi-l/2> t) clt ~ 1Jwi~11/2 + (1 - iJ) wL1;2,

tj

OU

81: '

where ~ designates approxin1ation, IJ is a numerical parameter and the

coefficients a; are expressed through the values of k(x) for X;_ 1 < x < X;

by means of pattern functionals A[k(s)], -1 < s < 1, so that

a( x;) = a; = A [ k( X; + sh)] or

1

• A [

1

a; - k( X; + sh) ] ·

Here A[k(s)] is linear nondecreasing functional, for which the conditions
A[l] = 1 and A[s] = -0.5 hold.