1549301742-The_Theory_of_Difference_Schemes__Samarskii

462 Hon1ogeneous Difference Schen1es for Time-Dependent Equations

From such reasoning it seems clear that the operator

L ll = _!}___ (k OU)

ox ox

is approxi111ated to second order by the difference operator Au = (au,,),,,

meaning

In so doing 0 < c 1 < a(x) < c 2.

Upon substituting the resulting expressions into (5) and replacing u

by y we obtain the difference scheme for the grid function y( x;, ti):

(6)

i=l,2, ... ,N-1, j>O,

where Ay = ( ay,, ),,.

The simplest formulas

Cli = 0.5 (ki-1 + k;)'

'P; j -- 1·i i +l/2 , 'Pi = 0.5 (Jj + 1]+^112 )

suit us perfectly for determination of t.p{ and CL;.
When the difference equation (6) is put together with the supplemen-
tary conditions y~ = u. 0 (:i:;), y~ = μ 1 (tj ), lfN = μ 2 (tj ), there arises naturally
the difference boundary-value problem

y( x, 0) = tl 0 ( x) , x E w h ,

within the usual notations