1549301742-The_Theory_of_Difference_Schemes__Samarskii

Conservative schemes

where

(15)

1/2

clx

k(x)

d; = d; = j q(xi +sh) ds,

-1/2

1/2

'Pi= &i = j f(x; +sh) ds.

-1/2

__ cl_s_ )-l

k(a:;+sh)

153

We have written the difference equation (14) at a fixed node x = X;. With

an arbitrarily chosen node x, it is plain to derive equation (14) at all inner

nodes of the grid. Since at all the nodes x;, i = 1, 2, ... , N - 1, the

coefficients a; and b; are specified by the same formulae (15), scheme (14)-

(15) is treated as a homogeneous conservative scheme. Because of this,

we may omit the subscript i in formulae (14)-(15) and write down an

alternative forn1 of scheme (14):

(a y, 0 )x - cly = -<p.

In the general case the coefficient ai built into the formula for the heat flow

is some functional of the values of k(x) on the segment [x;_u x;].

Observe that the conservation law in the entire grid domain wh known

as the "integral conservation law" is an algebraic corollary to equation (14)

for any conservative scheme of the form (14) with arbitrary ingredients a,

d and <p. Indeed, with the notation wi_ 112 = -a; (Y; - y;_ 1 )/h for the

difference expressi9n of the heat flow at the point x = X;_ 112 , we can

rearrange (14) as w;_ 112 - w;+ 1 ; 2 + h<p; = hdiyi, which becomes after

summation over i = 1, 2, ... , N - 1 the difference conservation heat law

within the entire grid domain:

N-l N-l

w1 12 -wN-1;2 + L h<p; = L hdi Yi,

i=l 1:=1

meaning the difference approxirnation of the integral conservation law for
equation ( 1) from Section 1.