1549301742-The_Theory_of_Difference_Schemes__Samarskii

152 Homogeneous Difference Schemes

heat being en1itted on the segn1ent [xi_ 112 , xi+ 1 ; 2 ] by heat sources with the

distribution density f(x). The integral on the right-hand side of (11) is the

a111ount of heat being transferred to the external environment by the heat

exchange on the lateral surface.

In order to develop a difference equation from ( 11), we substitute

linear combinations of the values of u at the grid nodes in place of w and

the integral with 11 that can be obtained through the interpolations in some

neighborhood of the node X;. The simplest interpolation gives

u = const = ui for xi_ 112 < x < X;+ 1 ; 2 ,

xi+1/2 "'i+1/2

(12) j q(x) u(x) dx R::! h di 1l;, d

1

J

1. 
   
   
   
   • q(x) dx,
     h
     Xi-1/2 "°i-1/2
   
   
   
   

where d; is the mean value of the function q(J;) on the seg111ent x,_ 112 :S:

x < X;+ 1 ; 2 of length h. Upon integrating the equality v' = w/k over the

segment xi-l :S: x :S: xi we arrive at

Xi

j

. w(x)
u,:-1 - ui = k(x) dx'
.'Ci-1

for which the substitution w(x) = W;_ 112 = const on the seg111ent X;_ 1 <

x < X; yields

Xi

Ui-1 - U; :::::;: Wi-1/2 J

Xi-1

dx

k(x)

From what has been said above it is clear that the approximate value w;_ 112
of the flow is

(13) wi-1/2 =-a,: a· l

dx

k(x)

Here J:.i Z-1 dx/k(x) is the heat resistance of the seg111ent [xi-u xi].

)

-1

Substituting (12) and (13) into (11) and denoting by Yi the unknown
function, we obtain the conservative difference scheme

(14) Yi - Yi-1
h )

• dy· i ·z = -/!"J. r i i