1549301742-The_Theory_of_Difference_Schemes__Samarskii

Methods for designing difference schemes 219

scheme

( 12)

a;= ~ 7 k(i) di - ~ 7 q(i) (xi - i) (i - xi_ 1 ) di,

x; x;+ 1

d; = h\ ( ;· (i-xi_ 1 )q(i) di+ j (x;+ 1 -i)q(i) di).

X£-l Xi

The right-hand side tp; can be determined by the same formula as we used

for d; with .f(i) standing in place of q(i). The formulae for a; and d; can

be rewritten as

0 0

ai = j k(xi +sh) ds+h^2 j s(l +s)q(xi +sh) ds,

-1 -1

( 13)

0 1

d;= j(l+s)q(xi+sh)ds+ j(l-s)q(xi+sh)ds.

-1 0

If k and q are constants, then

( 13')

h2

ai = k- 6 q, di= q.

In the sequel we will show that scheme (12) is identical with the scheme
e111erged in variational difference 111ethods (the finite element n1ethod).
In what follows we share our practical experience of the design of
difference schemes for problems with lumped parameters by means of the
IIM. Suppose, for instance, that a single heat source of capacity Q is located
at a point x =~so that a solution of problem (1 )-(2) satisfies the conditions

(14) [u] = 0, [k du] = -Q
dx

for X< • -- ~. c