Homogeneous difference schen1es for the heat conduction 471
making it possible to find thath ('l/Jn + 'l/Jn+1) = 2 h (it - f)~
0
2~ + h (<pn + <pn+I - ut,n - ut,n+1) + O(h^2 )= 0((0"-0.5)rh+r^2 h+h^2 )
and deduce for any weighted scheme of the form (7) that(30) h 'l/Jn = 0(1),
Under the special choice(32) a(x) = ~(x) =(Jo ds )-1
k(x+sh)
-]it is straightforward to verify by analogy with the available procedures (for
n1ore detail see Chapter 3, Section 3) thatQ,, = O(h), h'l/Jn = O(h).From such reasoning it seems dear that the following estimates are valid:(33) h 'l/Jrn = 0(1),
'
( ;34)thereby justifying the representations0
'ljJ = 'ljJ + 'ljJ * ,
(35)
'l/J7 = 0 for i f:. n , i f:. n + 1 ,where D; n is, as usual, Kronecker's delta.
The ' intervention of a new grid function
i -I o
17i = 2.::: h 'ljJ k , 1)i = 0 , i = 2, 3,. ,. , N ,
k=I