1549301742-The_Theory_of_Difference_Schemes__Samarskii

Methods for designing difference schemes 227

Adopting those ideas to problem (1), (2), (14) concerning a point heat

source, an excellent start in this direction is to replace the function f ( x)

involved in formula (40) by J(x) + 8(x - ~) Q, where 8(x - ~) is Dirac's

delta-function. Recall that 8( x - ~) = 0 if x f ~, 8( x - ~) = oo if x = ~ and

J((~,' 8(x - ~) dx = 1 for any c > 0. As a final result we get

( 43)

1

I[u] = j [k(u')^2 + qu^2 - 2fu] dx - 2Qu(~).

0

If~= xn + ()h, 0 < () < 1, then u(~) must be replaced by un for() < ~

and by un+i for () > ~, whose use permits us to establish

N N-1

(44) Ih[u] =Lai (Yx,;)^2 h + L (d; Yi - 2<.pi Yi) h,

and 8ik is, as usual, Kronecker's delta. Equating olh/oyi to zero we arrive
at the scheme ( ayx )x - dy = -<p with the right-hand side specified by
formulae ( 45).
In the case of a non-equidistant grid w h = { X; , i = 0, 1, ... , N, x 0 =
0, xN = 1} we obtain instead of (44)

N N-l
I h [y] = L a; ( Yx, i)^2 hi + L (di Yi - 2 'Pi Yi) Iii ·
' i = 1 i=l
In particular, it is al ways possible to choose a grid so that ~ = xn would be
one of the nodal point and

Q

'Pn = fn + fi'

n

'Pi = f; , i I n.

We are led by equating the derivatives olh[u]/oy; to zero to the scheme

( 46) (ay.,Jx,i - di Y; = -<p;, i = 1, 2, ... , N - 1.