1549301742-The_Theory_of_Difference_Schemes__Samarskii

Hornogeneous difference schernes for the heat conduction 495

If CJ # 0, then the difference equation related to Yi can be solved by

the standard elimination nwthocl.

The statement of the problem for the error z = y - u amounts to

z 1 =A(i)z(")+i/!, O<x=ih<l, tJ>O, z 0 =0, z(x,0)=0,

where 1jJ = A( i) uC ") + <p - u 1 • By exactly the same reasoning as in Chapter

3, Section 5 the resicl ual 1jJ is representable by

1/J = x x^1 (-'I) ) '" + 'lj· * + V' ** ' 1) -- a tlx (a) - (.k 1l ') :r=,z:, t=t, -

with the members

17 = O(h^2 +Tm~), ijJ* = O(h^2 /x),

Having no opportunity to touch upon this topic, we refer the readers to

the aforementioned chapters of the manograph "The Theory oof Differ-

ence Schemes", in which the method of extraction of "stationary nonho-

1nogeneities" was employed wit.Ii further reference to a priori esti111ates of

.::. The forward difference scheme with CJ = 1 converges unifor111ly with the

rate O(h^2 + r) clue to the 111axirnum principle.

Being concerned with the heat conduction proble111 in the case of

a spherical symmetry, we are now in a position to produce on the same

grounds the difference scheme associated with problem (81')-(82):

Yt = A(i) y(a) + <p for 0 < x = ih < 1, tJ > 0,

where

A(t) - Yi= -^1 (-2 - )

x.^2 x z a(xi, t) Yx ) i :i; ,z. for i > 0,

z

. (i
A(t) Yo = h a1 Y:1;,11.

<p.; = f( xi, t .. ) 01' lfl; r, = f,l") for^0 < i < N.
This is also consistent with the results expounded in Chapter 3, Sec-
tion 5. The standard elimination method applies equally well to such a
setting. vVe 0111it here more a detailed exploration of the residual and the
accurate account of accuracy of the describing scheme. The reader is invited
to do this on his/her own in line with established priorities for difference
schemes on "flowing" grids w h.