1549301742-The_Theory_of_Difference_Schemes__Samarskii

Ho1nogeneous difference schen1es for the heat conduction 477

u(x, t) is a solution of the preceding difference problem. Upon substituting

y = z + u into ( 49) the complete posing of the problem for the error z is

z(x,0)=0, xEwh, z(O,t)=z(l,t)=O, O<t=jr<T,

(50)

z(x,0)=0, xEwh, z(O,t)=z(l,t)=O, O<t=jr<T,

where

( 51) 'l/J(x, t) =A (1Ju + (1 - 1J) u) + <p - u

is the error of approximation of problen1 (1)-(3) by scheme (49). In such a

setting the balance equation on the segment xi-o. 5 < x < X;+o. 5 with the

ends X;_ 0 5 = X; - 0.5 h; and X;+o 5 = X; + 0.5 hi+i gives

.r1:+0.5

( 52) j f(a:,f)dx.

Xi-0.5 Xi-0.5

Here

• au -
  w(x, t) = k(x) -(x, t).
  at
  Some modification of the residual 'ljJ is possible with minor changes. Divid-
  ing both sides of identity ( 52) by Iii = 0.5 (h.; + hi+l) and then substracting
  the resulting expressions from ( 51) reveal

where v = v(x;_ 0 5 , t) and

0 l

'l/J, = 'P1 - <pi + -,-

';

Xi+0.5

J

a11 -

-at (.r, t) d:c - ut,i.

X'i-0 5

Using an expansion arising from Chapter 3, Section 4 such as

~ 1. = • f· z + l 8 (h2 1 .• f' z- (^0). 5) £: )i + 0(1i2) z
and allowing an alternative fonn of writing
(53) ·1/J = T/i: + 'lj;* '
(54) 1)j z =a Z u~a),j .T,Z - (ku')j+1;2 z-1/2 + h; 8 (u' - J')j+1;2 i-1/2