1549301742-The_Theory_of_Difference_Schemes__Samarskii

594 Economical Difference Schemes for Multidilnensional Problems

we find that

z)+l/2 - zJ

-----+ a1 zj = -u,i

T • J ,

where

1tj +^1 - 1tJ uJ + uJ +^1

1/J2 =----+a?---- 2 T - 2

By substituting here the expressions

we are led to

2

UJ _ -U j+l/2 - T · j+l/2 T ··j+l/2 0( 3)

2

u +

8

u + T ,

"{.J+l/2 ' = 'U (t j + (^0). ,) r; T ) , ·u=. -, du
clt
.. c[2 ·u
·u=-
dt2
Whence it follows that. 1/; 1 = 0(1), 4J 2 = 0(1) and 1/; 1 +1/; 2 = O(r), ineaning
that none of the auxiliary schemes concerned provides an approximation,
but the triplex composition generates a sum1narized approximation of 0( T).
Example 2 Of special interest is the heat conduction equation
au_ L
at - u'
8
,, ~u
L1l=8x2'
0 < x < l, u(O, t) = u 1 (t), ·u(l, t) = u 2 (t).
In dealing with the grid wh = {x.; = ih, i = 0, 1, ... , N, Nh = 1} and
the operator Att = 1c;;,r ,...., Lu, it see1ns reasonable to ernploy the explicit