1549301742-The_Theory_of_Difference_Schemes__Samarskii

556 Econ0111ical Difference Sche1nes for Multidimensional Proble1ns

where un = u(x, tn), ,un+l = u(x, tn+i) and fi is calculated by the formula

(36)

This is consistent with (35) as stated before, Under such a choice of ft we

obtain the homogeneous boundary conditions for specifying i,

By substituting into equation (33) yn = zn + 1ln, ~Q = z + ft and

yn+l = zn+l + un+l we are led to the problem statement

(37)

(38) zl 'Yh = 0, z(x,0)=0,

where '1/J~ and 1/J;' are the appropriate errors of approxi1nation:

• n

(^0) '1"1 ;,n = A ( l in+l/2 ) 1t+ - A ( ) 2 in 1l n +<p n - 1l 0 - ~ 1l '
,QT
n+l -
(^0) '1"2 ;,n - A ( l in+l/2 ) 1l -+ A ( 2 in+l ) 1l n+l + <p n - U 0 - U ,
,5 T
Before giving further n1otivations, it is worth noting here that 4•;' =
1/J~, This fact can readily be verified by substituting expression (36) into
the formula
1/J? n -^0 'f"l ;,n - h2 ' ( in+l ) 1l n+l - A ( ) 2 t,, 1l n - lln+l -^2 ft+ lln = 0,

• 0,5 T

At the 11ext stage expression ( :3()) is needed in the formula for the residual

1/J" 1