1549301742-The_Theory_of_Difference_Schemes__Samarskii

584 Economical Difference Schemes for Multidimensional Problems

On the strength of ( 46) the operator A is self-adjoint, meaning
(Ay, v) = (y, Av).
Fr01n ( 47) we derive the inequalities
0
( 53) yEr2h,
where
p j1 '//
A(O)y = L Yxa.t'a' (-J(ll)Y, Y) = L L (l, (y~a) "] '
a= 1 a=l s=l
As further developments occur, the operator
p
(54) R = L Ra, Ra Y = -(J' Yx-Ck'"' "' Q' , Cl=l,2, ... ,p,
a=l
will be declared to be the regularizer R. Here O' is a numerical para1neter,
the choice of which is stipulated by the stability criteria in the sequel.
In view of this, a reasonable form of the primary two-layer econ01nical
scheme is
( E + r R) Yr = Ay + 1.p,
where
t.p = [1 + O(lhl^2 + r^2 ).
Upon replacing E +TR= E + T L~:=l Ra by the newly formed factorized
operator

p

II (E + T Re,)= E +TR,

CY=!

Qp = L RaR/3 + · · · ,

CY </3

we are led by exactly the same reasoning as before to the econ0111ical fac-
torized scheme

p

( 5 5) II ( E + T Ra) Y t = A Y + t.p , :r E W h , t E W 7 ,
a=l

Y( x, 0) = U 0 ( x) , x E C,·h.