1549301742-The_Theory_of_Difference_Schemes__Samarskii

458 Stability Theory of Difference Schemes

For equation (113) we accept

/2 = C2 , Ay = -( ayx) ,

and take

With regard to problem (114) we thus have

p

1

p

1

Ry=(! L h Yxo: or Ry= -CJ I: h Y:ro: 1 12 = C2 I

o:=I 0: a:= l 0:

It is important to note that while constructing the Du Fort--Frankel
scheme as an original scheme the explicit unstable schen1e yo = Ay generat-
t
ing an approximation of 0( r^2 +h^2 ) has been taken with further modification
corresponding to the regularizator of the simplest type ( R = h\ E, CJ = h\
in (112)).
Afterwards when the sweep formulae became customary, one began
to analyze in full details two-layer implicit schemes (weighted schemes) for
which R = CJ A. These schemes obviously represent a particular case of the
scheme with R = CJ Ao.
We point out one more selection rule for R. Let Ao = A 1 + A 2 ,
A 2 = Ar > 0. Choose R in such a way that the two-layer scheme possesses
the factorized operator

so that
R =CJ Ao+ /^2 T A1 A2.

Since (A 1 A2 y, y) (A2y, A2y) = II A2y 112 > 0, this scheme 1s stable if
CJ> /2C!o·
Various schemes with the factorized operator

are in common usage as iteration schen1es for solving equations of the form
Ay = <p.