1549301742-The_Theory_of_Difference_Schemes__Samarskii

Economical factorized schen1es 585

In what follows the computational algorithm capable of describing

p

(E+rR 1 )w(i) =Ay+i.p,

p

w(l) = IT (E + T R13) μt for x 1 = 0, 11 ,

/3=2

(E + T Ra) w(a) = w(cr-l), Cl'= 2, 3, ... ,p,

w(<')= IT (E+rR13)μt for xa=O,la, Cl'=2,3, ... ,p-l,

/3=cr+l

will be performed for determination of a vector-function yj +^1 = y. Scheme

(55) is absolutely stable for(}< 0.5 c 2 and converges with the rate O(r +

lhl^2 ).

( 56)

The three-layer scheme

Yo+ t T^2 RY[t = Ay + i.p

is of second-order accuracy in T, so there is some reason to be concerned

about this. Rewrite it in the form

(E+2rR)Yt =F, F=2(Ay-i.p)-(E-2rR)Yt

with further replacement of the operator

p

E + 2 T R = E + 2 T L Ra

er= 1

by the newly formed factorized operator

p

IT (E + 2 T Ra)= E + 2 TR+ 4 r^2 Qp.

o·=l

By these changes we are led to the economical factorized scheme

p
(57) IT (E + 2 T Ra) Yt = F,
a=l