1549301742-The_Theory_of_Difference_Schemes__Samarskii

570 Econmnical Difference Schein es for M ultidin1ensional Proble1ns

For this, we have occasion to use the two-layer factorized scheme

(24)

xEwh, t=nr, n=O,l, ... ,

yfl'h = f-l(X, t), t =TIT> Q,

Y( x, 0) = U 0 ( x) , x E w h ,

where wh ={xi= ('i 1 h 1 , i 1 h 2 ), ·io: = 0, 1, ... , No:, ho:No: =lo:, Cl= 1, 2} is

the grid in the rectangle G with the boundary rh.

The operator Lo: is approxin1atecl to second order by the difference

operator

Let

0

0

A=-(A1+A2), Rot=-O'c2Aa, Cl'=l,2,

where A a'IJ ~ = ·1;,,. ~•'U'O ., ..
Stability analysis is mostly based on the assumptions that tllP bouncl-
o
ary conditions are h0111ogeneous and H = r2 h is the space of all grid func-
tions given on the grid w h and vanishing on the boundary rh. The inner
product structure is the same as suggested before for problem (22). A brief
survey of the properties of the operators A and Ro: as operators in that
space H is presented below:

0 0 0 0

A=A*>O, Ao:<c 0 Ao:, Ac,y=-A 0 ,y forany yEr21i,

(25)

0

Ra = O' C 2 A a, Ro: = R: > 0 , Cl' = l, 2, Ri R2 = R2 Ri.

As can readily be observed, the stability condition

will be ensured if O' > 0 .. 5 or even if O' > U.5 - 1/(rff A[[). It seems clear
that the three-point difference operators Ba = E + T R 0 with constant
coefficients are "econon1ical", since the equations

Bo: W = ( E + T Ro:) W = Fa, Cl' = l, 2,