1549301742-The_Theory_of_Difference_Schemes__Samarskii

The alternating direction method 547

where the operator B = ( E+T A-)( E+T A+) is a product of two self-adjoint

"triangle" operators, since A+ = (A-)*.

Observe here that the factorized operator B also is self-adjoint. By

means of these operators a sufficient condition of stability becomes

in light of the relations (A-A+:r,:r) = llA+:r 112 > 0 and A-A+> 0. This

supports the view that scheme (6) is absolutely stable and it is of seconcl-

order accuracy.

Let A± be new tridiagonal matrices differing from A± solely by the

zero elements on the main diagonals. While solving equation ( 4) and equa-

tion (5) we should save in the storage the vectors A'-y 2 n+l and A+y 2 n+ 2 ,

respectively. All this enables us to evaluate the number Q 1 relating to the

necessary operations in passing from the layer t 2 n to the layer t 211 + 2. For

scheme (4)-(5) we have Q 1 = 2m^2 + 12m, while for the explicit scheme -

Qo = 4m^2 + 4111, that is, Qi < Qo form> 4.

2. An alternating direction scheme. Further developments are concerned
   with the heat conduction equation of two independent variables that can
   serve as test vehicles for the difference schemes to be presented:

(7)

01l.

ot =Lu+f(x,t), xEG02. tE(O,t 0 ],

ulr = μ(x, t), u(x, 0) = u 0 (x),

321l

L~.u= C< ox 2 , a:=l,2.

Cf

Here 002 =Go= {O < x°' <la, CY= 1,2} is the rectangle of sides l 1 and l 2
with the boundary r.
As a first step towards the solution of this problem, it will be sensible
to introduce an equidistant grid w h in the direction x a with steps h 1 = l 1 / N1
and h 2 = l 2 / N2 and the boundary ih containing all the nodes lying on the
sides of the rectangle except its vertices, w h = wh + ih. We will use them
for later approximation of the operator Lo: by the difference opera.tor

Aay = Yx-n• x n '

Recall that in the case of the one-dimensional heat conduction equation
a similar implicit scheme is associated on every layer with the difference