On the application of disintegrating difference operators

ON THE APPLICATION OF DISINTEGRATING JIIFFERENCEOPERATORS* Ye.G. D'YAKONOV (Moscon) (Received

9

Noveaber

1962)

The concept

of a disintegrating difference operator is introduced in together with a computing algorithm based on the disintegration of an operator in the case of a rectangle. This computing algorithm can be used to find the solution of the system of finite-difference equations* *

[II,

[21,

nvW+i) = p(s) (n(n), Vc+-l), , . . , v(n-_q), p),

0)

which occur both in the case of difference schemes with a disintegrating operator for non-stationary problems and in the case of certain iterative methods for the solution of difference stationary problems. In the case p = 2 in order to find uCnfl) only x i,fhZ arithmetical operations are required. In this note we show that the computing algorithm can be modified for certain non-rectangular regions so that the solution of system (1) with the disintegrating operator A can be found at-the cost =iJh’ arithmetical operations; for the arbitrary region 8 we give a method for =: i/ha arithmetical operations. To simplify solving (1) which requires the explanation we have only considered the case p = 2, since nothing essentially new is required to generalise the results to the case of a larger number of space variables. 1. Notation. adjacent if

The points

max

1i!‘) -

i(“h

ii2)j = 1;

and

of the net are said

i(’

$1) -_ (ip),

it’) 2 ),

6=1.2

* l

Zh.

vych.

nat.,

* Our notation

3, NO. 2, 385-388. is everywhere

that of 511

1963. 121.

it2) z

(ip),

id2)),

to be

512

Ye.G. D’yakonov

The set of net points belonging jacent points is called the set &, denotes

the set of points

one adjacent

point

!d together with all their eight adof internal points and denoted by Q,. to

ih E 5 such that

is internal.

r,, = Rh \

for each point

at least

Rh is the set of boundary

points. rh = Yl u YP u Y1,2 where y1 is the set of points ih E Th such that either ((i, + I)$, i2h2) or ((il - l)hl, i2h21 is an internal Point; y2 is the set of points ih E fh such that either (ilhl, (i2 (i2 + l)h2 is an internal point; yl,2 = rh \ (yl u y2) l)h2) or !ilhl, is the set of “angular” points (the internal angles at these points are right angles). 2.

The

computing

with

atgorithlx

choice

of

fhtl(;rder

of

successive

substitution. Let us assume that the values v n at the Points are given, and let us consider the system with the disintegrating ator A = A,A2:

which we assume to be defined,

Atii = bi,

Of rfi oper-

where

1;:+ bi,gi + bi, -Ii:,

22E ah

u ?-I,

(4)

where TJ(;~+‘) = Q+R~““’ for ihgE rh; a, 6 are given coefficients such that when zi is given on yi the system Alzi = (Di, ih E Rh is defined and, similarlg, when Yi is given on y2 the system A2yi = cbi, ih E !+, is also the computing algorithm is as defined. If the region 6 is a rectangle, follows: Carrying equations

out substitutions

with respect

to x1, we solve

the system of

where (*+‘M = Azlpin+r) for ih E TX; vi then, using the value of v (n+%) thus found we find substitutions with respect to x2: &&“fl) 1

=

.~+‘hf,

(5’)

v(~+‘).

carrying

out

ih E Q,.

0%

ih Era.

(6’)

where @+l)

= cpy+l)

for

Disintegrating

difference

operators

513

Let us coy$dyr the region shown in Fig. 1. In this case, starting from given v n on r,, we can find v!~+%) = A&n+1 ) at all the points of y1 apart from the points 1 and 2. iherefore the substitutions for x1 can be made along all the horizontals, apart from that passing through the points 1 and 2 (this Fo:;yontal is shown by a dotted line). using we can find v(*+l) making the subst ituthe resulting values of vin tions for x2 on all the verticals not pass& further to the right than the vertical I or further to the left than the vertical II (first group of vertical substitutions). we can find As a result, having obtained v (n+l) on these verticals, .(n+W) at the points 1 and 2 and, making a substitution with respect to xl along the horizontal passing through the points 1 and 2 we find v (ntK) at all the points of R,. Using the values of v(“*) we have found, substituting along the verticals lying between I and II (the second group of vertical substitutions), we find the values of v (n+l) on these verticals, i.e. ~(~+l) has now been found in the whole net region. As we can see, the characteristic feature of this algorithm is that the substitutions with respect to XI and with respect to x2 are done in a strict order, and v (n+l) is not found in the whole region 8, at once, but step substitutions has been carried out. by step, after each group of vertical The order of the substitutions does, of course, depend on the shape of for the region shown i’; 7::. 2. after carrying the region; for instance, out the first group of vertical substitutions, v n is found on the marked sub-region marked I, and after the second group on the sub-region II, and so on. This method is applicable to quite a wide class of regions, but this class is rather awkward to describe. and we shall not do so here. 3. Computing algorithm with the introduction of parametric points. There are regions for which, after the groups of horizontal substitutions have been carried out, there is nowhere any possibility of making the vertical substitutions (Fig. 3), or even in which it is impossible to begin the computation and make the horizontal substitutions (Fig. 4). Let us cons:dt;j for instance, the region shown in Fig. 3. We denote the values of v n at the points 1 and 2 by v1 and vg, and call the points 1 and 2 parametric points. Then taking v1 and v2 to be parameters, and using the algorithm we have described above, we can express all the values of vi(n+l) in terms of these values at the parametric points. Substituting the expressions for the points adjacent to 1 and 2 in the equations .4vI = F1,

Aus = Fzr

we obtain a system of two equations with two unknowns which is equivalent to the initial system and is therefore defined. Finding u1 and vg. which excluding all the other unknowns, takes = l/ha arithmetical operations,

514

Ye. G. D’yakonov

we can then find

= l/ha

operations.

the remaining There will

values of vi(n+l) which will also be the same asymptotic value for

take the

-_f_ _ _ _ -_ _ _ _ _ E&k FIG. 2.

FIG. 1.

niI

FIG. 4.

FIG. 3.

number of arithmetical operations for regions in which the number of parametric points does not depend on h. But in the case of an arbitrary region we can take as our parametric points all the boundary points; there will be Xl/h such points. Excluding all the other unknowns, using the algorithm we have described, at a cost of xl/ha arithmetical operations Te+yive at a system of equations which contains only the at the parametric points and requires xl/h8 arithvalues of v n metical operations for its solution. It will take zl/h’ operations to express the values of u(n+l) at the other points in terms of the known arithmetical opervalues of v (n+l) . Thus in this case it takes xl/h3 ations to solve system (1). and this is better than the ordinary straight line method. for

Note. It is easily verified a non-uniform net:

that these

algorithms

can also

be used

Ilis integrat

1n conclusion,,

ing

difference

we wish to thank V.I.

515

operators

Lebedevl for, his valuable

Translated

comments.

by R. Feinstein

RR~R~~S 1.

D’yakonov,

Ye. G. , I)okl.

2.

D’sakonov,

Ye. G., 2%. vych.

Akod.

.%mik SSSR, mat.,

144,

2, NO. 4,

HO. 1, 29-32,

549-568,

1962.

1961.