TWO-DIMENSIONAL SPLITTING SCHEMES FOR HYPERBOLIC EQUATIONS Текст научной статьи по специальности «Математика»

UDC 519.6 10.23947/2587-8999-2020-1-2-71-86

TWO-DIMENSIONAL SPLITTING SCHEMES FOR HYPERBOLIC

EQUATIONS*

A.I. Sukhinov

Don State Technical University, Rostov-on-Don, Russia

The article considers splitting schemes in geometric directions that approximate the initial-boundary value problem for p-dimensional (p > 3) hyperbolic equation by chain of two-dimensional-one-dimensional problems. Two ways of constructing splitting schemes are considered with an operator factorized on the upper layer, algebraically equivalent to the alternating direction scheme, and additive schemes of total approximation. For the first scheme, the restrictions on the shape of the region G at p = 3 can be weakened in comparison with schemes of alternating directions, which are a chain of three-point problems on the upper time layer, the region G can be a connected union of cylindrical regions with generators parallel to the axis OX3. In the second case, for a three-dimensional equation of hyperbolic type, an additive scheme is

constructed, which is a chain «two-dimensional problem - one-dimensional problem» and approximates the original problem in a summary sense (at integer time steps). The stability and convergence of the constructed

schemes are proved: with the factorized rate O|||h||2 + r2 j, and with the additive rate h||2 + r j, where ||h||

is the norm of the step of the spatial grid, r is the time step, under the appropriate restrictions on the smoothness of the functions included in the statement of the initial-boundary value problem. For the numerical implementation of the constructed schemes - the numerical solution of two-dimensional elliptic problems -one can use fast direct methods based on the Fourier algorithm, cyclic reduction methods for three-point vector equations, combinations of these methods, and other methods. The proposed two-dimensional splitting schemes in a number of cases turn out to be more economical in terms of total time expenditures, including the time for performing computations and exchanges of information between processors, compared to traditional splitting schemes based on the use of three-point difference problems for multiprocessor computing systems, with different structures of connections between processors type «ruler», «matrix», «cube», with universal switching.

Keywords: splitting schemes, hyperbolic equation, additive scheme, stability and convergence of schemes, two-dimensional splitting schemes, multiprocessor computing systems.

Introduction. The method of fractional steps splitting is widely used for numerical solution of mathematical physics multidimensional problems. The beginning of its development in the fifties-sixties of the XX century was laid by the works of researchers, primarily A.A. Samarskiy, G.I. Marchuk, N.N. Yanenko, A.N. Konovalov, J. Douglas, D.W. Peaceman, H. Rachford, R.D. Richtmyer, P.O. Lax and others. Currently, the list of works related to this topic continues to grow.

Useful splitting schemes in spatial variables, in particular, additive schemes, imply the replacement of the original multidimensional problem by a chain of one-dimensional problems. In many respects, this splitting method was due to the fact that the sweep method was for a long time

* This paper was supported by the Russian Foundation for Basic Research (RFBR) grant No. 20-01-00421.

the only economical method for solving difference problems. The conditions for the applicability of the sweep implied the expediency of reducing the multidimensional problem to a chain of three-point difference equations, which, in turn, required replacing the multidimensional problem with a chain of one-dimensional problems. Progress in the development of fast direct methods, primarily for two-dimensional grid elliptic equations, provides an alternative opportunity in splitting a multidimensional problem. Another factor that motivated the emergence of two-dimensional splitting schemes is the increasing use of multiprocessor systems and parallel computing.

The introduction of parallel computing systems leads to the need to expand and, to a certain extent, rethink the concept of an algorithm's economy. In most cases, the user is concerned with the total cost of solving the problem, the main part of which is the cost of performing arithmetic, logical operations and information exchange operations. It seems expedient to consider such a parallel algorithm as economical, which, among others, has the minimum total time costs. It turned out that one-dimensional splitting schemes, being economical in terms of time spent on performing computations, are not in some important cases economical in terms of time for performing information exchanges between processors.

The study of parallel implementations of fast direct methods - cyclic reduction (CR), decomposition in basis (Fourier algorithm - FA), their combination (methods like FACR) and other methods shows that for multiprocessor computing systems, with different structures of connections between processors, such as «ruler», «matrix», «cube», with universal switching and with a wide range of relative performance of information exchange channels, two-dimensional splitting schemes require less total time expenditures, including the time of performing calculations and exchanges of information between processors as compared to traditional splitting schemes based on the application of three-point difference problems. In this article, we consider one of the families of splitting schemes, factorized and additive, which imply the replacement of the original p-dimensional (p > 3) chain of one-dimensional and two-dimensional problems.

1. Statement of the problem

Consider the equation

with initial conditions

(1)

, . , . du ( x, 0) _ , . —

u ( x, 0) = uq (x), —--- = uq (x), x e G,

(2)

and boundary conditions

u(x, t) = x, t), x er, 0 < t < T ,

where G is 3D area with border r.

First, assume that G is a parallelepiped, that is

G = {0<xa<la, a = 1,..., p} .

Assume that u (x, t) is the solution of the problem (1)-(3) has continuous in Qt = G x(0 < t < T] derivatives with respect to xa, a -1,2,3, up to the fourth order inclusive, derivatives, dau/dta , a - 1,2,3 satisfy the Lipschitz condition in the variable t, and there is also a derivative d4u/ dt4 in Qt . Let build a uniform time grid cdr - {tj - jr, j - 0,1,..., | on the segment 0 < t < T with step r ,

as well as a uniform spatial grid cc>h in the region Go with steps ha -la/Na , a-1,2,3 . Let yh — is the grid area boundary cc>h, which contains all the nodes of the faces (rectangles) of the parallelepiped Go, except for those that lie on his ribs, a>h - c uyh.

2. 2D factorized schemes.

Let assum that

p I

a-1

A - I Aa, Aa - Aa,

A = I Aa,

a-1

fi -~C2ap(a0° 1, p +(1-¿2fi-1, pa)A00, fi- 1,2,...,p'- 1, C2°p' (A°p_ 1 (1 -¿2p'- 1, p (1" a)) + A0 ), fi -1,2,..., p' -1

Rp =-C a

where p'-[(p +1)/2], [q] means the largest integer not exceeding the given number q, ¿a, p is the

Kronecker symbol, a is the weight parameter, 0<a< 1, up (fi-1,...,p') are the coefficients that

are selected based on the stability and accuracy conditions. Let us construct the factorized operator

D -(( E + r2 Rp

P-V

and consider the factorized scheme

Dytj -Ay + q>, x ea>h; t ecr y - ju at x e yh, t e cor,

2

p +1

y (x,0) = u0 (x), yt (x,0) = u0 +r

A

I Lau0 + f (x, 0) ya-1

Let us prove the unconditional stability of scheme (4) - (6). Obviously

.2 p

(4)

(5)

(6)

D - E + r 2 I Rfi + Qp , fi-1

where Qp is the operator polynomial containing the degrees of operators Rp from two to p'. It is not hard to see that Qp > 0 . Let up > (1 + s)/4, where s - const > 0. Then

D -1A - E + r2

p

Z ufiC2 (A2°fi-1 +(l— ¿2fi+1, p ) A<°fi)-fi-1

1 + s

I Aa> E

a-1

whence the stability of scheme (4)-(6) follows. The considered scheme has an approximation error O(||hi2 +r2 j, if each of the expressions Aa, yt , p approximates the corresponding continuous

expressions with the second order, and, therefore, converges at rate O(||h||2 + r2 j. To obtain y +1, can use the following algorithm:

(E + r2R1 j w(1 j = n (E + r2Rp j yj + r (Ay + pj, (7)

\ 7 jg=1y '

(E + r2Rpj w(pj = w(p-1j, p = 2,•••, P', (8)

yj+1 = yj +rw( pj (9)

For functions wp, p = 1,..., p-1 boundary conditions are used

p'

w(1j = (E + r2R2 )(e + r2 Rp' j^ at X1 = 0, X1 = ¡1, X2 = 0, X2 = l, w(pj = n (E + rlRa)vt,

a=p+1

X2p-1 = X2p-1 = l2p-b x2p = x2p= l2p> P = 2,...5 P'-!. (10)

To solve systems of difference equations (7) - (8), one can use fast direct methods for solving two-dimensional grid elliptic equations. The restrictions on the shape of the region G can be weakened: at p = 3, a = 0 is the region G can be a connected union of cylindrical regions Gp with

generators parallel to the axis OX3 .

3. Locally-two dimensional schemes.

Let assume that the following conditions are satisfied with respect to the shape of the region

Intersect an area with any plane 5*2^-1,2/3= {^2/3-1-hp), where hp~\, hp are ur|it vectors of numbers 2p-1, 2p respectively, 1 <p<[(p + 1j/2j, consists of a simply connected two-dimensional area.

In the area G it is possible to build a connected grid ®h with steps hp, p = 1,..., p .

The set of nodes of -dimensional grid consists of a collection of interior and boundary nodes. The set of internal nodes ®h is formed by the intersection points x = (X1,X2,...,XpjeG of the hyperplanes xp= Jp- hp, p = 1,..., p, Jp= 0, ± 1, ± 2,....

The set of boundary nodes yj., is obtained as a result of the intersection of all possible lines Cp, such that each of them is parallel to the corresponding base vector (unit vector) /p and passes through at least one internal node, with a boundary r. To construct a p -dimensional mesh, characteristic sets of nodes are determined, as in the case of VOD. Introduced: the set of all boundary nodes y h; set of nodes, boundary for all kinds of 2D planes intersecting ^2p-\2p, 1 <p<[( p + 1j / 2]

intersecting G, y (2p-lj 2p; a lot of border nodes belong to planes S^p^, 2p, ®h (2p-1 2pj; many of all border nodes o>h; many nodes, irregular in direction xa -®h*a, a = 1,..., p; a lot of irregular nodes lying in planes; S2p-1,2p -®hh**(2p-1,2p); the set of all irregular knots ®h*, the set of all regular

knots (o\. Here and below the symbol [x] means the largest whole, not exceeding x.

For simplicity of presentation, we will further consider the case of a three-dimensional domain (p = 3). Consider an additive LTS of the form

yj+V2 _ 2yj + y-1/2 Al +ÀW j+V2 + j_1/2\ M

T2 4 ly +y ) + " 2

(11)

yj+1 _ 2 ^+12 + yj = iA3 ( yj+1 + yj ) +1^2, j -1.2..... (12)

y (x. 0) = u0 (x).

\

E _T(A1 + A2 )

J

yj+12 = F1 at t = j. (13)

F1 = u0 + j_ u0 + j (A1 + A2 ) u0 + j2 ^f1 _1 (A1 + A2 + A3 + f )|. at t = 0 The boundary conditions have the form

y

j+1/2 _

m(x, tj+12 j, at x ur2, (14)

yj+1 =^( x, tj+1 j, at x eyh, (15)

In scheme (11)-(15) for approximation of relations of the form (L[u + Lqu + / j/2, (L3U + /2j/2 use homogeneous difference operators of the second order of approximation,

respectively, (A1 +A2j(yj+1/2 + yJ-1/2j^4 + p//2 and A3(yj+1 + yj+ ({/2. In this case, the coefficients of the operators A1, A 2 and the right-hand side (1 are taken at the time instant tj ; the

operator A3 and the right-hand side p 2 are determined at the time tj + r/2. The function yj+12 can be found from the equation

2

yj +12-^(A1 +A2 jyj +1/2 =oj , (16)

where $j is the known right-hand side, and the function yJ +1 is found from the equation

y

j +1 A3 yJ+1 = *2+12. (17)

where the function ®j+12 is known. with the boundary conditions (14). (15).

Let z} +a2 = yJ+a2 _u (x. tj+a/2 ). a = 0.1 is the error of scheme (11) - (15). For the error. we have the problem:

Zj +12 _2Zj + ^_12 =1 (A1 +A2)(zj+12 + zj_12) + ^1 (18)

j 4 /

zj+1 — 2 zj+V2 + zj

-1 A3 (zJ +1 + zj j + y2, t >r,

E — T(A1 + A2 )

z12 T

at t =r,

t2 2

z (x, 0)- 0 at x eo>h, zj+12 - 0 at x eyh uy2

zj+1 - 0 at x <

yh

where

j+12 — 2u j + uj—12

y -A1 + A2 (uj+12 + uj—V2 ) — u

A3/ j+1 A uj+1 — 2uj+12 + uj p2 y - (u +1 + uJ I----+ 2

p1

4 V > r2 2

The scheme (11) - (15) is additive, since y - yy + y - o(t2 + Indeed, since the equalities hold

1 (A: + A2 )(u j+12 + uj—12) - ((L + L2) u )j + O (h2 + h2)

1 (Aj + A2 )(uj+12 + —12) - ((L + L2) u )j + O (h + h2) at x e ®h,l u ah

0

h2 + h2 ) at x e C , u C

0 , , ^0 '0,1 uah,2,

j+12

+ O

at x e a

h, 3'

1 A3 (uj+1 + u )-(L3u) 1 A3 (uj +1 + u )-(L^u)j +12 + O(h3), at x e®h,3,

+12 — 2 u + —12 1 fa2u Y

+1 — 2uj+12 + uj 1 fa2u

vat2 , V +12

+ O (r

vat2 ,

n

j

1 d2u

+ O(r2 ), then

V

(L1 + l2 ) u — 1 + f1

y +1/2 -1 2 "2

1 ^ V+V2

1 5 u

L3u — + f2 2 5t2

O(h2 + h| +r2), xe®0i u®0 2,

>( h12 + h2 +r2 ),

)(h1 + h2 +r2 ),

O ( o + ho + r ), x ea, 1 ua

^0 , -0

0,1w h, 2'

(19)

(20)

(21) (22) (23)

(24)

2

T

T

T

¥2

O ( A2 +r2 ), x

1 00 * * * *

and y/= y/\+ ¥2+ ¥l+ ¥l= ¥l+ ¥2+0\T

O( h3 + r2 ), x eö0

h,3

that is, the scheme has a total approximation.

Let further assume that the region G is a parallelepiped G - {0 < xa < la, a -1,2,3| and the

grid &>h is uniform in each direction: a>h — ^ (¿1h1, ¿2h2, ¿3h3 ) ; ¿a

— 0,1,..., Aa; ha —

a

N

a,

In the space of grid functions °Q defined on the grid cojx and vanish at its boundary y ¡t, we

introduce the scalar products

N1 —1 N2 —1 N3 —1 N1 N2 —1 N3 —1

(u,«9)- I I I ui1i2i3&i1i2t3h1h2h3; (u-I I I ut1t2t3^t1t2t3h1h2h3;

'1=1 '2=1 '3=1

'1 =1 '2 =1 '3 =1

N1 N2 -1 N3 -1 N1 -1 N2 N3 -1

(us\ = Z Z Z u11t2t3911t2t3h1h2h3; (us]2 = Z Z Z ui1t2t3311t2t3h1h2h3;

'1 =1 '2 =1 t3 =1

'1 =1 '2 =1 t3 =1

N1 -1 N2 -1 N3

(w,,9]3 — Z Z Z u11i2i3911i2i3h1h2h3.

¿1—1 ¿2 —1 ¿3 —1

Let us prove that scheme (11)-(15) is absolutely stable and converges at a rate no worse than O(|\h\\2 +r). The solution of problem (18)-(23), can be represented in the form z — 7 +9, where 77 is

the solution of problem (18)-(23) with right-hand sides y/a - °y/a , a = 1,2; S is the solution of the

same problem, but with right-hand sides, ya, i.e. we have a problem for the function ij:

V 12 — 21 + ^ 12 = A1 + a2 j+1/2 + „ j—12)

J+V2 - 2r' +VJ-12

r2 4

r

'+1 -+A1+A2 _ A3/ y+1

4

(rj+1 + rj )

r/J +rjJ + l//2, />z\

^-_(A! +A 2 )

\A1 +A2

2-= ¥\ at t=~,

r/(x, 0) = 0, x e®h ,

and also the task for the function S :

+12 - 2&' +&j-12 _ A1 +A:

2 (s'+V2 +s'-V2 ) + *¥1

s'+1 - 2S +12 +s A3 / ni+1 ' *

-= -j3 (&+1 + S')+V2, t

r2 ( 2

IT WA A >V2 * r

E -"r(A1 + A2)

— = ¥1 at t = -, 2 2

S(x, 0) = 0, x eö/j

(18) (19) (20") (21)

(18") (19")

(20") (21")

2

= ^

2

T

2

4

r

Let us first estimate in some energy norm the function ij. To do this, we multiply equation (18') by j+1/2 - jj-1/2, and equation (19') on j+1-j j scalarly. Instead of relation (18'), obtain

_L(Li+1/2 _VJ)s L+V2 V) + LV-V2

(L-L-V2 ) • [L+V2-L )+L-L-1/2

), [L+12-U )+(

4VV /4

Further. we take into account that

/+12 j j+1/2 j j -12 J

L -U = tL , L L ~tLt

\ '2

((Al +A2 +12

(25)

(26)

a12 ' (Ui/1+

((A1 + a2

i-V2LJ -1/2 L-

a12 ' (j V2

a2 ' (U3/2+ a2' (j 1/2 '

Let get the energy identity, which call the first

J+1/2

hi '2

a/ ' T )2 - 4 ( a2' (J12 f + ( a/ ' (uxJ"12 f + L± ) (27)

In a similar way. obtain the second energy identity

J+1

U

'2

J +12

U

'1

1

J +1/2 I J +1

ai ' IL

x3

1

+ — 4

a3 +12' (J

+ r

>2'+1/2UJj+1 + LJ+12 ). (28)

Let assume further that the coefficients aa( x. t). included in the operators La . a = 1.2.3.

satisfy the Lipschitz condition with respect to the variable t. i.e. the inequalities

a

(x, t + r) < (1 + C^t)aa (x, t)' C4 = const, a = 1,2,3.

Using inequality (29), from identities (27) and (28) we obtain the inequalities

J+12 U 1

L

2 1'

+ —

4

V

■J-)21+ ï^('X/;1'2 )21 < ï(C4T +1)( a!-1, J2 )

-I1 V -I2 V

+1 (C4T +1)

J-1 ^ j-12 '2

a2 , L

+ r ( >1, Lttj+12 +lJ2 ).

L

2+1

L

2+12

1

+ — 4

V

a3+12, ('

1

< - (C4T +1)

( -_\j2 t ' ^2 a3-/ , (LX3

+ r( >2, Ltj+1 "L+12

(29)

(30)

(31)

Let add inequalities (30) and (31) and arrive at the relation

J+1

2 1' +—

4

1'-L+^l +4(a2- J2)

1

+ -4

(

J +12

2+ + (1 + Qr)>

zv

r-(LX1-12 )21 +( a2-1j (LéT12 )

(32)

+1 (1 + C4T)

t*.L )2 +t{>J.L^ +L{2^(vf1^' )

Inequality (32) can be rewritten as

4

2

+

1

t

t

2

x

Jj+1 < (C4T +1) J

i J +12 ^ iLf.J +12 j+1 j+12

Jj =

t2

a/ 1, fë 12

where

aj 1, (r21/2

J 12 ' rj

'x3

■ \2

(33)

(34)

Let us consider an estimate for the last two terms on the right-hand side of inequality (33): aj — t ( v/', +12 + ' ) + r ( v +12 ' j 1 + '' +12 ) — 7 ( v ' '' +12 ) + 7 ( v +12 ' ''+1 ). Let transform the last relation using the equalities

o / o /

v{ =- v2>

■ i + V Vj+1/J -

^2 =

^=vr1/2-rlvf1/2

As a result, have

Aj —-<(>+"2-j ) '+12 ) + t(>2+V2,r/'+1 ) —t2 (j2'+12 )

Note that the equalities:

(v'1'2-'''+12L—(Vf-12 )+( j'2'+12

2^ j+1/2 jV

+ r-| '//2

(»V^2, jV2 «/

'2 11

j+1/2 „j+1 •2

^2 ' >%t

considering which, have

Aj = r2 ( Vj+12. rfL -r2 ( V2?12, rj2 ) + r2 (•v^2. +12

rr2 (jv -12

In view of the Cauchy-Bunyakovsky inequality, from (35) we obtain the estimate

(35)

Aj

<rJ

VJ +12,rj+1 )+(V2+V2,rj +12

j+VJ „i+AJvj1/2 jVJ

^2

ni

+ T

+ T

otJ+V2 „j h

j-yz

Vit'

VJ2+f '

i+VJ

n TuT

n

> j+1/2 VV

rd '2

Let us strengthen the last inequality using s -inequality

Aj <rJ

vj+1/J, j )+(vjf2, j12

+ r

¥

j+V2

2t1, t

,7-1/2

4s

o j +1/2

¥J7

e + —

r

>1T

Further, take into account that

r

j-VJ

J-VJ

2 /2 2 < i-16

2 -J

< —

16

j - VJ

2 -

<

2 f

16q

a1-1, (j VJ

r

j - VJ

<

* jir^2

16c

(36)

(37)

(38)

and also the inequality

V±12

V2 \t

n

J-1/2

<

«1

n

J-1/2

r

j2

(39)

Let put in the last inequality q = 4qq/l2 , having previously multiplied both sides by 1/2, and

using inequality (37), obtain

j2

J-l/2

<

<

£

4t

fL 2r

n

j~ 1/2

+ -

or

j "I '-V2

8s1 2

j2

rl2

4q £

J-l/2

: 8 • 4qs

2t12

j2

rlt

+ -

8 • 4qs

V2txt

(40)

Setting in inequality (39) q = 4qe/ l| and applying inequality (38), as a result of reasoning

similar to the previous ones, we arrive at the estimate:

°,J+1/2

2 tt

n

7-1/2

4t

°2-1, ('X^

xli

■4c1£

°.J +V2

2tt

(41)

Adding inequalities (40) and (41), we obtain the inequality

j2

J-l/2

<-

4t

j-1 -V2

j-1 -12

T l(+l

2 , ,2 2

4s 8c1

j2

(42)

Substituting inequality (42) into relation (36), we obtain the desired estimate for Aj

Aj <t {('vi+1/2,j MjV +12

> j+12 V2F

2 /2 , /2 % l1 +l2

8q

j+1/2 2rxr

(43)

+ st

U2

2 1 r -+ -

4

1(C12 )21+ 4 r oj-1, ('C12)

-1 {

Substitute estimate (43) for a, into inequality (33), we arrive at the inequality

J j+1 < (C4T +1) J j + st

U2

2 i r , w , ,/„\2'

j-1 I j-12

a , i'

x1

+t

2 rUj +12 j+1w 0,^+12 j +1/2 \ \ , t_

4s

1 4 {

j2

oj 1, ('j 12

x2

2 I2+I2 , 1 2

8c

v2txt

Taking into account expression (34) for Jj, and, strengthening the resulting inequality, we arrive at the energy inequality

Jj+1 <((Q +S)T + 1) Jj

+ t

2 |7o i +12 j ,+12 j +12

^2

4s

j +12

^It

2 12+12 1 2

8c

W2 txt

(44)

Let us sum up the resulting inequality over j = 1,..., Jo, arrive at the estimate

<

2

£

Jo

Jj +1 <( C4 +s)ZTJJ +r((V20 VJ

J=1

VJo +12 „Jo +^°„/Jo +V2 „Jo V

-r

1/2 „1

1/2 1/2

.2 Jo

J=1

J +12

2 /2+/2 , 1 2

8ci

¥

J±1/2 2/,/

r +Jl

Identities (27) and (28) for the first time layer (j - 0) take, respectively, the form

12 Vj h

0 i 1/2

a1 , V

a2, (V

0 /„,12

+ rrwH4 2

(45)

4 '2

12 Vj h

12 / 1 T / I V_

x3

, . - 12 1 , 1/2

Adding the last relations and, taking into account that r,12 - ,12, r (,1 +l12) arrive at equality

j-(W )+(>?,,12 ^vf,,1 )+r(v2f.,1/2).

Substituting the expression for J1 from (46) into inequality (45), obtain

= V

Jo

Jjo0 +1 <( C4 +-)ZrJJ +r[(V20 +12, V J=1

°,> +l^Jo + iVJo +12 VJo +12

2 jo

— I

/1 „

4s

j =1

' J+12

2 i2+12 , 1 2

8ci

J+12

1

^lüt

To complete the functional assessment Jj +1 we use the inequality (see (42))

<-

8r

and also inequality

V+12,

f Jo L Jo +1^2 a1 , (Vx1

^o+l2,<

(

J o_+12 1

y'lui

n

Jo +1/2

<

1

Jo L Jo +12 2

a2o, (V

/ 2 + / 2

r /1 + /2

2 8ci

Jo+12 1

°„Jo +12 Jo +1

¥2

rg <

)< VJo +12

io+1

Vj

h

J_ 2r

io+1

Vt

t + —

2

VJo +12 ¥2

substituting them into estimate (47), we arrive at the inequality

1

Jo

I

J=1

JJo +1 <(C4 + s)IrJJ + 2 JJo +1

+

2 i2 +12 r l1 +12

Jo +12

T

2 8q

From the last inequality we obtain

VJo +12

_2 Jo

— I

4s ^ 4S J=1

v-V2

2 I2 +12 , 1 2

8^

J+12 1

(46)

(47)

2

1

(

Jo

Jj0 +1 < 2 (C4 + q j X rJj +r

J=1

12 +12 l1 +12

8c1

Jo +12

r 2 j0

X

j=1

3 J+12 V2F

2 /2 + /2 , 1 2

8c1

^2 \T

V^2 v2txt

V{o +12

(48)

Notice, that Fj = O (r2 j,

F{ = r Jo

(12 +12

8c

'1

^2/j/

+12

where

2

V0 +12

-2 7b

-—X

J =1

J+12

^F

2 /2+/2 , 1 2

8c

V^2

^2txt

(49)

if V2 (x1,x2,x3,tj = (L3U-12 •d2ujdt2 + /2 j^2 has a bounded second derivative with respect to t. Moreover, from (46) and the Cauchy-Bunyakovsky inequality we have

T /0 12 1 \ , /0 1/2 1/2( o 1/2

J1 = r( vi 'j + r( v21r1 j<r( V{

rh

O 12

21

,12

(50)

From problem (4.18') - (4.21'), applying the theorem on the estimation of the solution to the inhomogeneous grid equation [1], obtain

J/2

1

>h

< Mr, M = const

2 c ■- - / 2\ obtain Jj +1 < Fj + 2e"v"4 'Jo (C4 + - j X rFj , and therefore, Jj = O (r j.

Therefore, taking into account relations (50), we arrive at the estimate

Fq -J1 = O(r2j, Fj = O(r2j, j = 1,..., Jo. The generalized Gronwall lemma is applicable to inequalities (48) and (49), from which we

2(C4 +^0 (c4 +qj J0

J=1

Task (18")-(20") for S differs from the corresponding task for the function j only the right-hand sides included in the equations - instead of the functions Vi and V2 must be substituted Vi and V2. Therefore, have

Pj+1 <(C4r + 1jPj + r( Vj, S+12 j + r(*Vj +12, S'+1 j, (51)

where

PJ =

h

2 1 (

+ —

4

J-1, (sJ-12 j2] +1 (aJ-1, (SJ-12 j2] +1 f-J-12, (sJ j

1 \ x1 4 2 \ x2 4 3 \ x3/

J1 V J 2 V

(52)

The course of further reasoning is the same as for the assessment J j , therefore, we will describe it briefly. Let

=r[vj S+12 j+r^V+1/2,S'+1 H^V SJ+12 )+(VJ'+12,SJ+1 ))T-r{VJj SJ-12 ^rCV j then the estimate

Bj <rf(*vJ, +12 j + (VJ+V2>+1 j

8c1

* j +12 V2F

% l12 +122

8c1

* 7

Vi

v

2

+s

2 1

+ —

4

J-'. (NJ-V2 )2 + La2'-'(NXJ-12 )2 +L°33'-V2' NN )2

(53)

Substituting (53) into inequality (51), we arrive at the estimate

Pj+1 <(C4 r +1)PJ +a

V L 1V ( 1 VV

: J '

J

S 1. J 2

j + 2

S 2

J J+1

JJ

1

+ — 4

(f ( 1 V2

J-

aJ 1.

N 2

v J

JJt

f ( 1 V2

j

l3 8c

1

J+2 S 2t

( i 2 + i 2 V l1 +12

8c

v J

j S it

J1

N 2

vJ

(a3-2. Ji )2

The resulting inequality, taking into account the expression for the functional Pj will take the

form

Pj +1 <((C +s)r +1) Pj +a

' ' 1V (*J+1

*J J

S 1

J 2

S 2

2. J

JJ t

8c1

Summing up inequality (54) over j, j = 1,.. . /'q , obtain

Jo

f

PJo +1 < (C4 +s) I rpj

J=1

1V

*Jo Jo +1 S 1 . J 2

* J

S 2J'

( . 1

(№V 8q

v J

* J S U

(54)

* j o + S 2

2, JJo +1

v

( o 1V

S 1

J 2

( 1 V

* 9

S 2. j1

Jo

Ai

s j=1

8ci

1

* J + 2

S 2F

( /12 + /22 ^

8 c

1

v J

*J

s n

(55)

A + P

The identities are valid

( 1 1 ^ ( 1 ^ #2 — #0 #2 —

1

J2 J2

A2

J

'2

1 '1

' ( 1 V2'

adding which we have P1 -

(* o 1V S1. J2

v J

1 J2

( 1 V * 2

S2, J1

1 4

( 1

a32, K

J2

x1

( ( 1 V2'

J2

x2

(* o 1 S 1.J2

v J

( 1 V * 2

S2, J1

v J

v /

The last equality allows us to simplify inequality (55)

2

Jo

j +1 <(C4 +s)^TPj +

J=1

* > +1 ¥ 1 , & 2

( . i

Jo + ¥ 2

2, & Jo +1

_ Jo 7 j=i

8c1

¥ 2t '

8c1

v

* i ¥ it

(56)

It is not difficult to obtain the inequality

1 1

„Jo +1 ¥ 1 , & 2

( . i

* Jo + ¥2

2 &Jo +1

v

< 2 PJ0 +1 + 16c

2

/ >■> rs \ * J o

('1 + '2 ) ¥1 +'32

1 * Jo + 21

¥2

Substituting this inequality into estimate (56), we arrive at the relation

Pj0 +1 < 2(C4 +7)X tPJ +J-J= 1

J 8c1

( 2 . 1 * Jo + 2 ¥2 21 1 * J + 2 ¥ 2t 2

('? + '2 ) * Jo ¥1 + '32 2r ' 2 '3 8c1 ('i2 + '21 + --2 8c1 V J *j ¥ 11

V J

(57)

Let us further assume that |h| < cor, co = const

With regard to the functional Pjo +1 the reasoning given when evaluating the functional is

valid if we take into account that

, 2

* J+ ¥ 2t'

*j

¥ it

( („ ,2 11

Mi

- + r

V v JJ

( (1, i2 112

<

Mo

- + r

< M;

1 (+1)2r2,

:(co2 + 1)2r2,

M1 = const,

Mo = const

V v JJ

Therefore, inequality (57) can be represented as:

Jo

Pjo +1 < 2 (C4 +e)YdrPj + Fj

J=1

Ft =

Jc

1

o 8c

2

* Jo

('12 + '2) ¥1 + '32

* Jo +7

¥2 2

2 1

2r

jo

8c

J

j+

¥ 2 i

( '1 + '2 1

8c

1

v J

*j

¥ 1f

r = OI Ihl4 +r2

According to the generalized Gronwall lemma, we have

2(c4 +7)t Jo

PJo +1 < FJo + 2e

Jo

(c4 + e)£ rFj <M(|h\4 + r2)

j=1

(58)

Considering that Jj = O (r 2), Pj = O (| h|4 + r 2) arrive at an estimate of the convergence of the locally two-dimensional scheme (11)-(15) in the energy norm, i.e.

<

2

j+1/2 \zj + j+1/2 \z- + j+1 \Zj

x1 ■ 1 x2 ■ 2 x3 ■

< M ||A||2 +r) j - 0,1,2,... j — 1, where M - const > 0, M depends only on C4.

If u(x,t) has continuous in QT derivatives with respect to xa ,a-1,2,3 up to the fourth order inclusive, coefficients aa (x, t), included in operators La , a -1,2,3 , satisfy the Lipschitz condition in the variable, i.e., the inequalities aa(x, t + r) < (1 + C^r) aa(x, t), C4 - const, a-1,2,3, f (x, t) twice differentiable with respect to t, then LTS (11) - (15) converges at a rate no worse than O(| h\\3 +r).

References

1. A.A. Samarskiy. Introduction to the theory of difference schemes. M. Nauka, 1971, 552 p.

2. A. N. Konovalov. Method of fractional steps for solving the Cauchy problem for the multidimensional equation of oscillations, DAN USSR 147. No. 1, 1962, pp. 25-27.

3. G.I. Marchuk. Splitting and Alternating Direction Methods // Handbook of Numerical Analysis. Amsterdam, North-Holland, 1990, Vol. 1, pp. 197-464.

4. A.A. Samarskiy, P.N. Vabishchevich, P.P. Matus. Difference schemes with operator factors. Minsk.-1998-442 p.

5. A.I. Sukhinov. Locally two-dimensional schemes for solving multidimensional parabolic equations on matrix-type computing systems. Proceedings of higher educational institutions. Maths. 1984. No. 11. pp. 45-53.

6. A.I. Sukhinov. Additive schemes for modeling three-dimensional equations of heat conduction in cylindrical and spherical coordinates Differential Equations. 1987. Vol. 23. No. 12, pp. 2122-2132.

7. A.I. Sukhinov, V.S. Vasiliev Local-two-dimensional schemes for approximating the three-dimensional heat equation in toroidal coordinates

Proceedings of higher educational institutions. Maths. 1996. No. 3. pp. 58-67.

8. E.S. Nikolaev. Methods for Solving Grid Equations), Moscow: MAKS Press, 2018, 424 p.

9. A.I. Sukhinov. Parallel algorithms based on two-dimensional splitting schemes for multidimensional parabolic equations // Parallel Computational Fluid Dynamics, Proceedings of the Parallel CFD 2002 Conference, Kansai Science City, Japan, (May 20-22, 2002) ELSEVIER, 2003, pp. 345-352.

10. A.I. Sukhinov. Two-dimensional splitting schemes and some of their applications. M. MAKS Press, 2005. 408 p.

Authors:

Sukhinov Alexander Ivanovich, Don State Technical University, (Gagarin square, 1, Rostov-on-Don, Russia), Doctor of Science in Physics and Maths, Corresponding Member of RAS. Email address: sukhinov@gmail.com, ORCID: 0000-0002-5875-1523

УДК 519.6 10.23947/2587-8999-2020-1-2-71-86

ДВУМЕРНЫЕ СХЕМЫ РАСЩЕПЛЕНИЯ ДЛЯ ГИПЕРБОЛИЧЕСКИХ

УРАВНЕНИЙ *

А.И. Сухинов

Донской государственный технический университет, Ростов-на-Дону, Российская Федерация

В статье рассмотрены схемы расщепления по геометрическим направлениям, аппроксимирующие начально-краевую задачу для р-мерного (р > 3) уравнения гиперболического типа цепочкой двумерных-одномерных задач. Рассматриваются два способа построения схем расщепления - с оператором, факторизованном на верхнем слое, алгебраически эквивалентные схеме переменных направлений и аддитивные схемы суммарной аппроксимации. Для первой схемы ограничения на форму области О при р — 3 могут быть ослаблены по сравнению со схемами переменных направлений, представляющими собой цепочку трехточечных задач на верхнем временном слое - область О может быть связным объединением цилиндрических областей с образующими, параллельными оси ОХ3 . Во

втором случае для трехмерного уравнения гиперболического типа построена аддитивная схема, представляющая собой цепочку «двумерная задача - одномерная задача» и аппроксимирующая исходную задачу в суммарном смысле (на целых временных шагах). Доказаны устойчивость и

сходимость построенных схем: со скоростью О|||Щ2 + т21 - факторизованной и со скоростью

й|| + т| - аддитивной, где ||Л|| - норма шага пространственной сетки, т - шаг по времени, при

соответствующих ограничениях на гладкость функций, входящих в постановку начально-краевой задачи. Для численной реализации построенных схем - численного решения двумерных эллиптических задач - можно применять быстрые прямые методы, базирующиеся на Фурье-алгоритме, методах циклической редукции для трехточечных векторных уравнений, комбинациях данных методов и других методах. Предлагаемые двумерные схемы расщепления в ряде случаев оказываются более экономичными в смысле суммарных временных затрат, включающих время выполнение вычислений и обменов информацией между процессорами по сравнению с традиционными схемами расщепления, базирующимися на применении трехточечных разностных задач для многопроцессорных вычислительных систем, с различными структурами связей между процессорами — типа «линейка», «матрица», «куб», с универсальной коммутацией.

Ключевые слова: схемы расщепления, гиперболическое уравнение, аддитивная схема, устойчивость и сходимость схем, двумерные схемы расщепления, многопроцессорные вычислительные системы.

Автор:

Сухинов Александр Иванович, Донской государственный технический университет (344000, Ростов-на-Дону, пл. Гагарина, 1), Член-корреспондент Российской академии наук, доктор физико-математических наук, профессор, заведующий кафедрой, sukhinov@gmail.com, ОЯСГО: 0000-0002-5875-1523

* Исследование выполнено при финансовой поддержке РФФИ в рамках научного проекта № 20-01-00421.