TO NUMERICAL METHODS FOR SOLVING MULTIDIMENSIONAL INTEGRO-DIFFERENTIAL EQUATIONS Текст научной статьи по специальности «Математика»

ВЫЧИСЛИТЕЛЬНАЯ МАТЕМАТИКА

Научная статья УДК 519.64

DOI: 10.18101/2304-5728-2023-3-34-52

TO NUMERICAL METHODS FOR SOLVING MULTIDIMENSIONAL INTEGRO-DIFFERENTIAL EQUATIONS

© Beshtokova Zaryana Vladimirovna

junior researcher,

Institute of Applied Mathematics and Automation KBSC RAS Russia, 360000, Nalchik, str. Shortanova, 89a [email protected]

Abstract. The third boundary value problem for a multidimensional convection-diffusion equation with memory effect and non-local (integral) source is investigated. To solve numerically the multidimensional problem, a locally one-dimensional difference scheme is constructed, the essence of the idea of which is to reduce the transition from layer to layer to sequential solving of a number of one-dimensional problems in each of the coordinate directions. Using the method of energy inequalities for the solution of a locally one-dimensional difference scheme, an a priori estimate is obtained. The main research method is the method of energy inequalities. An a priori estimate of the LOS solution is obtained, from which follow uniqueness, stability, and convergence of the solution of the difference problem to the solution of the original differential problem at a rate equal to the approximation error. Numerical experiments were carried out.

Keywords: third initial-boundary value problem, locally one-dimensional scheme (LOS), a priori estimate, difference scheme, parabolic equation, integro-differen-tial equation, equation with memory, equation with non-local (integral) source.

For citation

Beshtokova Z. V. To Numerical Methods for Solving Multidimensional Integro-Differential Equations // Bulletin of Buryat State University. Mathematics, Informatics. 2023. N. 3. P. 34-52.

Introduction

In the mathematical modeling of many processes in mechanics, physics, biology, economics, there are such systems with memory, the behavior of which depends on the entire «history» of the [1] system and is not entirely determined by the state at the moment, therefore it is necessary to describe such systems by integro-differential equations containing the corresponding integral over the time variable, i.e. when an unknown function is included in the differential expression and, at the same time, appears under the integral sign. Partial differential equations with memory are studied in

[2-5] - when describing the thermomechanical behavior of [2-3] polymers, viscoelastic fluids at low temperatures [4-5]. Boundary value problems for parabolic equations with a non-local (integral) source arise when describing the mass distribution function of drops and ice particles, taking into account the microphysical processes of condensation, coagulation (combining small drops into large aggregates), crushing and freezing of drops in convective clouds [6-9].

From the point of view of numerical implementation, multidimensional (in terms of spatial variables) problems are considered the most complex. The difficulty lies in the significant increase in the amount of calculations that occurs when moving from one-dimensional problems to multidimensional ones. In this regard, the problem of constructing economical difference schemes that have the ability to sufficiently effectively stabilize solutions (stability) and require Q arithmetic operations proportional to the number of grid nodes, so that Q = O (hp), where h = min hi, p is dimension of

space, hi are grid steps in direction xi.

The work is devoted to the construction of a locally one-dimensional difference scheme for the numerical solution of the third initial-boundary value problem for a multidimensional differential equation in partial derivatives of parabolic type of general form with memory effect and non-local linear source, the main idea of which is to reduce the transition from layer to layer to the sequential solution of a number of one-dimensional problems in each of the coordinate directions. Moreover, although each of the intermediate problems may not approximate the original differential problem, in the aggregate and in special norms such an approximation takes place. These methods are called splitting methods, which were developed in the works of Douglas J., Peaceman D.W., Rachford H.H. [10-11], N.N. Yanenko [12], A.A. Samarsky [13-14], G.I. Marchuk [15], E.G. Dyakonova

[16] and others.

In the works [13-14, 17-24] for the numerical solution of multidimensional parabolic equations, a LOS was constructed

Thus, in [13], in an arbitrary domain G, a locally one-dimensional scheme is considered for solving linear and quasilinear parabolic equations. The stability of the difference scheme with respect to the right-hand side, boundary and initial data is proved, as well as convergence with a rate O(h2 + t). In [14], locally one-dimensional difference schemes are considered on arbitrary "nonuniform grids"for linear and quasilinear equations of parabolic type with "heat conductivity coefficient" ka = ka(x,t,u) depending on the "temperature"u = u(x,t).These schemes converge on arbitrary nonuniform grids

In the work [17] locally-one-dimensional difference schemes for the frac-

tional diffusion equation in multidimensional domains are considered. Stability and convergence of locally one-dimensional schemes for this equation are proved. In [18] for a fractional diffusion equation with Robin boundary conditions, locally one-dimensional difference schemes are considered and their stability and convergence are proved. In the [19] locally one-dimensional difference scheme for a general parabolic equation in a p—dimensional parallelepiped is considered. To describe microphysical processes in convective clouds, non-local (nonlinear) integral sources of a special type are included in the equation under consideration. An a priori estimate for the solution of a locally one-dimensional scheme is obtained and its convergence is proved.

In the article [20, 21] discusses the construction and study of parallel algorithms for solving the multidimensional diffusion-convection problem. Schemes of a special type are constructed - explicit-implicit difference schemes with weights, which assume the representation of the original problem as a chain of two-dimensional and one-dimensional problems.

In the work [22] the analysis of the initial-boundary value problem with a multidimensional space variable belonging to the Euclidean space Rn, (n >

2) for the transport equation of a continuous medium with distributed parameters on a network-like domain is considered. An algorithm for the numerical solution of the problem under consideration is proposed.

This work is a continuation of the author's series of works [23-25] devoted to the study of local and nonlocal boundary value problems for multidimensional parabolic equations.

1 Problem statement

In a cylinder QT = Gx[0 < t < T], the base of which is a p - dimensional rectangular parallelepiped G = {x = (x\,x2, ...,xp) : 0 < xa < la,a = 1,2, ...,p} with boundary r, G = G U r, consider the problem

t

du f

K(x,t,T)u(x,r)dr = Lu + f (x,t), (x,t) e QT, (1)

Qu

ka(x,t) -- = в-aU - V-a(x,t), Xa = 0, 0 < t < T,

OXa

-ka (x,t) dXa = в+aU - V+a(x,t), Xa = la, 0 < t < T,

u(x, 0) = u0(x), X £ G,

(2)

(3)

where Lu = J2 Lau,

a=l

LaU =

д 0x„

ч

du \ du f ka(x,t) — ) + Ta(x,t) —--I pa(x,t)u(x,t)dx0

0 <co < ka(x,t) < ci, lra(x,t)l, lkXa(x,t)I, lrXa(x,t)I,

IK(x, t,T)|, |p(x, t)|, IP±a(x,t)I< C2, (4)

u(x,t) e C4,2(Qt), ka(x,t) e C3,1 (Qt) ,

ra(x,t), K(x,t,T), pa(x,t), f(x,t) e C2,1(Qt), 0 < T < t,

c0,c1,c2 are positive constants, a = 1, 2,...,p, ^±a(x,t) are continuous functions.

Further, we will use positive constants Mi,i = 1,2,..., depending only on the input data of the problem under consideration

2 Locally one-dimensional scheme

We divide the interval [0,T] into equal parts UT = {tj = jT, j = 0,1, ...,j0} with a step t = T/j0. The interval [tj,tj+1] is divided into p equal parts by the following points tj+a = tj + ta, a = 1,2, ...,p, and we

^ p P

denote by A

a — (tj+ a—i ,tj+r ]. j+ p p

For each direction Oxa, we construct a uniform grid with a step ha =

Na, a = 1,2,...,p:

P

Uh = n , Wha = {x= iaK, ia = 0, 1, ...,Na}, a = 1, 2, ...,p,

a=1

h l ha, ia = 1, 2, ...,Na — 1, na = 1 hr, ia =0,Na.

Equation (1) can be rewritten as YlPa=1 £au = 0, £au = 1 — Lau — fa,

where fa(x, t), (a = 1, 2,..., p) are functions that have the same smoothness

P

as f (x, t) and satisfy the normalization condition ^ fa = f.

a=1

On each half-interval Aa, a = 1,2,...,p we will successively solve the problems

19§{a) _

£a^(a) =--^ — La^(a) — fa = 0, x e G, t e Aa, a = 1, 2,...,p,

y ' p dt K '

(5)

(6)

dti(a)

ka— fi-o$(a) , xa — 0

dxa

ka dX-.r — (a) , xa — la,

wherein

ft(1)(x, 0) = uo(x), &(1)(x,tj)= ft(p)(x,tj), j = 1,2, ...,jo — 1,

tf(a)(x,tj+*=1 )= a-l)(x,tj+o=1), a = 2, 3, ...,p, j = 0, 1, 2,...,jo - 1,

where La-d(a) = La-d(a) + pf K(x, t, r)$(a) (x, r)dr.

0

Using the technique of Samarsky A.A. constructing a monotone circuit [26, p. 401], we thus obtain for each equation of number a a monotone scheme of the second order of accuracy in ha, then we rewrite the equation with a perturbed operator La for a fixed a:

--dr = La$(a)+ fa, t G A«, a = 1,2, ...,p, (7)

p dt y

0xa

t 1a

— Pf K (x, t, T )•&(«) (x, T )dr — / p«(x, t)&(«)dx«,

where La^(a) = Xagxa( ka(x,t) ^щО}) + ra(x,t)

00 X« = , R« = °'5hia^ra 1 is the difference Reynolds number, x(-0-5a) =

(xl, ..., x«-1, x« 0.5h«, xa+1, ■ ■■, xp),

x = (xi,x2, ...,xp), r+ = 0.5(r« + \r«\) > 0, r- = 0.5(r« — \r«\) < 0,

b+ = 'L h- = r = r+ + r- a = k fr(-0-5a) A

ba = i, , ba = i, , 1« = 'a + 'a , aa = ka , b ,

ka ka v '

ra = ra(x,t), Pa = pa(x,t), <£a = fa(x,t), t = tj+1 .

Approximating on the half-interval Aa = [tj+ a—i,tj+a each equation

\ j p P J

(7) of number a implicitly we obtain p one-dimensional equations [26, p. 401]:

a a—1

y P — y P ~ jL a j+a

y y - =Kyj+a + <plP , a = 1, 2, ...,p, (8)

т

j+a

A ay = Xa( ^¿a) x + K^»1 a) yZ P + b-aaV'x a P

1 j Na jLa

— pY,K (x,tj ,tj' )y(x,tj'+P )r payj +P h«,

pj'=0 ia=0

j+^ j+P j+^ j+-

yj+ P = yia + 1 — yia P yj + P = yia " — yia -"l

yxa = h , yx a = h ■

ha ha

The difference analogue (6) takes the form

(1a) j+P R j+P fl

aa a yxa,0 = P-ay0 — a, xa = 0,

(N ) j+a j I a

—aa yxa,Na = P+ayNaV — №+a, xa = la.

Let us increase the order of accuracy of the boundary conditions (9) to O(h2a), then, using the equation (1), we obtain

j+a j+a

41 a)$xa,0 = R-a$Q P - V-a + O(ha).

From the latter, by the Taylor formula, we obtain

j+p j+a-i . a . a

0 5h yQ - yQ _ X n(1a)i,j+p R yj+p

°.5ha- _ X-aaa jyx q — P-ayQ —

T '

1 j 3 '+a Na (a)

-0.5ha1 K (x,tj ,tj ' )yQ P T - 0.5ha^2 Pay.!* K + V-a, Xa _

-j'=0 ia=Q

j+P Q+ — pp

n ri yNa — yNa (Na ) j+p o j+p

°.5ha -T - _ -X+aaa ^Na - p+oLvN« -

1 j Q '+a Na (a)

-0.5ha K (x,tj ,tj ' )yNa p T - 0.5ha^2 -ayi^ K + V+a, Xa _ la,

-j'=0 ia=0

where

V-a _ V-a + 0.5hafa,Q, V+a _ V+a + °.5hafa,Na , V±a _ V±a(tj), X _ _1_ r(Q) < 0 X . _ _1_ r(Na) > 0

0.5ha |r(0)| ,, 0.5ha >\

1+ , (0.5) 1+ , (N„-0.5)

The integral over the space variable is approximated by the trapezoid formula to achieve second order accuracy.

Thus, we obtain the following difference scheme

j+r j+ a — 1

V P — V P ~ a j+r

V-V- =AaVj+P + Pa P , a = 1, 2,...,p,xa e Uha , (10)

T

j+- j+—1

V p — V p _ j+r _

0.5ha- — A- y p + V-a, xa = 0,

a T a -a a (11)

j+aa j + a—1 a

0.5hay p -— = A+Vj+ p + ~p+a, xa = la,

y(x, 0)= uo(x), (12)

where

^ay _ Xalaa^Z x + ^a^ vZ ? + baaayx+a "

1 j N--£ K (X,tj ,tj ' )y(x,tj'+p )T Pa Vt^^a, Xa G Wh,

P j ' = 0 P i- = 0

j

a — (i ) j+P /-> j+P 0.5ha s , , x j '+P

Л- V = X-aa{aa>yXoi;0 - P-aVo P--p— K (x,tj ,tj ' )Vo T

P j '=0

N-

-0.5haJ2 PaVil^a, Xa = 0, i =0

Л+У = -x+aaONa- в+avlP - ^ £ K(x,tj,tj')у£PT

P =0

j '=0

Na j + % j+ ^

n,, va (a)k J 1 (a) У p - y' p

-0.5ha У j Payi ha, Xa = la, ~УТ = -

^^^^ " a p t t

i =0

3 LOS approximation error

j+a j+a j i a

Replacing in (10) — (12) yJ p = zJ p + U p we get the problem for

j+% z p :

j+- j+-=1 .

z - - z % r j+- j+

- = Л aZ % + Wa

j+a j+a j+a j+a j+a

where z p = yJ+ p —uJ+ p , u p is the solution of problem (1)-(3), ^a p = A auj+ i + j 'p — jji.

° ( 1 f) \ j+1/2 p ° Denoting by ^a = [l«u + fa — pdtj and noticing that ^a =

aa

a=1

p j+-

p

a

a=1

0, if £ fa = f, we represent ^a p = pa + :

j+a j + a-1

j+P À j+a j+p U p - U p ° ° (~ j+a j+ 1\

a P =AaU P +<£a P----+pa-pa = (AaUJ+ p - LaUj+ 2 J +

+ ( j+P j+1A (Uj+p - Uj+V 1 ( du V+l/2\ « + ,

+ {VaP - fa 2J ^^-T--p[g^J )+ a = ^ + 4>a.

Obviously

= O(hl + T), iJa = O(1), £ jP = £ ^Pa + C = O(lh2 + t).

a=l a=l a=l

We write the boundary condition for xa = 0 as follows:

j+P j+—

y p y p

0.5ha = X-aaa a^yxa,0 — fi-ay0 —

T

1 j j '+a Na (a)

-0.5ha P^K (x,tj ,tj' )y0 p T — 0.5ha^2 -aVia fra + 0.5hafa,0 + V-a.

Pj'=0 ia=0

Substituting Vj+p = zj+ p + uj+p , we have

j+p j+— pp

05h ^_z0_ = X n(1a)z(a) - 6 Z(a)-

°j.oha — X-aaa zXa,0 p—azo

1 j j'+a Na

-0.5ha K (x,tj ,tj' )z0 p T — 0.5ha^2 'Paz(i°)fra + p j '=0 i =0

( ) j+a 1 v^ j '+a

+X-aa(;a1 a)uXr,0 — 66-au0 p — 0.5haK(x, tj, tj')u0 p T

p j '=0

Na j+ p _ j+ ^

(a) u0 - u

— 0.5haJ2 Pauia fra — 0.5ha u0-0--+ 0.5ha fa,0 + V-a.

i =0

To the right-hand side of the resulting expression, we add and subtract

0.5ha1p-a = 0.5ha

1 f K(x,t,T)udT — f pa(x,t)udxa + fa — 1

p J 0 J p dt

0

d /, du \ , , du

»-\kaj-,- + ra(x,t) —

dxa V OxaJ dxa

j+1/2

Then, due to the boundary conditions (2), we obtain

a = 0.5ha1p-a + a, a = O(h2a + T) + O(haT).

j+a

So, the problem for the error zJ p takes the form:

j+a j+a — 1

z p — z p ~ I \ j+a

Zz-z- — Aaz(a) + j p , (13)

T

j+p j+— z p z p

0.5ha- = A-z(a) + a, xa = 0,

T

j+p j+ — z p z p

0.5ha- = A+ z(a) + ^+a, xa = la,

T

z(x, 0) = 0, 41

0

where

Фа = Фа + Фа, Фа = O(l), ф*а = O(k2a + Т), ф-а = 0.5ЬаФ-а + Ф-а,

о о » о

Ф+а = 0.5На1р+а +Ф-а = O(hа +Т), Ф±а = O(1)^ }^Ф±а = 0

а=1

4 Stability of a locally one-dimensional scheme

Let us multiply equation (10) scalarly by V(a) = Vj+p :

1 (а) (а)

-y\ ) ,У(а) Lp ъ

Лау(а),у(а) = Ф(а) ,у(а)

(14)

where

u, v

= EN:=0 Uia Via Па, II У(а)\\12{а) = £ У2^а,

г a =0

u, v

£ UVH, H = П Па, \1У(а)1\12Ш = £ \1У(а)ИЫа) H/Па xeuih а=1 г p=ia

Based on the Cauchy inequality with e, the Cauchy-Bunyakovsky-Schwartz inequality, Lemma 1 [27] and transformations

Na

У(а)^РаУ^)П

г =0

<

+

1

N

£ РаУ^Ь

г =0

<

< l\У(а)И1(а) + ПаЕ У*а Па <

2 la

N N

а / j У г a'hа < г =0 г =0

1 ц

< 11\У(а)111(а) + c2 2 £ У1 Па < М1\\У(а)1Ц2(

а),

г =0

У(а)^К (x, tj ,tj' )у{х,Ьjp )т j '=0

<

<

Ea ' I a

К (x, tj ,tj' Mx,tj )т

p ^

1 j '=0

< Т «хУ ' +P )В12(а) + 2I

Р j '=0

Ь2(а)

1

(а)

Ьъ(а)

<

p 112

\\Ь2(а)'

a

2

2

Oi

OL

Oi

a

1

after summing over ip = ia,p = 1,2, ...,p, from (14) we get

1 (\\yj+'\\l(,h))-t + Ms\\ysa ]\l(Qh) < MAs\\yja ' ]\l2(Qh) +

+M5(e)\\yj+'\\l2(th) + M6 £ \\yj'\\l2(tth)T+

j '=0

+ 2 [W^WUth) + £ (v-a + tia)H/ha\ , (15)

V iP =i a /

where e > 0, c(e) = 1 + 1.

1 a £

Choosing e < tMj^ , from (15) we find

-a 2 \ ' \\L2(th)) t+

2p \

j

+M\\yita!]\Uth) < M5 £ \\y(x,tj)\\l2(th)T + M6\\y(aHl2th) +

j '=0

+ 1 ( \\<f(a)\\l2th) + £ (£a(tj)+ v\a(j)) H/ha ) ■ (16)

ip =i a

Let us sum (16) first over a from 1 to p and then, multiplying both sides by 2r and summing over j1 from 0 to j, we get:

j P •' + a j P

\\y3+1\\l2(th) + £ T £ \\y3x' ' ]\l(th) < M7 £ T £ \\yj +'\\l (th) +

j '=0 a=1 j '=0 a=1

/ j P / \

j'+-112 i V^ (,,2 , ,,2

+Ms[£ r£ \ j'\\U) + £ [V-a + via) H/ha +

\j '=0 a=1 \ ip =i a )

+ \m\Uth) i (17)

where M7 = TM5 + M6. From (17), we have

\W+1\\2Mth) < M7 £ T £ \\yj'+'\\L2(th) + M$Fj, (18)

j '=0 a=1

Fj = £ T £ (yWlw + £ (v-a + via)H/hA + mi^y

j '=0 a=l\ 2( h) ip =ia V ' ) 2( hJ

Let us show that the following inequality holds

j-i

j+aa o \—^ j '+aa o ■

max \\y p \\L2{Qh) < T \\y p W^) + v2Fj, - j'=0 - —

where ,vo are known positive constants.

In view of this, we rewrite the inequality (16) as

_1 j

\\yj+p\\l(,h) < \\yj+v\\U,h) + M5T E t\\yj +p\\U,h)+

j'=0

+2TM6\\yj+p\\2L^h) + + T [\\jp\\l2^h) + E {v-a + V+a)H/ha ) . (19)

\ iP =i a J

Summing (19) over a1 from 1 to a, then we get

a j '

\\yj+p\\U,h) < \\yj\\U,h)+M E T E T\\yj'+"\\L2{,h)+

a'=l j '=0

+т E I \\j"\Ц2Ш + E {v-a' + V+a') Н/Па I +

a>=l\ ip =i'a J

a , p j

+2тМб E \\yji'\\1Ш < \Wj\\1Ш + M E т E т\\yj +p\\1Ы) +

a' = 1 a=1 j '=0

+2тМб£ \\yjip\\ЪШ +

a=1

+ T El \\j p\\L^h) + E {v-a + via) H/ ha ) . (20)

a=1 \ ip =ia

Without loss of generality, we can assume that

ms^ \\yj+"WL^h) = \\yj+p\\L2{-.h),

otherwise (19) will be summed up to such a value of a that \\yj+ p \L2(Qh) reaches its maximum value for a fixed j. Then (20) can be rewritten as

ma*\\yj+p\\L2(Qh) - \\yj\\L2^h) + j

+2-TM6ima<x \\yj+p WL^h) + -M5T £ max \\yj'+p - -p j '=q - -p

+ T E ( \\j p\\l2^h) + E {ß-a + ß\a )H/ha J . (21)

a=1 \ iß =ia J

We rewrite (21) once again in the following form

j

(1 - 2pM6t) max \\yjip ) < PM5T £ max \\yj p f^)t+ --p j '=o - -p

+ y WU*H) + T E I № + p + E {v-a + V+a )H/ha ) .

a=l \ iß = ia )

Choosing t < t0 = 4pM6, from (22), we find

^ Wyjip Wl2{,h) < M E ^ Wyj + p Wl2{,h)T + MioFj, (23)

- -p j '=0 - -p

where Fj = \\yj W^+r £ (\j W^) + £ (v-a + vla)Hßa).

a=1 \ iß =i a x /

Based on Lemma 4 [28, p. 171] from (23), we get the estimate maXp Wyj+p ^< MAW ^(*h) +

+ tMw ET I \\jp\\Î2^h) + E (»-« + via)H/ha I . (24)

=1 iß =i

Since it follows from (18) that

\\yj\\U*h) < M7 E T ^ \\yj + p \\l(,h) + M*Fj

j '=o--p

then, from (24), we have

j-1

j+a 2 \—^ j ' + 9L 2 '

max*\\y p < t maaxp \\y p \\L^h)+v2Fj■

- — j /=Q -

Introducing the notation gj+\ — max \\yj+ p \\L2(-h), the last inequality can be rewritten as follows:

9j+1 < rgt + v2f j, (25)

k=1

where ,v2 are known positive constants.

Applying Lemma 4 [28, p. 171] to (25), from (18), we obtain the estimate

„j+1\\2

WU*h) < M

f Wl ш + E ^ US'+p WU*h)+

j'=0 a=1

+ E TE E {f-a (0,x',tj ')+ via (la,x',tj ')) H/H0

j '=0 a =1 i@ =i a

(26)

where M — const > 0 does not depend on ha and t , X — {Xli X2i ..., xa —1, Xa+\} ..., , Xp1).

Theorem 1. Let conditions (4) be satisfied, then the LOS (10)-(12) is stable with respect to the right-hand side and initial data, so estimate (26) is valid for the solution of the difference problem (10)-(12) with t < t0.

5 Convergence of a locally one-dimensional scheme

The solution Z(a) of the problem (13) can be represented as Z(a) —

j+a

v(a) + ), Z(a) — z p , where n(a) is defined by the conditions [26]

—-(-1 = фа, x e Wha + Yh,a, a = I, 2,...,p, (27)

фа, X a e Wh c П(Х, 0) = 0, Ipa^ Ф—a, Xa = 0

Ф+a, xa — la

We represent the solution n in (27) as nj+1 — n(p) — nj + t(4\ + ip2 +... + tpp) — nj — Vj—1 — ... — n0 — 0. For n(a) we have n(a) — t(4\ + 4)2 + ... + 4a) — O(T).

The function U(a) is determined by the conditions

U h —\ — -|\ _ _

- = AaV(a) + Ipa, ta = n(a) + t*a, xa ^ wha , (28)

- - „ ua

T

^KU^-^ = A-V(a) + tp-a, t-a = A-n(a) + t-a, xa = 0, (29)

O.'ohaU^—U^—^ = A+V(a) + 1p+a, 1p+a = A+^(a) + , xa = la, (30)

u(x, 0) = 0. (31)

If there exist continuous in the closed domain QT derivatives

d2 u d4u d3u

dt2 ' dxaOx2' dxidt'

1 < a,fi < p, a = fi,

then Aan(a) = TAa (jpa+l + ... + %) = O(T), A±n(a) = O(T). We estimate the solution of problem (28)-(31) with the help of (26)

\\V \\L2(ph) <

j

;/ , a

P

(32)

< MY.T E P ^h) + E E a + t+a )H/Ha

j '=0 La =1 a=1 ip =ia

Since Vj+1 = 0, n(a) = O(t), \\zj+1\?2j2(,hh) < \\vj+1 ^^ then it follows from estimate (32)

Theorem 2. Let problem (1) - (3) have a unique solution u(x, t) continuous in QT and there exist derivatives also continuous in QT

d2u d4u d3u ^ ^ ^

dt2' dxa dx2' dxa dt < ' < ' '

and conditions (4) are satisfied, then the locally one-dimensional scheme (10)-(12) converges to the solution of the differential problem (1)-(3) with the rate O(lhl2 + t), so that for sufficiently small t the following estimate is valid

"j+1 —uj+1\\L2(Fh) < M (|h|2 +T), 0 <T < TO, Ihl2 = h2+h2 + ...+hp .

Таблица 1: The error in the norm || ■ \\ь2(п>кт) when decreasing grid size for problem (1)-(3)

h Maximum error COi CO2

1/10 1.937583553e-1

1/20 7.249175016e-2 1.418369799 0.876007024

1/40 1.848616435e-2 1.971370905 1.081827896

1/80 4.662492757e-3 1.987272525 1.225050757

1/160 1.172124523e-3 1.991975638 1.329794326

1/320 2.948753635e-4 1.990948648 1.409241578

Таблица 2: The error in the norm || ■ Hc(whT) when decreasing grid size for problem (1)-(3)

h Maximum error COi CO2

1/10 6.222675505e-1

1/20 1.716194235e-1 1.858322172 0.588328911

1/40 4.432645471e-2 1.952972959 0.844748018

1/80 1.120724198e-2 1.983736692 1.024912780

1/160 2.821637800e-3 1.989826572 1.156696892

1/320 7.136216269e-4 1.983301564 1.256025397

6 Numerical experiment

Let us define the coefficients and boundary conditions of problem (1)-(3) so that the exact solution of the problem in the two-dimensional case is the function

u(x, t) = t3(x\ + x2).

Below in Tables 1-2, we present the maximum value of the error (z = у — u) and the computational order of convergence (CO) in the norms || • |l2(WwhT)

and || ■ Hc(whT), where ЦуЦс) = max |y|, when h = hi = h2 = л/т,

(xi ,tj )ewhT

while the mesh size is decreasing. The error is being reduced in accordance with the order of approximation O(lhl2 + т).

The order of convergence is determined by the following formulas:

COi =logh! M = log2 Щ, CO2 = ln||Z211

h

Ш1 1Ш1' ln h

where zi и z2 are the errors corresponding to steps 0, 5h, h.

Conclusion

We study the third boundary value problem for a multidimensional integro-differential convection-diffusion equation with a memory effect and a nonlocal (integral) source. Problems of this kind arise in the study of natural processes, for which it is necessary to take into account the prehistory (memory, hereditary properties) of the process. From physical considerations, a nonlocal source in the integral form arises in mathematical modeling in cases where there are sources (or sinks, depending on the sign of p(x, t)) and it is impossible to obtain information about the ongoing process using direct measurements, or when it is possible to measure only some of the averaged (integral) characteristics of the desired value. For the problem under study, a locally one-dimensional difference scheme is constructed. The main research method is the method of energy inequalities. An a priori estimate of the LOS solution is obtained, from which follow uniqueness, stability, and convergence of the solution of the difference problem to the solution of the original differential problem at a rate equal to the approximation error. Numerical experiments were carried out.

References

1. Grasselli M. Uniform attractors of nonautonomous dynamical systems with memory // Progress in nonlinear differential equations and their applications. Basel: Birkhauser Verlag. 2002. Vol. 50. P. 155-178. URL: https://link.springer.com/chapter/10.1007/978-3-0348-8221-7_9.

2. Coleman B. D, Gurtin M. E. Equipresence and costitutive equations for rigid heat conductors // Math. Phys. 1967. Vol. 18. P. 199-208. URL: https://link.springer.com/article/10.1007/bf01596912.

3. Gurtin M. E, Pipkin A. C. A general theory of heat conduction with finite wave speeds // Arch. Rational Mech. Anal. 1968. Vol. 31. P. 113-126. URL: https://link.springer.com/article/10.1007/bf00281373.

4. Fabrizio M., Morro A. Mathematical problems in linear viscoelasticity. Philadelphia: SIAM Studies Appl. Math. 1992.

5. Renardy M., Hrusa W. J., Nohel J. A. Mathematical problems in viscoelasticity. New York: Longman Scientific and Technical, 1987. 273 p.

6. Ашабоков Б. А., Шаповалов А. В. Конвективные облака: численные модели и результаты моделирования в естественных условиях и при активном воздействии. Нальчик, 2008. 252 с.

7. Численное моделирование облаков / Е. Л. Коган, И. П. Мазин, Б. Н. Сергеев, В. И. Хворостьянов. Москва: Гидрометеоиздат. 1984. 185 с.

8. Berry E. X. Cloud Droplet Growth by Collection // J. Atmos. Sci. 1967. Vol. 24. P. 688-701.

9. Berry E. X., Reinharolt R. L. An Analysis of Cloud Drop Growth by Collection: Part 2. Single initial Distributions //J. Atmos. Sci. 1974. Vol. 31. P. 1825-1837.

10. Douglas J., Rachford H. H. On the numerical solution of heat conduction problems in two and three space variables // Trans. Amer. Math. Soc. 1956. Vol. 82, № 2. P. 421-439. URL: https://www.ams.org/journals/tran/1956-082-02/S0002-9947-1956-0084194-4/.

11. Peaceman D. W., Rachford H. H. The numerical solution of parabolic and elliptic differential equations //J. Industr. Math. Soc. 1955. Vol. 3, № 1. P. 28-41. URL: https://epubs.siam.org/doi/10.1137/0103003.

12. Яненко Н. Н. Метод дробных шагов решения многомерных задач математической физики. Новосибирск: Наука. Сиб. отд-ние, 1967. 196 с.

13. Самарский А. А. Однородные разностные схемы на неравномерных сетках для уравнений параболического типа // Ж. вы-числ. матем. и матем. физ. 1963. Т. 3, № 2. C. 266-298. URL: https://www.mathnet.ru/rus/zvmmf7801.

14. Самарский А. А. Локально-одномерные разностные схемы на неравномерных сетках // Ж. вычисл. матем. и матем. физ. 1963. Т. 3, № 3. C. 431-466. URL: https://www.mathnet.ru/rus/zvmmf7777.

15. Марчук Г. И. Методы расщепления. Москва: Наука, 1988. 263 с.

16. Дьяконов Е. Г. Разностные схемы с расщепляющимся оператором для многомерных нестационарных задач // Ж. вычисл. матем. и матем. физ. 1962. Т. 2, № 4. C. 549-568. URL: https://www.mathnet.ru/rus/zvmmf7874.

17. Лафишева M. M., Шхануков-Лафишев М. Х. Локально-одномерная разностная схема для уравнения диффузии дробного порядка // Ж. вычисл. матем. и матем. физ. 2008. Т. 48, № 10. C. 1878-1887. URL: https://www.mathnet.ru/rus/zvmmf102.

18. Баззаев А. К., Шхануков-Лафишев М. Х. Локально-одномерная схема для уравнения диффузии дробного порядка с краевыми условиями III рода //Ж. вычисл. матем. и матем. физ. 2010. Т. 50, № 7. С. 1200-1208. URL: https://www.mathnet.ru/rus/zvmmf4901.

19. Ашабоков Б. А., Хибиев А. Х., Шхануков-Лафишев М. Х. Локально-одномерная схема для параболического уравнения общего вида, описывающего микрофизические процессы в конвективных облаках // Доклады АМАН. 2021. Т. 21, № 4. С. 45-55. URL: www.mathnet.ru/links/c5c255e075d6952d8b261b288ecc10eb/aman30.pdf.

20. Экономичные явно-неявные схемы решения многомерных задач диффузии-конвекции / А. И. Сухинов, А. Е. Чистяков, В. В. Сидоря-кина, Е. А. Проценко // Вычисл. механика сплошных сред. 2019. Т. 12, № 4. С. 435-445. URL: https://doi.org/10.20948/mm-2023-03-03.

21. Локально-двумерные схемы расщепления для параллельного решения трехмерной задачи транспорта взвешенного вещества / А. И. Сухинов, А. Е. Чистяков, В. В. Сидорякина [и др.] // Математическая физика и компьютерное моделирование. 2021. Vol. 24, № 2. С. 38-53.

22. Тран З. Локально-одномерный метод для уравнения переноса сплошной среды с распределенными параметрами на сетеподобной области. Моделирование, оптимизация и информационные технологии. 2022. Т. 10, № 2. URL: https://moitvivt.ru/ru/journal/pdf?id=1141.

23. Бештокова З. В., Шхануков-Лафишев М. Х. Локально-одномерная разностная схема для третьей краевой задачи для параболического уравнения общего вида с нелокальным источником // Дифференц. уравнен. 2018. Т. 54, № 7. С. 891-901. URL: https://www.elibrary.ru/item.asp?doi=10.1134/S0374064118070051.

24. Бештокова З. В., Лафишева M. M., Шхануков-Лафишев М. Х. Локально-одномерные разностные схемы для параболических уравнений в средах, обладающих «памятью» // Ж. вычисл. матем. и матем. физ. 2018. Т. 58, № 9. С. 1531-1542. URL: https://doi.org/10.31857/S004446690002531-5.

25. Бештокова З. В. Численный метод решения нелокальных краевых задач для многомерного уравнения параболического типа // Вычислительные методы и программирование. 2022. Т. 23, № 2. С. 153-171. URL: https://num-meth.ru/index.php/journal/article/view/1215/1190.

26. Самарский А. А. Теория разностных схем. Москва: Наука, 1983. 656 с.

27. Андреев В. Б. О сходимости разностных схем, аппроксимирующих вторую и третью краевые задачи для эллиптических уравнений // Журнал вычисл. матем. и матем. физ. 1968. Т. 8, № 6. С. 1218-1231. URL: https://www.mathnet.ru/rus/zvmmf7186.

28. Самарский A. A., Гулин A. B. Устойчивость разностных схем. Москва: Наука, 1973. 416 с.

The article was submitted 10.06.2023; approved after reviewing 07.07.2023; accepted for publication 27.09.2023.

К ЧИСЛЕННЫМ МЕТОДАМ РЕШЕНИЯ МНОГОМЕРНЫХ ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

Бештокова Зарьяна В. младший научный сотрудник,

Институт прикладной математики и автоматизации КБНЦ РАН Россия, 360000, г. Нальчик, ул. Шортанова, 89а

Аннотация. Исследуется третья краевая задача для многомерного интегро-дифференциального уравнения конвекции-диффузии с эффектом памяти и нелокальным (интегральным) источником. Для численного решения поставленной многомерной задачи строится локально-одномерная разностная схема, основная идея которой состоит в сведении перехода со слоя на слой к последовательному решению ряда одномерных задач по каждому из координатных направлений. С помощью метода энергетических неравенств для решения локально-одномерной разностной схемы получена априорная оценка. Из полученной априорной оценки следуют единственность, устойчивость, а также сходимость решения локально-одномерной разностной схемы к решению исходной

дифференциальной задачи со скоростью, равной порядку аппроксимации разностной схемы. Проведены численные расчеты.

Ключевые слова: третья начально-краевая задача, локально-одномерная схема, априорная оценка, разностная схема, параболическое уравнение, интегро-дифференциальное уравнение, уравнение с памятью, уравнение с нелокальным (интегральным) источником.

Для цитирования

Бештокова З. В. К численным методам решения многомерных интег-ро-дифференциальных уравнений // Вестник Бурятского государственного университета. Математика, информатика. 2023. № 3. С. 34-52.

Статья поступила в 'редакцию 10.06.2023; одобрена после рецензирования 07.07.2023; принята к публикации 27.09.2023.