Стохастическое моделирование фильтрации диффузии Текст научной статьи по специальности «Математика»

STOCHASTIC SIMULATION OF DIFFUSION FILTERING

Arutyunyan Robert Vladimirovich,

candidate of physical and mathematical sciences, docent, Moscow Technical University of

Communication and Informatics (MTUCI), Moscow, Russia, Keywords: filtration, diffusion, kinetics, stochastic

[email protected] equation, existence, uniqueness, numerical method.

Most of the works devoted to the study relevant to many technical areas filtering problem, based on the hypothesis of continuity, which limits the accuracy of the calculations for a large size dispersion of the particles in the filtered stream. In this paper to study the process of overgrowth of holes in the lattice structure, acts as a filter, used a stochastic approach. In contrast to [1], where absorption parameter was calculated by computer by means of statistical tests, the basic characteristics of the filter structure in the current study were determined by deterministic methods.

In this paper to study the process of overgrowth of holes in the lattice structure, acts as a filter, used a stochastic approach. We formulated and investigated the system of kinetic equations describing the process of diffusion filtering based on this approach. In contrast to the well-known investigations, where the absorption parameter was calculated by computer by means of statistical tests, the basic characteristics of the filter structure in the current study were determined by deterministic methods. The theorem of existence and uniqueness of solutions for the case of a continuous density is proved. Representation of solution in the form of a uniformly convergent and asymptotic series, and explore the nature of its behavior at infinity studied.

The concrete particular cases such as the density of the delta function and a uniform distribution investigated. It constructed and justified finite-difference scheme for the solution of the corresponding Cauchy problem on a finite time interval. The results of computer simulation obtained. It is shown that the finite-difference scheme of the first order is almost acceptable in calculations on a computer. Investigated model, despite a number of simplifying assumptions, it gives an overview of the filtering process in the lattice structure. The results can be developed, especially in respect of the functional classes density filter size distribution of the particles and the method of asymptotic estimates on the segment.

Для цитирования:

Арутюнян Р.В. Стохастическое моделирование фильтрации диффузии // T-Comm: Телекоммуникации и транспорт. - 2015. - Том 9. -№10. - С. 69-72.

For citation:

Arutyunyan R.V. Stochastic simulation of diffusion filtering. T-Comm. 2015. Vol 9. No.10, рр. 69-72.

T-Comm Vol.9. #9-2015

1. The mathematical model and its research

The structure of the one-dimensional type of "sieve". The length of the hole (holes) is 1 and the impenetrable part - 2. Through the sieve the flow of one-dimensional particles (sticks) of random sizes z with a density p{z\z e (0,2] sifted.

i§=-?(*)C(x, 0+ \P(y-*)C(y, t)d; Vjce(0, l], Vi>0;

v/>o;c(*,o)=£(x-i+o), v.v6(o, i]; cM=Qïp<w)=-)p(z)m oiwïv,

2

y

<K4=f

_0 x 2x

Vxe (0, l); c0(/) - the probability of a zero-holes,

C(jr,/) = C1(x, i}+<5(jr-l + 0)e ■

^ = ~q{x)c, + )p(y - x]c,(y, t)dy + P(l - > 0,

5t i <?,{*, 0)=0, Vx s (0.1].

<p{z,t)=p{z)\R{y,z)c{y,i)dy Vr e [O, 2), />0,

i/2

R{}\ z)=( 2y - 2 ) / 3, z / 2 < v < m in(z, I ); z) = y / 3, m i n<z.1) < >' < 1.

is unique, and CB(0«C*(a*)/ P f (x,i)eC(0, l) v/>0,

dt*

k = 0,1,...

Property 2. We have the lower estimate

"(i(jf>f _ -r/(lji

C,(*, /)> P(1 -XWr—pr-i J>(| - *)

Vxe[0,l]/ /¿0, </miv =|MIC(M>-Property 3. The upper estimate

Fig. 1. Scheme of the filtration process in orie-dimensional lattice structure

The unknown is the densities distribution of the size of the hole c(x, /) and sticks <p(z, ?) at the exit of "sieve" and for which a system of equations:

dC.

Property 4. The normalization condition

N/r (I - a:)

2. Special cases

2.1. The density distribution of the lengths of sticks such as the delta function

In this casep(z) = 5(z -l),

^I'flir)-

I + X

3

2-x

, re

-, xe

V

the exact solution of the Cauchy problem has the form:

cM=

3

i*4 x> 5) 1

3 192y

+a2{x, t

x\l)

I ^ 2 +3 /

„-//3

C\x, t) = e C0{t)=l-e~"3

\ (l — jc>2 2/

Six -1) + --—+ —

K ' 9 3

' 2'1

5,4 t2 t

- + -— + — +- +

186624 486 24 3

, V/>0;

81;

Substitution cM, t)=A(x, t]e~"U)' yields the equation:

— = -Q(x)A + |>(y- x)A(y, t)ify + P(l - x\ 4*> 0)=0/

dt J

X

Vxe(0,1], Q{x) = </(\)-qU).

Property 1, If the density distribution p(z) continuous on [o, 2], then the solution of the stochastic system exists,

' 0 " 0

0(.v) - the unit Heaviside function. The density of the conditional distribution of the lengths of sticks at the exit of "sieve" is defined by the formula

T-Comm Tom 9. #10-2015

<p(z, f)=S(z-l)/(f) where /(f) - the probability of finding a stick at the outlet:

( .2

/(f) =

-i/3

L 1

648 + 1S + 3

V /

The asymptotic behavior of the average width of the hole is given by

<*) = ■

559872

2.2. Uniform distribution of the lengths of sticks at the exit

We

have

/ \ X X 1

' 4 3 3

P(x) = ^{x-1), *(l)=-«(*,) = 4, q{\) = .\ the

coefficients of the Laurent series are a method of power series of an appropriate system of recurrence relations. Moreover, the leading asymptotic term in the interval

f I 11 determined by the formula

U' J

C, x, /

J_« if J_

9720

3x 1

9x2 \2x 19 F a,b,c,~ 1 x 2

where F(a, b, c, z) - hypergeometric function [2];

a — — 2

, Î35Y , if, [35

3+\T/i"213"VT

; c = 6.

Asymptotics of the segment

<4

are expressed from

the general relations, from which it follows that

—>00.

' (-=i Bi=Potje-<«<-, a^qil-z,)) Pt=P{?t), z(=ih) Rj = Pie-""' ; C,(z, f,)=+ r, h 0,

/ = 0, M ; h = — ; t = jr ; y - o, J ; j = M

e"'T -1

Stability of the scheme:

IK1 --sfl sHK" -^IL' r==

= max|zi/|, □ = {(/,,/): i = 0, M; y = 0, /}.

|C, (z, , f,)- Bi_t \</at + fsh, V(i, y)e £2. Test the problem at fl(r)=l + (l-x)/6, />(*) = 2/3. Parameters of the scheme h = 0.025, x = 0.5. (Fig. 2, a£x, t)=Ah(x, t)!\A,\h, «2(je, t)= A(x, f|^||); settings scheme h = 0.025, T = 0. (3)) - Fig. 3

(«,(*, f)=4MM,)-

.o *

Fig. 2.1 -t = 10; 2 -t = 30; 3 -t = 60; FDM_; exact solution____

The simulation based filtering stochastic equations has the advantage that the finite-difference methods are more economical in comparison with statistical methods.

a(x,t) 1.0-

3. Finite-difference method for the approximate solution of the Cauchy problem

On a finite interval (o, T), where T >0 - the order of

the time constant of the process, one of the most effective ways of solving stochastic equations is the use of FDM. Driving 1st order:

0.5-

0

._!__ j_ 1 1 1 1 _l__1 j__

__ 71 ^ __l — 1/it* ji _l__

__lj 1. _/l 1 _ i., '11 \ 1 -1 -\l _ _l_ 1 _l__

l j_ i \ vl__ia cir j__

a _zi__ 1/ ' 1 _ j. _ \ 1 jl__i _ 1 -1— 1

% 7t 1 -x- a i -1+-1- __ 1 _l__

__ a i -i. _l_ 1 \ 1 j -u- j_ jv _j__

i j_ i 1 — j.—

— i 4- 1 -4- 1 __v- 1 4-- 1 l\ 4__ij j—

--j-- 4-j_ -x. [ _1 1 -x- 1 i 1 -i — 1 _!_ 1 1 4--11 1 i l j-

0.5

1.0 *

Fig. 3, A uniform particle size distribution: 1 -t = 3 (3); 2 -t= 13 (3); 3 -t = 60

Conclusion

Research article model, despite a number of simplifying assumptions, gives an overview of the filtering process in the lattice structure. The results can be developed, especially in respect of the functional classes density filter size

ПУБЛИКАЦИИ НА АНГЛИЙСКОМ ЯЗЫКЕ

distribution of the particles and the method of asymptotic estimates on the segment.

The assumption of fixed sign of the second derivative of the function was made solely to reduce the cumbersome calculations.

The properties of density and size distribution of holes in the case of a two-dimensional square lattice as a whole are similar.

The modeling filtering based on the relevant kinetic equations for probability distribution function of aperture size of the filter structure has the obvious advantage that the finite difference methods are more economical compared to the Monte Carlo.

If it is necessary to take into account the three-dimensional effects, caused in particular sticking of particles to each other and their separation, reflection and diffusion in the tangential and reverse directions, complex topology structure and many other factors, the output of the corresponding kinetic equations becomes difficult, and

the structure is considerably complicated, since that is the only available method of statistical computer modeling.

References

1. Reznikov G.D., Zhikhar А.С. Numerically - analytical approach to modelling transfer particles at filter // Mathematical Modeling, 1995. V .7. № 6. Pp. 118-125.

2. Handbook of Mathematical Functions / Ed. M, Abramowitz and I. Steagall - M.: Science, 1979. 832 p.

3. Trenogin V.A. Functional analysis. M.: Nauka, 1980. 496 p.

Gavlch IK Fundamentals of hydrogeology, Gidrogeodinamika.

1983.

4. Gurevich A.E. A practical guide for the study of groundwater movement in the search for minerals. 1980.

5. Kolesnikov A. V. Mathematical modeling of fluid flow in heterogeneous porous and periodic bodies: автореферат Dis. Candidate ... technical Sciences: 05.13.18 / North Caucasus federal University - Stavropol, 2014.

СТОХАСТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ФИЛЬТРАЦИИ ДИФФУЗИИ

Арутюнян Роберт Владимирович,

к.ф.-м.н., доцент, Московский технический университет связи и информатики (МТУСИ), Россия, Москва,

[email protected]

Аннотация

Для изучения процесса зарастания отверстий в структуре решетки, используется стохастический подход. Нами сформулирована и исследована система кинетических уравнений, описывающих процесс фильтрации диффузии на основе этого подхода. В отличие от известных исследований, где параметр поглощения, рассчитан компьютером с помощью статистических тестов, основные характеристики структуры фильтра в данном исследовании были определены детерминированным методом. Доказана теорема существования и единственности решения для случая непрерывной плотности. Представлено решение в виде равномерно сходящегося и асимптотического ряда, а также изучен характер ее поведения на бесконечности. Исследуются конкретные частные случаи, такие, как плотность дельта-функции и равномерного распределения. Построена и обоснована разностная схема для решения соответствующей задачи Коши на конечном отрезке времени. Получены результаты компьютерного моделирования. Показано, что разностная схема первого порядка почти приемлема в расчетах на компьютере.

Ключевые слова: фильтрация, диффузия, кинетика, стохастическое уравнение, существование, единственность, численный метод.

T-Comm "Гом 9. #10-2015