ALGORITHM OF ATMOSPHERE CONTAMINATION NONLINEAR MODEL NUMERICAL REALIZATION

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1 UDC Математичне та комп ютерне моделювання L. A. Мitko* Ph. D. of Physical and Mathematical Sciences N. G. Serbo** Ph. D. of Technical Sciences Jo Sterten*** Assistant Professor *NASU Pukho Institute for Modeling in Energy Engineering Kyi ** Odessa State Enironmental Uniersity Odessa *** Gjøik Uniersity College Høgskolen i Gjøik Noray ALGORITHM OF ATMOSPHERE CONTAMINATION NONLINEAR MODEL NUMERICAL REALIZATION The article presents the algorithm for soling nonlinear partial differential equations that describe the process of air contamination. The algorithm is based on an appropriate modification of the grid method. Boundary conditions and border alues are considered and a difference scheme construction method is proposed. As a result problem is degenerated in to-dimensional one and reliable algorithms for its solution are deeloped and approed. Key ords: atmosphere contamination nonlinear model grid method. Computing experiment in contamination modeling. Reealing tendencies of enironmental contamination and determination of antropogeneous influence on it are extremely actual problems. The research of natural and chemical processes and defining the gien problem are not alloed be conducted as the location experiment in most cases. Therefore important issues acquire the possibility of realization of a computing experiment for hich the construction of mathematical models adequate to the natural processes researched and realized on modem computer facilities is necessary. The problem of calculation of contamination concentration q(x y z t) in: he atmosphere oing to their ejection and transposition can be represented as folloing [1]: 3 3 q q q q ui aq ki s f (1) t i1 xi i1 xi xi z here (x l x 2 x 3 ) = (x y z) cartesian coordinates hose plan XOY corresponds to a terrestrial surface t temporary coordinate (u l u 2 u 3 ) components of the ind field a factor of disintegration of substance taken as a boundary alue for the equation. Taking into account chemical transformation of substance and ashing the partials aay by settlings ki factors of turbulent transposition s the settled elocity of the graitational settling of partials of substance f = f(x y z t) is the knon function circumscribing the density function of contamination sources. The solution of the equation (1) is discoered for ant of certain initial and boundary conditions. The entry conditions are usually formulated as knon concentration of substance in researched area for ant of t = : L. A. Мitko N. G. Serbo Jo Sterten 214

2 Серія: Фізико-математичні науки. Випуск 11 Qxyz ( ) q ( xyz ). (2) The boundary conditions at first should correctly reflect the process researched and secondly ensure correctness of the deliered problem. If the researched area is infinite on a horizon it is natural to assume that the concentration decreases up to zero for the ant of rushing x y to infinity. Hoeer hile applying numerical methods for the solution of the considered problem the research area should be limited that is to hae for example some kind of a parallelepiped: D ( x y z): a1 xb1 a2 y b2 z z H. On the boundaries side it is possible to put q a 1xb. 1 n a2yb2 On the upper boundary of the area the modeling of arious processes is possible: The concentration of substance equal to zero: q z H ; Absence of diffusion transposition proided that ertical component of the elocity of ind is equal to zero: q zh ; (3) z Aailability of the substance stream through the upper bound: 3 ( ) q k z sq zh z. (4) It is possible to simulate processes taking place on a spreading surface also by arious images. For example in [2; 3] it is considered that impurities poorly interact ith the surface of the ground usually and hitting on it are not accumulated and are again carried aay in the atmosphere ith turbulent curls. As such curls at the surface are insignificant it is possible to put: q k3 zz. (5) z Hoeer hile modeling such processes as shaping of spots of increased concentration of impurity on the spreading surface secondary rise and sedimentation of partials the condition (5) becomes unsatisfactory. In such condition on the spreading surface it is offered: 3 ( ) q Q k z sq zz zz (6) z z here Q(x y t) the concentration of the ground field of impurity * * Z z z z leel of roughness z the thickness of the salta- 153

3 154 Математичне та комп ютерне моделювання tion stratum. As this concentration can ary in time the condition (6) is supplemented by the ratio Q j d q( x y z t) Q( x y t) (7) t here d the elocity of the dry settling of impurity; the intensity of ind rise. The use of conditions (6) (7) requires in turn the giing of initial concentration of impurity on the surface: Qxy ( ) Q ( xy ). (8) If the spreading surface represents a mirror of ater space that assuming the ater to sallo all impurities e receie a boundary condition q. z z Boundary alue problem in contamination process mathematical modeling. The correctness of statement of boundary alue problems for the equations of type (1) is proed ith certain restrictions superimposed on diffuseness and conection and for some combinations of boundary conditions. It is natural that the most adequately considered processes describe threedimensional boundary alue problems. The difficulties accompanying realization of such mathematical models are also obious. The problems of choosing the solution method and correctness of the appropriate difference problem (hen apply the different methods of solution) maintenance of practical computing stability of algorithm the necessity of ork ith large scale arrays of data concern them. Many scientists try to sole these problems by simplification of the model more often at the expense of diminution of dimensionality of the problem. In this ork the method of solution of a three-dimensional boundary alue problem is stated hich statement is indicated. Let us consider a problem about eolution of the ground field of impurities Q(x y t) and field of impurities of near-the-land stratum of an atmosphere q(x y z t). The gien problem is described by the equation (1) q = f(x y z t)= = ith the gien initial concentration of fields of impurity (2) (8) and boundary conditions (3) (4) (6). The last boundary condition is supplemented by ratio (7). With alloance for the last parity (ratio) the problem of impurity transposition ith the aailability of ind rise and dry sedimentation is essentially non-stationary and the dynamic equilibrium beteen the concentration of impurity on the surface and in near-the-surface stratum arises in time (t ~ -1 ) as is rather a small number according to [4]. With alloance for (7) let us copy boundary condition (6) as: q k3 sq dq( x y z t) Q( x1 y t) z or q k3 q Q( x y t) x3 z s d. z

4 Серія: Фізико-математичні науки. Випуск 11 By designating u 3 = u 3 s let us copy the equation (1) in the deeloped ay: _ q q q q q q q u 1 u 2 u 3 k 1 k 2 k 3 t x y z x x y y z z. (1`) For the solution of the problem (1') (2) (4) (8) (9) e shall use the difference method offered in [5]. For it in area Ω = D[ T] D ( x y z) a1 x b1 a2 y b2 z z H e shall enter a grid x xi : xi a1ihx i n hx ( b1a1)/ n y : y a jh j m h ( b a )/ m y j j 2 y y 2 2 z : z z kh k l h ( H z )/ l t : t T / z k k z z here in after e shall deal ith the grid area h D. z h Dh x y z In the grid area h (problem (1 ') (2) (4) (8) (9) e shall put in the orrespondence the difference scheme of the second order of approximation on space ariables: _ 1(( d x ) x ) r1 ( x ) r1 ( ) 2(( d y ) y ) x x y y 2 3 z z 3 z 3 y z z ( ) ( _) (( _ ) ) ( ) ( _) r r d r r i1 n1 j 1 m1 k 1 l Here is designated: = (x i y j z k ) the grid function approximating function q(x y z t) in knots of grid (1) 1 1 (1 R1) R1 5 hx u1 r1 5( u1 u1) r1 5( u1 u1 ) ( dx) K1( xi5 yj zk) 1 (1 R ) R 5 h u r 5( u u ) r 5( u u ) y ( d ) K ( x y z ) 2 y 2 i j y h (11) 155

5 156 Математичне та комп ютерне моделювання 1 3 R3 R3 hz u3 r3 u3 u3 r3 u3 u3 (1 ) 5 5( ) 5( ) ( d ) K ( x y z ). z 3 j k y dz ij1 z ij ijij yijqij j m 1 1 z z s ij ( ) ( ) 1 1 ; 1... ( d ) ( ) ( ) i 1 n1 ; j 1 m1 ; jk jk 2 (12) 1 j 1 m1 ; k 1 l1; 1... (14) 1 1 h1 jk njk 1 1 ik 1 i k j 1 m1 ; k 1 l1 ; 1... (15) _ i 1 n1 ; k 1 l1; 1... (16) 1 1 in1 k ink _ i 1 n1 ; k 1 l1 ; 1... (17) Difference scheme construction. The conentional designations in theory of different schemes [6] are used here. Thus the constructed difference scheme is monotonous. That is hy the grid of a maximum principle ithout restrictions on pitches of a grid on space ariables is executed. The difference analogue of conditions (9) (8) looks as folloing: 1 ij ij 1 1 d ij ij ij ij ( ) i n; j n; 1... (18) ij Q ij i n; j n (19) here ij = (x j y t ) grid function approximating function Q(x y t). Let us copy a difference scheme (11) (19) as: [ A B C D E F i1jk i1jk ij1k ij1k 1 1 G 1] i 1 n1 ; j 1 m1 ; k 1 l1 ; here A 1 ( B C D E F G ) ( 1) ( r1 ) ( 1) ( r1 ) B ( d ) ( ) 2 x C d 2 x i1 jk h h 1 1 h h 1 1 ( 2) ( r2 ) ( 2) ( r2 ) D ( d ) ( ) 2 y E d 2 y ij1k h h 2 2 h h 2 2 ( 3) ( r3 ) ( 3) ( r3 ) F ( d ) ( ) 2 z G d 2 z 1 h h 3 3 h h 3 3

6 Серія: Фізико-математичні науки. Випуск ij aij[ h3 ijij ( dz ) ij1ij1 ] i 1 n1; j 1 m1; 1... (2) Where a (( d ) h ) b i1 n1; j 1 m1; 1... (21) ij z ij1 ij 3 ij1 ij ijl1 1 bij (( dz ) h3 ( s ) ij ) ( dz ) 1 1 ij ij d ij ij ij Where C [ ( ) ] i 1 n1; j 1 m1; 1... (22) 1 Where cij (1 ij ). We shall discoer an approximate solution of the difference problem using :he relaxation method [7]. The settlement formulas look as follo: s 1 1 s 1 s 1 1 s 1 (1 ) A ( B i 1jk C i1jk s1 1 s 1 s1 1 s 1 D E F G ) (23) ij1k ij1k 1 1 i 1 n1; j 1 m1; k 1 l1; 1...; s 1... s 1 1 s 1 s 1 (1 ) a[ h3 ( dz ) ] i 1 n1; j 1 m1; 1...; s 1... s (1 ) s s jk jk 1jk j 1 m1; k 1 l1; 1..; s 1.. s 1 1 s 1 s 1 njk njk n1 jk (1 ) j 1 m1; k 1 l1; 1..; s 1.. s 1 1 s 1 s 1 ik(1 ) ik ik j 1 m1; k 1 l1; 1..; s 1.. s 1 1 s 1 s 1 ink (1 ) ink in1k j 1 m1; k 1 l1; 1..; s 1.. s (1 ) s s ijl ijl ijl1 s1 1 s 1 s 1 ij ij c d ij ij ij (24) (25) (26) i 1 n1; j 1 m 1; (27) (1 ) [ ( ) ] ij (28) i 1 n1; j 1 m1; 1...; s 1... Here s the number of iteration parameter of a relaxation. As an initial approximation on ( + 1)-e the temporary stratum the significance of the solution obtained on the preious temporary stratum is selected. The criterion of completion of iteration process: max s s s A R i j k 157

7 Математичне та комп ютерне моделювання i j s 1 1 s 1 s 1 1 ij ij A R max. unites monitoring absolute А and relatie R errors. Deised solution description. One of the features of the considered problem is the absence of model examples permitting to be coninced in a regularity ork of the computing algorithm. As one of the schemes of check it is possible to offer the folloing. Let's assume that the transposition of impurity along one of the coordinate axis for example along the axis OY is absent that is k 2 u 2. Then our problem is degenerated in to-dimensional one and reliable algorithms for its solution are deeloped and approed. Conducting calculation on these algorithms and on offered algorithm of calculation of the three-dimensional problem and comparing outcomes ith their concurrence to the exactitude required it is possible to consider the offered algorithm as acceptable. The program realization of circumscribed algorithm in language PASCAL allos to sole problems on the Pentium type COMPUTER ith olume of the main memory 1 in the protect mode on a grid by a size up to 1x1x7 knots. References: 1. Marchuk G. I. Mathematical simulation in the problem of enironment / G. I. Marchuk. M. : Nauka Berljand М. E. Prognostication and regulation of atmosphere pollution / М. E. Berljand // L. Gi drometeoizdat Buiko М. V. About boundary conditions for the equation of turbulent diffusion on spreading surface / М. V. Buiko // A meteorology and hydrology Slinn W. Formulation and Solution of the Diffusion-Deposition-Resuspension Problem / W. Slinn // Atm. Enironment Vol Buiko М. V. Turbulent transposition of radioactie impurity ith alloance for of processes of ind rise and dry sedimentation / М. V. Buiko // Information AN. Physics of the atmosphere and the ocean Vol Samarski A. A. The theory of the difference schemes / A. A. Samarski. M. : Nauka p. 7. Ortega J. Iteratie solution of nonlinear equations seeral ariables / J. Ortega W. Rheinbold. Ne York ; London : Academic press p. У роботі розглянутий алгоритм розв язання нелінійного рівняння з частинними похідними який описує процес забруднення атмосфери. Алгоритм оснований на відповідній модифікації сіткового методу. Ключові слова: забруднення атмосфери нелінійна модель сітковий метод. Отримано: