Algorithm of atmosphere contamination nonlinear model numerical realization
The article presents the algorithm for solving 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 values are considered, and a difference scheme constructio...
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| Veröffentlicht in: | Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
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Інститут кібернетики ім. В.М. Глушкова НАН України
2014
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| Zitieren: | Algorithm of atmosphere contamination nonlinear model numerical realization / L.A. Мitko, N.G. Serbov, Jo Sterten // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2014. — Вип. 11. — С. 152-158. — Бібліогр.: 7 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860086759151370240 |
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| author | Mitko, L.A. Serbov, N.G. Sterten, Jo |
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| citation_txt | Algorithm of atmosphere contamination nonlinear model numerical realization / L.A. Мitko, N.G. Serbov, Jo Sterten // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2014. — Вип. 11. — С. 152-158. — Бібліогр.: 7 назв. — англ. |
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| container_title | Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки |
| description | The article presents the algorithm for solving 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 values are considered, and a difference scheme construction method is proposed. As a result, problem is degenerated in two-dimensional one, and reliable algorithms for its solution are developed and approved.
У роботі розглянутий алгоритм розв’язання нелінійного рівняння з частинними похідними, який описує процес забруднення атмосфери. Алгоритм оснований на відповідній модифікації сіткового методу.
|
| first_indexed | 2025-12-07T17:20:20Z |
| format | Article |
| fulltext |
Математичне та комп’ютерне моделювання
152
UDC 004.942
L. A. Мitko*, Ph. D. of Physical and Mathematical Sciences,
N. G. Serbov**, Ph. D. of Technical Sciences,
Jo Sterten***, Assistant Professor
*NASU Pukhov Institute for Modeling in Energy Engineering, Kyiv,
** Odessa State Environmental University, Odessa,
*** Gjøvik University College, Høgskolen i Gjøvik, Norway
ALGORITHM OF ATMOSPHERE CONTAMINATION
NONLINEAR MODEL NUMERICAL REALIZATION
The article presents the algorithm for solving nonlinear partial
differential equations, that describe the process of air contamina-
tion. The algorithm is based on an appropriate modification of the
grid method. Boundary conditions and border values are consid-
ered, and a difference scheme construction method is proposed. As
a result, problem is degenerated in two-dimensional one, and reli-
able algorithms for its solution are developed and approved.
Key words: atmosphere contamination, nonlinear model, grid
method.
Computing experiment in contamination modeling. Revealing ten-
dencies of environmental contamination and determination of antropogeneous
influence on it are extremely actual problems. The research of natural and
chemical processes and defining the given problem are not allowed be con-
ducted as the location experiment in most cases. Therefore, important issues
acquire the possibility of realization of a computing experiment, for which the
construction of mathematical models adequate to the natural processes re-
searched and realized on modem computer facilities is necessary.
The problem of calculation of contamination concentration q(x, y, z, t)
in: he atmosphere owing to their ejection and transposition can be repre-
sented as following [1]:
3 3
1 1
i i s
i i ii i
q q q q
u aq k v f
t x x x z
, (1)
here (xl, x2, x3) = (x, y, z) — cartesian coordinates, whose plan XOY corre-
sponds to a terrestrial surface, t — temporary coordinate, (ul, u2, u3) —
components of the wind field, a — factor of disintegration of substance
taken as a boundary value for the equation.
Taking into account chemical transformation of substance and wash-
ing the partials away by settlings ki — factors of turbulent transposition,
vs — the settled velocity of the gravitational settling of partials of sub-
stance, f = f(x, y, z, t) is the known function circumscribing the density
function of contamination sources.
The solution of the equation (1) is discovered for want of certain ini-
tial and boundary conditions. The entry conditions are usually formulated
as known concentration of substance in researched area for want of t = 0:
© L. A. Мitko, N. G. Serbov, Jo Sterten, 2014
Серія: Фізико-математичні науки. Випуск 11
153
0( , , ,0) ( , , )Q x y z q x y z . (2)
The boundary conditions, at first, should correctly reflect the process
researched, and secondly ensure correctness of the delivered problem. If
the researched area is infinite on a horizon, it is natural to assume, that the
concentration decreases up to zero for the want of rushing x, y to infinity.
However, while applying numerical methods for the solution of the con-
sidered problem, the research area should be limited, that is to have, for
example, some kind of a parallelepiped:
1 1 2 2 0( , , ) : , ,D x y z a x b a y b z z H .
On the boundaries side it is possible to put
1 1
2 2
0a x b
a y b
q
n
.
On the upper boundary of the area the modeling of various processes
is possible:
The concentration of substance equal to zero:
0z Hq ;
Absence of diffusion transposition provided that vertical component of
the velocity of wind is equal to zero:
0z H
q
z
; (3)
Availability of the substance stream through the upper bound:
3 ( ) 0s z H
q
k z v q
z
. (4)
It is possible to simulate processes taking place on a spreading surface,
also by various images. For example, in [2; 3] it is considered that impurities
poorly interact with the surface of the ground usually and, hitting on it, are not
accumulated, and are again carried away in the atmosphere with turbulent
curls. As such curls at the surface are insignificant, it is possible to put:
03 0z z
q
k
z
. (5)
However, while modeling such processes as shaping of spots of in-
creased 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:
0 03 ( ) s z z z z
q Q
k z v q
z z
, (6)
where Q(x, y, t) — the concentration of the ground field of impurity,
*
0Z z z , *z — level of roughness, z — the thickness of the salta-
Математичне та комп’ютерне моделювання
154
tion stratum. As this concentration can vary in time, the condition (6) is
supplemented by the ratio
0( , , , ) ( , , )d
Q
j v q x y z t Q x y t
t
, (7)
where vd — the velocity of the dry settling of impurity; — the intensity
of wind rise. The use of conditions (6), (7) requires, in turn, the giving of
initial concentration of impurity on the surface:
0( , ,0) ( , )Q x y Q x y . (8)
If the spreading surface represents a mirror of water space, that, assum-
ing the water to swallow all impurities, we receive a boundary condition
0
0z zq .
Boundary value problem in contamination process mathematical
modeling. The correctness of statement of boundary value problems for the
equations of type (1) is proved with certain restrictions superimposed on dif-
fuseness and convection and for some combinations of boundary conditions.
It is natural that the most adequately considered processes describe three-
dimensional boundary value problems. The difficulties accompanying realiza-
tion of such mathematical models are also obvious. The problems of choosing
the solution method and correctness of the appropriate difference problem
(when apply the different methods of solution), maintenance of practical com-
puting stability of algorithm, the necessity of work with large scale arrays of
data concern them. Many scientists try to solve these problems by simplifica-
tion of the model, more often at the expense of diminution of dimensionality
of the problem. In this work the method of solution of a three-dimensional
boundary value problem is stated, which statement is indicated.
Let us consider a problem about evolution of the ground field of impuri-
ties Q(x, y, t) and field of impurities of near-the-land stratum of an atmosphere
q(x, y, z, t). The given problem is described by the equation (1) q = 0, f(x, y, z, t)=
= 0 with the given initial concentration of fields of impurity (2), (8) and
boundary conditions (3), (4), (6). The last boundary condition is supplemented
by ratio (7). With allowance for the last parity (ratio), the problem of impurity
transposition with the availability of wind rise and dry sedimentation is essen-
tially non-stationary, and the dynamic equilibrium between the concentration
of impurity on the surface and in near-the-surface stratum arises in time (t0 ~ -1),
as is rather a small number according to [4].
With allowance for (7) let us copy boundary condition (6) as:
3 0 1( , , , ) ( , , ),s d
q
k v q v q x y z t Q x y t
z
or
3 3 0( , , ), , s d
q
k vq Q x y t x z v v v
z
.
Серія: Фізико-математичні науки. Випуск 11
155
By designating 3u = u3 – vs, let us copy the equation (1) in the devel-
oped way:
_
31 2 1 2 3
q q q q q q q
u u u k k k
t x y z x x y y z z
. (1`)
For the solution of the problem (1'), (2)–(4), (8), (9) we shall use the
difference method offered in [5]. For it in area Ω = D[0, T]
1 1 2 2 0( , , ), , ,D x y z a x b a y b z z H ,
we shall enter a grid
_____
1 1 1
_____
2 2 2
____
0 0
: , 0, , ( ) / ,
: , 0, , ( ) / ,
: , 0, , ( ) / ,
: , / ,
x i i x x
y j j y y
z k k z z
v v
x x a ih i n h b a n
y y a jh j m h b a m
z z z kh k l h H z l
t t v T
here in after we shall deal with the grid area
, .
zh h h x y zD D
In the grid area h
(problem (1 '), (2)–(4), (8), (9) we shall put in the
orrespondence the difference scheme of the second order of approximation
on space variables:
_ _ _
_ _ _
1
1 1 1 1
1 1 1 2
1 1 1 1 1
2 2 3 3 3
(( ) ) ( ) ( ) (( ) )
( ) ( ) (( ) ) ( ) ( ) ,
1, 1, 1, 1, 1, 1, 0,1,2...
v v
ijk ijk
v v v v
x x ijk x ijk ijk y y ijk
x x y
v v v v v
y ijk ijk z z ijk z ijk ijk
y z z
w w
d w r w r w d w
r w r w d w r w r w
i n j m k l v
(11)
Here is designated: wijk = w(xi, yj, zk) — the grid function, approxi-
mating function q(x, y, z, t) in knots of grid (10),
1
1 1 1 1 1 1 1 1 1 1
1 0,5
(1 ) , 0,5 , 0,5( ), 0,5( ),
( ) ( , , ),
x
x ijk i j k
R R h u r u u r u u
d K x y z
2
1
2 2 2 2 2 2 2 2 2 2
2
(1 ) , 0,5 , 0,5( ), 0,5( ),
( ) ( , , ),
y
y ijk i j y h
R R h u r u u r u u
d K x y z
Математичне та комп’ютерне моделювання
156
2
1
3 3 3 3 3 3 3 3 3 3
3
(1 ) , 0,5 , 0,5( ), 0,5( ),
( ) ( , , ).
z
z ijk j k y
R R h u r u u r u u
d K x y z
(12)
__________
1 1 1
1 0 0 0
_________ __________
1 1
( ) ( ) , 1, 1 ; 0,1,...
( ) ( ) ( ) 0, 1, 1 ; 1, 1 ; 0,1,...
v v v
z ij z ij ij ij ij ij
v v
z ijk z ijk s ij ijk
d w v w y Q j m v
d w v w i n j m v
__________ ________
1 1
1 0 , 1, 1 ; 1, 1 ; 0,1,...v v
jk jkw w j m k l v (14)
1 1
1, ,v v
h jk njkw w
__________ ________
1, 1 ; 1, 1 ; 0,1,...j m k l v (15)
_________ ________
1 1
1 0 , 1, 1 ; 1, 1 ; 0,1,...v v
i k i kw w i n k l v (16)
1 1
1 ,v v
in k inkw w
_________ ________
1, 1 ; 1, 1 ; 0,1,...i n k l v (17)
Difference scheme construction. The conventional designations in
theory of different schemes [6] are used here. Thus, the constructed differ-
ence scheme is monotonous. That is why the grid of a maximum principle
without restrictions on pitches of a grid on space variables is executed.
The difference analogue of conditions (9), (8) looks as following:
1 _____ _____
1 1
0( ) , 0, ; 0, ; 0,1,...
v v
ij ij v v
d ij ij ij ij
v v
v w v i n j n v
(18)
0
0 ,ij ijv Q
_____ _____
0, ; 0, ,i n j n (19)
where vv
ij = v(xj, y, tv) — grid function, approximating function Q(x, y, t).
Let us copy a difference scheme (11)–(19) as:
1 1 1 1 1 1
1 1 1 1 1
_________ __________ ________
1
1
[
], 1, 1 ; 1, 1 ; 1, 1 ; 0,1,...,
v v v v v v v
ijk ijk ijk ijk i jk ijk i jkw ijk ij k ijk ij k ijk ijk
v
ijk ijk
w A w B w C w D w E w F w
G w i n j m k l v
where 1 1 ( ),ijk ijk ijk ijk ijk ijk ijkA B C D E F G
1 1 1 1
12 2
1 11 1
( ) ( ) ( ) ( )
( ) , ( ) ,
ijk ijk ijk ijk
ijk x ijk ijk x i jk
r r
B d C d
h hh h
2 2 2 2
12 2
2 22 2
( ) ( ) ( ) ( )
( ) , ( ) ,
ijk ijk ijk ijk
ijk y ijk ijk y ij k
r r
D d E d
h hh h
3 3 3 3
12 2
3 33 3
( ) ( ) ( ) ( )
( ) , ( ) ,
ijk ijk ijk ijk
ijk z ijk ijk z ijk
r r
F d G d
h hh h
Серія: Фізико-математичні науки. Випуск 11
157
1 1 1
0 3 1 1[ ( ) ], 1, 1; 1, 1; 0,1,...v v v
ij ij ij ij z ij ijw a h v d w i n j m v (20)
Where
1 1 1
1 0 3 1 1(( ) ) , , 1, 1; 1, 1;v v
ij z ij ij ij ij ijla d v h w b w i n j m
0,1,...v (21)
Where 1
3(( ) ( ) ) ( )ij z ijk s ij z ijkb d h v d ,
1 1
0[ ( ) ], 1, 1; 1, 1; 0,1,...v v v
ij ij d ij ij ijv C v w v i n j m v (22)
Where 1(1 )ij ijc .
We shall discover an approximate solution of the difference problem
using :he relaxation method [7]. The settlement formulas look as follow:
1 1 1 1 1 1
1 1
1 1 1 1 1 1
1 1 1 1
(1 ) (
),
1, 1; 1, 1; 1, 1; 0,1,...; 0,1,...
s v s v v s v s v
ijk ijk ijk ijk ijk i jk ijk i jk
s v s v s v s v
ijk ij k ijk ij k ijk ijk ijk ijk
w w A w B w C w
D w E w F w G w
i n j m k l v s
(23)
1 1 1 1
3(1 ) [ ( ) ],
1, 1; 1, 1; 0,1,...; 0,1,...
s v s v s v
ijk ijk ijk ijk z ijk ijkw w a h d w
i n j m v s
(24)
1 1 1 1
0 0 1(1 ) ,s v s v s v
jk jk jkw w w
1, 1; 1, 1; 0,1,..; 0,1,..j m k l v s
1 1 1 1
1(1 ) ,s v s v s v
njk njk n jkw w w
1, 1; 1, 1; 0,1,..; 0,1,..j m k l v s
1 1 1 1
0 0 0(1 ) ,
1, 1; 1, 1; 0,1,..; 0,1,..
s v s v s v
i k i k i kw w w
j m k l v s
(25)
1 1 1 1
1(1 ) ,
1, 1; 1, 1; 0,1,..; 0,1,..
s v s v s v
ink ink in kw w w
j m k l v s
(26)
1 1 1 1
1(1 ) ,s v s v s v
ijl ijl ijlw w w
1, 1; 1, 1;i n j m (27)
1 1 1 1
0(1 ) [ ( ) ],
1, 1; 1, 1; 0,1,...; 0,1,...
ij
s v s v s v v
ij ij d ij ij ijv v c v w v
i n j m v s
(28)
Here s — the number of iteration, — parameter of a relaxation. As
an initial approximation on (v + 1)-e the temporary stratum the signifi-
cance of the solution obtained on the previous temporary stratum is se-
lected. The criterion of completion of iteration process:
1 1 1 1 1
, ,
max s v s v s v
ijk ijk A R ijk
i j k
w w w
Математичне та комп’ютерне моделювання
158
1 1 1 1 1
,
max .s v s v s v
ij ij A R ijk
i j
v v v
unites monitoring absolute А and relative R errors.
Devised solution description. One of the features of the considered
problem is the absence of model examples permitting to be convinced in a
regularity work of the computing algorithm. As one of the schemes of
check it is possible to offer the following. Let's assume that the transposi-
tion of impurity along one of the coordinate axis, for example, along the
axis OY is absent, that is k2 0, u2 0. Then, our problem is degenerated in
two-dimensional one, and reliable algorithms for its solution are developed
and approved. Conducting calculation on these algorithms and on offered
algorithm of calculation of the three-dimensional problem and comparing
outcomes, with 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 allows to solve problems on the Pentium type COMPUTER
with volume of the main memory 1 in the protect mode on a grid by a size
up to 100x100x70 knots.
References:
1. Marchuk G. I. Mathematical simulation in the problem of environment /
G. I. Marchuk. — M. : Nauka, 1982.
2. Berljand М. E. Prognostication and regulation of atmosphere pollution /
М. E. Berljand // L. Gi drometeoizdat. — 1985.
3. Buikov М. V. About boundary conditions for the equation of turbulent diffu-
sion on spreading surface / М. V. Buikov // A meteorology and hydrology. —
1990. — № 9.
4. Slinn W. Formulation and Solution of the Diffusion-Deposition-Resuspension
Problem / W. Slinn // Atm. Environment. — 1976, Vol. 10. — № 3.
5. Buikov М. V. Turbulent transposition of radioactive impurity with allowance
for of processes of wind rise and dry sedimentation / М. V. Buikov // Informa-
tion AN. Physics of the atmosphere and the ocean. — 1993. — Vol. 29, № 2.
6. Samarski A. A. The theory of the difference schemes / A. A. Samarski. — M. :
Nauka, 1983. — 616 p.
7. Ortega J. Iterative solution of nonlinear equations several variables / J. Ortega,
W. Rheinbold. — New York ; London : Academic press, 1970. — 558 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, сіт-
ковий метод.
Отримано: 20.06.2014
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| id | nasplib_isofts_kiev_ua-123456789-86570 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 2308-5878 |
| language | English |
| last_indexed | 2025-12-07T17:20:20Z |
| publishDate | 2014 |
| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Mitko, L.A. Serbov, N.G. Sterten, Jo 2015-09-21T17:07:36Z 2015-09-21T17:07:36Z 2014 Algorithm of atmosphere contamination nonlinear model numerical realization / L.A. Мitko, N.G. Serbov, Jo Sterten // Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки: зб. наук. пр. — Кам’янець-Подільський: Кам'янець-Подільськ. нац. ун-т, 2014. — Вип. 11. — С. 152-158. — Бібліогр.: 7 назв. — англ. 2308-5878 https://nasplib.isofts.kiev.ua/handle/123456789/86570 004.942 The article presents the algorithm for solving 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 values are considered, and a difference scheme construction method is proposed. As a result, problem is degenerated in two-dimensional one, and reliable algorithms for its solution are developed and approved. У роботі розглянутий алгоритм розв’язання нелінійного рівняння з частинними похідними, який описує процес забруднення атмосфери. Алгоритм оснований на відповідній модифікації сіткового методу. en Інститут кібернетики ім. В.М. Глушкова НАН України Математичне та комп'ютерне моделювання. Серія: Фізико-математичні науки Algorithm of atmosphere contamination nonlinear model numerical realization Article published earlier |
| spellingShingle | Algorithm of atmosphere contamination nonlinear model numerical realization Mitko, L.A. Serbov, N.G. Sterten, Jo |
| title | Algorithm of atmosphere contamination nonlinear model numerical realization |
| title_full | Algorithm of atmosphere contamination nonlinear model numerical realization |
| title_fullStr | Algorithm of atmosphere contamination nonlinear model numerical realization |
| title_full_unstemmed | Algorithm of atmosphere contamination nonlinear model numerical realization |
| title_short | Algorithm of atmosphere contamination nonlinear model numerical realization |
| title_sort | algorithm of atmosphere contamination nonlinear model numerical realization |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/86570 |
| work_keys_str_mv | AT mitkola algorithmofatmospherecontaminationnonlinearmodelnumericalrealization AT serbovng algorithmofatmospherecontaminationnonlinearmodelnumericalrealization AT stertenjo algorithmofatmospherecontaminationnonlinearmodelnumericalrealization |