Development of information modeling technology for atmospheric pollution monitoring
Table of contents: The KazakhAmerican Free University Academic Journal №9  2017
Authors: Rakhmetullina Saule, D. Serikbayev East Kazakhstan state technical university, Kazakhstan
Bugubayeva Alina, D. Serikbayev East Kazakhstan state technical university, Kazakhstan
Introduction
Strengthening the impact
on the natural environment has generated a number of related problems, the most
acute of which is the state of atmospheric air. At present, the use of
information technologies in the environment is topical, as they are widely
intended to provide storage, processing, interpretation, access, delivery,
integrity and relevance of empirical and theoretical information using
innovative, analytical, empirical methods and information processing models.
The tools of information
technologies are the systems for calculating and forecasting atmospheric
pollution. In most cases, these systems use computer modeling techniques of
great practical importance. Since they provide an assessment, forecast and
control of changes in the state of natural resources under the influence of
anthropogenic factors. The mathematical support of such systems is based on
models of physical processes. At the stage of numerical realization, the
transition from the model to the finitedifference analogue is carried out on
the grid in time and space.
There are two main
classes of grids used to solve problems in multidimensional domains: uniform
grids whose nodes in the region under consideration are equidistant from each
other and the cells have a rectangular shape; and nonuniform or adaptive grids
characterized by irregular distribution of nodes and cells of arbitrary shape,
configuration and location.
At present, there has
been a significant increase in interest in constructing adaptive grids and
carrying out numerical calculations on them. As research shows, the method of
adaptive grids can significantly increase the accuracy and profitability of
computational algorithms. It allows you to obtain a result of high accuracy
even with a relatively small number of grid nodes. High accuracy is achieved
due to an increase in the concentration of grid nodes in the zones of location
of the features of the phenomenon being investigated [1  4].
This article is devoted
to the development of mathematical support for the atmospheric pollution
monitoring system using the predictorcorrector method on uniform and adaptive
grids.
Mathematical model
The process of
impurities distribution in the atmosphere is carried out by wind currents of
air taking into account their smallscale fluctuations. The averaged flow of a
substance has advective and convective components, and their averaged
fluctuation motions can be interpreted as diffusion against the background of
the main averaged motion associated with it [5].
Consider an equation describing the
process of nonstationary impurity transport in a simplyconnected domain Ω:
_{ } (1)
_{}
Here _{} is the intensity of the aerosol substance
that migrates with the air stream in the atmosphere; _{} analogue of the component of the wind speed
vector in the direction of the X axis;_{ }_{}  the reciprocal of the time
interval over which the intensity of the substance has changed in comparison
with the initial intensity; _{} coefficient of turbulence; _{} power of the emission source
located at the point _{}; _{} is the delta function.
The predictorcorrector scheme on a
uniform grid
For the numerical solution of
problem (1), consider the predictorcorrector scheme on a uniform fixed grid
with nodes _{ }and step_{}.
At the "predictor" step,
splitting into the convective and diffusion parts occurs:
_{}
_{}
Thus, at the "predictor"
step
_{}
(2)
_{}
(3)
_{}
two auxiliary values are calculated _{} and _{}. The first of these is determined from the
equation with convective transfer (2). It refers to a halfknot _{}. In equation (2), the quantity _{} is a step in time. The second
quantity _{} is calculated in the diffusion transfer stage (3). To implement
this step, we use the sweep method, where we calculate the necessary quantities _{} и _{}.
At the stage of
"proofreader"
_{}
(4)
_{}
the required quantities _{} defined in whole nodes _{} are determined.
The predictorcorrector scheme on
the adaptive grid
In order to construct a scheme on a
moving grid, we must rewrite problem (1) in new coordinates_{} connected with the original
coordinates _{} by a smooth transformation
_{}
(5)
with a positive Jacobian _{}, which uniquely maps a unit
interval _{} on the solution domain _{}. In coordinates _{}, equation (1) can be written in divergent and
nondivergent forms:
_{ }(6)
_{ }(7)
At the predictor stage, the equation
(7) is split into two equations, the first of which describes convective
transport, and the second takes into account the diffusion process and the
source term.
The transport equation:
_{}_{ }(8)
is
approximated in halfinteger nodes of a uniform grid _{}
_{}
_{ }(9)
_{}
_{}
_{}
_{}
where _{}  step in time, _{ }
step of grid _{}, _{} number of grid nodes, _{} nodes
of a nonuniform moving grid _{}, which is the image when grid _{} is
displayed.
The second equation:
_{}
is approximated in integer nodes:
_{ }(10)
_{}
_{}
_{}.
At the step of the corrector,
equation (8) is approximated in a divergent form
_{ }(11)
_{}
_{}
_{}
Thus, the constructed
predictorcorrector circuit (9)  (11) will allow to obtain a numerical
solution without oscillations.
Conclusion
In this article, a method is
proposed for the numerical solution of the problem of transport and diffusion
of matter in the atmosphere from an emission source. The numerical method is
based on the explicitimplicit finitedifference predictor  corrector scheme.
In the simulation, splitting is carried out according to physical processes,
and the equation at the first step of the predictor is approximated explicitly
at halfinteger nodes, and on the second  implicitly in integers, while the
source term is included in the second step of the predictor. The application of
this method makes it possible to get rid of parasitic oscillations that appear
in numerical calculations using other schemes.
Since during the transfer and
diffusion of the pollutant from the source, the function characterizing the
concentration of the pollutant undergoes strong changes near the source, then
the use of adaptive grids is one of the optimal approaches for solving such
problems.
REFERENCE
1. Rakhmetullina
S.Zh. The forecasting subsystem of the information system for monitoring air
pollution // Search.  2010.  №2.
2. Khakimzyanov GS,
Shokin Yu.I. Difference schemes on adaptive grids: Part 1. Problems for partial
differential equations with one spatial variable. Novosibirsk. 2005.
3. Shokin Yu.I.,
Sergeeva Yu.V., Khakimzyanov G.S. Monotonization of the explicit
predictorcorrector scheme. 2005. № 2.
4. Degtyarev LM,
Ivanov TS The method of adaptive grids in onedimensional nonstationary
convectiondiffusion problems // Differential equations. 1993. T. 29, №7.
5. Marchuk G.I. / Mathematical modeling in the
environmental problem / / M: Science, 1982.
Table of contents: The KazakhAmerican Free University Academic Journal №9  2017
