Development of information modeling technology for atmospheric pollution monitoring

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 finite-difference 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 non-uniform 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 predictor-corrector 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 small-scale 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 simply-connected 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 predictor-corrector scheme on a uniform grid

For the numerical solution of problem (1), consider the predictor-corrector 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 half-knot . 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 predictor-corrector 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 non-divergent 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 half-integer nodes of a uniform grid

      (9)

where  - step in time,  - step of grid , - number of grid nodes,  nodes of a non-uniform 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 predictor-corrector 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 explicit-implicit finite-difference 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 half-integer 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 predictor-corrector scheme. 2005. № 2.

4. Degtyarev LM, Ivanov TS The method of adaptive grids in one-dimensional nonstationary convection-diffusion problems // Differential equations. 1993. T. 29, №7.

5. Marchuk G.I. / Mathematical modeling in the environmental problem / / M: Science, 1982.