Zahari Zlatev

Computer treatment of large air pollution models

1. The Air Pollution Problem. 2. Mathematical Models for Studying the Long-Range Transport of Air Pollutants. 3. Numerical Treatment of Large Air Pollution Models. 4. Testing the Reliability of the Numerical Algorithms. 5. Need for Efficient Algorithms. 6. Computations on High-Speed Computers. 7. Running Air Pollution Models on Vector Machines. 8. Running Models on Parallel Machines with Shared Memory. 9. Running Models on Massively Parallel Computers. 10. Numerical Experiments with the Danish Eulerian Model. References. Author Index. Subject Index.

Direct Methods for Sparse Matrices

The mathematical models of many practical problems lead to systems of linear algebraic equations where the coefficient matrix is large and sparse. Typical examples are the solutions of partial differential equations by finite difference or finite element methods but many other applications could be mentioned. When there is a large proportion of zeros in the coefficient matrix then it is fairly obvious that we do not want to store all those zeros in the computer, but it might not be quite so obvious how to get around it. We first describe storage techniques which are convenient to use with direct solution methods, and we then show how a very efficient computational scheme can be based on Gaussian elimination and iterative refinement. A serious problem in the storage and handling of sparse matrices is the appearance of fill-ins, i.e. new elements which are created in the process of generating zeros below the diagonal. Many of these new elements tend to be smaller than the original matrix elements, and if they are smaller than a quantity which we shall call the drop tolerance we simply ignore them. In this way we may preserve the sparsity quite well but we probably introduce rather large errors in the LU decomposition to the effect that the solution becomes unacceptable. In order to retrieve the accuracy we use iterative refinement and we show theoreticaly and with practical experiments that it is ideal for the purpose. Altogether, the combination of Gaussian elimination, a large drop tolerance, and iterative refinement gives a very efficient and competitive computational scheme for sparse problems. For dense matrices iterative refinement will always require more storage and computation time, and the extra accuracy it yields may not be enough to justify it. For sparse problems, however, iterative refinement combined with a large drop tolerance will in most cases give very accurate results and reliable error estimates with less storage and computation time.

Comparison of numerical techniques for use in air pollution models with non-linear chemical reactions

Abstract The coupled, non-linear system of continuity equations describing an air pollution model with non-linear chemistry is solved numerically using finite differences, finite elements and pseudo-spectral methods. A smoothing procedure is proposed to avoid negative concentrations. Several tests are performed: single puff transported parallel and not parallel to the co-ordinate axis, two puffs along parallel lines, a rotating puff and a rotating plume. The accuracy of the results of advection + chemistry + smoothing calculations is evaluated through the comparison with the results of box model calculations. The concentration at the peak of the puff is compared in the case with advection only, chemistry only and advection + chemistry after 24 h integration. If the advection is performed by a pseudospectral algorithm, then the relative errors made in the case where advection + smoothing + chemistry is applied do not exceed 5%. These errors are of the same magnitude as the errors at the peak of the puff for the case where advection only is performed. For runs with discretization by second order finite differences, it is well known that the advection algorithms are neither able to preserve the shape of the puff nor to preserve the maximum concentrations in the puff. Our runs only confirmed this conclusion. For runs with the Smolarkiewicz algorithm, the results are slightly better than with the algorithm based on second order finite differences. However, the improvement of the accuracy is negligible compared with the increase of the computing time spent. The runs with the finite elements (CHAPEAU) advection algorithm show that the accuracy of this advection algorithm is worse than that of the pseudospectral advection, but it is faster than the latter algorithm with regard to computing time. The second order finite differences algorithm is about 5 times faster than the pseudospectral algorithm when the advection time only is taken into account. In the same situation the Smolarkiewicz algorithm is only a little better than the pseudospectral algorithm, while the finite elements (CHAPEAU) algorithm is about 2.S times faster. The differences are less when advection + smoothing + chemistry is applied.

On Some Pivotal Strategies in Gaussian Elimination by Sparse Technique

Pivotal interchanges are commonly used in the solution of large and sparse systems of linear algebraic equations by Gaussian elimination (in order to preserve the sparsity of the matrix and to prevent the appearance of large roundoff errors during the computations). The Markowitz strategy (see [H. M. Markowitz, The elimination form of inverse and its applications to linear programming, Management Sci., 3 (1957), pp. 255–269]) is often used to determine the pivotal sequence. An efficient implementation of this strategy is given by Curtis and Reid (see [A. R. Curtis and J. K. Reid, Fortran subroutines for the solution of sparse sets of linear equations, A.E.R.E., Report R.6844, HMSO, London, 1971]) and improved by Duff (see [I. S. Duff, MA28—a set of Fortran subroutines for sparse unsymmetric matrices, A.E.R.E., Report R.8730, HMSO, London, 1977]). In this paper it is shown how the classical Markowitz idea can be generalized. Consider the following parameters: u —the stability factor and

Operational air pollution forecasts from European to local scale

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Studying cumulative ozone exposures in Europe during a 7‐year period

Abstract A new operational air pollution forecast system, THOR, has been developed at the National Environmental Research Institute, Denmark. The integrated system consists of a series of air pollution models, covering a wide range of scales (from European scale to street scale in cities) and applications. The system is designed to automatically produce 3 days air pollution forecasts of the most important air pollution species on different scales on a continuous basis. The various models, the coupling/integration and the configuration of the models, the visualizations and the real-time performance on fast workstations with parallel architecture will be described. Some examples of model results and validations on street level are presented.

Parallel runs of a large air pollution model on a grid of Sun computers

A long-range transport model with nonlinear chemical reactions is described. The model contains 35 pollutants and 70 chemical reactions. This is a Eulerian model defined on a space domain containing the whole of Europe. The spherical space domain (corresponding to the Earths surface covered by the model) is mapped into a square plane domain and discretized by using a 32×32 grid. The grid increments are equidistant (both along the Ox axis and along the Oy axis). The choice of values of the physical parameters involved in the model and the numerical treatment of the model are shortly discussed. The model is tested with meteorological data for 1985 and 1989. The numerical results are compared with measurements at stations located in different European countries. Extensive comparisons of ozone concentrations for July 1985 with measurements taken at 24 European stations are also carried out. Results concerning three episodes in July 1985 as well as results obtained in the study of the sensitivity of the ozone concentrations to variations of NOxand/or anthropogenic VOC emissions are presented. The advantages and the limitations of such a model are discussed. The model is continuously improved by adding new modules to it. The plans for improvements in the near future are outlined.

Use of Iterative Refinement in the Solution of Sparse Linear Systems

Ozone is one of the most harmful pollutants in the troposphere. High ozone concentrations can damage plants, animals and humans. The damaging effects depend on the magnitude of a critical level of a special parameter, the cumulative ozone exposure. This is why cumulative ozone exposures must be carefully studied. It is important to determine the relation- ships between relevant emissions (NOx emissions, human-made VOC emissions, and/or a combination of NOx emissions and human-made VOC emissions) and cumulative ozone exposures. All these issues are discussed in this paper. Meteorological data from seven consecutive years, from 1989 to 1995, have been used in the experiments with different scenarios for varying the emissions (the NOx emissions, the human-made VOC emissions, as well as both the NOx emissions and the human-made VOC emissions). The particular air pollution model used in this study is the Danish Eulerian Model. Several hundred runs with different input data (meteorological data and/or emission data) have been performed. Advanced visualization techniques are used to interpret the large amount of digital data collected in these runs and to show clearly different trends and relationships that are normally hidden behind millions and millions of numbers. The model results were compared with measurements taken at more than 80 stations located in different European countries. The experiments indicate that it is sufficient to carry out computations over 5 consecutive years in order to eliminate the influence of extreme meteorological conditions (very warm or very cold summer months) on the cumulative ozone exposures, while this effect is clearly seen if less than 5 years are used in the experiments. It is shown that the relationship between the emissions (NOand/or human- made VOC emissions) and the cumulative ozone exposures is in general nonlinear. Finally, it is illustrated that the critical values for ozone exposures are exceeded in most of Europe (in many areas by more than 7 times).

Implementation of a variable stepsize variable formula method in the time-integration part of a code for treatment of long-range transport of air pollutants

Large-scale air pollution models can successfully be used in different environmental studies. These models are described mathematically by systems of partial differential equations. Splitting procedures followed by discretization of the spatial derivatives lead to several large systems of ordinary differential equations of order up to 80 millions. These systems have to be handled numerically at up to 250,000 time-steps. Furthermore, many scenarios are often to be run in order to study the dependence of the model results on the variation of some key parameters (as, for example, the emissions). Such huge computational tasks can successfully be treated only if: (i) fast and sufficiently accurate numerical methods are used and (ii) the models can efficiently be run on parallel computers.The mathematical description of a large-scale air pollution model will be discussed in this paper. The principles used in the selection of numerical methods and in the development of parallel codes will be described. Numerical results, which illustrate the ability of running the fine resolution versions of the model on Sun computers, will be given. Applications of the model in the solution of some environmental tasks will be presented.