Computational and numerical challenges in environmental modelling

Many large mathematical models, not only models arising and used in environmental studies, are described by systems of partial differential equations. The discretization of the spatial derivatives in such models leads to the solution of very large systems of ordinary differential equations. These systems contain many millions of equations and have to be handled over large time intervals by applying many time-steps (up to several hundred thousand time-steps). Furthermore, many scenarios are as a rule to be run. This explains the fact that the computational tasks in this situation are enormous. Therefore, it is necessary to select fast numerical methods; to develop parallel codes and, what is most important when the problems solved are very large to organize the computational process in a proper way.The last item (which is very often underestimated but, let us re-iterate, which is very important) is the major topic of this book. In fact, the proper organization of the computational process can be viewed as a preparation of templates which can be used with different numerical methods and different parallel devices. The development of such templates is described in the book. It is also demonstrated that many comprehensive environmental studies can successfully be carried out when the computations are correctly organized. Thus, this book will help the reader to understand better that, while (a) it is very important to select fast numerical methods as well as (b) it is very important to develop parallel codes, this will not be sufficient when the problems solved are really very large. In the latter case, it is also crucial to exploit better the computer architecture by organizing properly the computational process.

Author Notes

Zahari Zlatev received his MSc from the Sofia University and his PhD from the Sct. Petersbourg University. He is a senior scientist and a project leader (Long-range Transport Modelling) at NERI (National Environmental Research Institute, Roskilde, Denmark) since 1980. He has developed the Unified Danish Eulerian Model (UNI-DEM), which has been used in many environmental studies and scientific projects (see also http://www.dmu.dk/AtmosphericEnvironment/DEM).

Zahari Zlatev published five monographs (two in Kluwer, two in Springer and one in MIR Publishers), has been editor of four proceedings volumes, has 92 papers in international journals, 123 papers in proceedings of international conferences and more than 200 institutional reports. He has been involved in training young specialists; including graduated students and PhD students. Zahari Zlatev organized five international conferences and nine mini-symposia at international conferences; in Stanford (USA), in Hamburg (Germany), in Varna (Bulgaria), in Toronto (Canada), three in Sozopol (Bulgaria), in Limassol (Cyprus), in Copenhagen (Denmark) and in San Francisco (USA). He has been an invited speaker at 24 international conferences and in the organizing committee of many international workshops and conferences (see also http://www.dmu.dk/AtmosphericEnvironment/zlatev.htm).

Ivan Dimov received his MSc in 1976, PhD in 1980 and the Doctor of Sciences degree in 1984 in Moscow, Russia. He is professor in Mathematical modelling since 1996. From 1996 to 2004 he was director of the Institute for Parallel Processing of the Bulgarian Academy of Sciences and Head of the Bulgarian Information Society Center of Excellence BIS-21. Currently he is a research professor at the University of Reading, UK.

Ivan Dimov published more than 90 papers in refereed international scientific journals and series, has been editor of seven special volumes published by Kluwer, Springer (Lecture notes in Computer Science), World Scientific, and IOS Press.

He has been involved in training young specialists: graduated students, PhD students and young post-docs. Ivan Dimov has organized nine international conferences. He is involved as a director or co-director in 12 international scientific projects (mainly EU projects). He has been an invited speaker at 35 international conferences and scientific institutions and served in program and organizing committees of many international conferences and workshops (see also http://www.reading.ac.uk/~sis04itd).

Preface	p. vii
Acknowledgements	p. xiii
Contents	p. xv
1 PDE systems arising in air pollution modelling and justification of the need for high speed computers	p. 1
1.1 Need for large-scale air pollution modelling	p. 2
1.2 The Danish Eulerian Model (DEM)	p. 5
1.3 Input data	p. 13
1.4 Output data	p. 18
1.5 Measurement data	p. 21
1.6 Some results obtained by DEM	p. 24
1.7 Applicability to PDEs arising in other areas	p. 35
1.8 Concluding remarks	p. 39
2 Using splitting techniques in the treatment of air pollution models	p. 43
2.1 Four types of splitting procedures	p. 44
2.2 Sequential splitting procedures	p. 45
2.3 Symmetric splitting procedures	p. 51
2.4 Weighted sequential splitting procedures	p. 52
2.5 Weighted symmetric splitting procedures	p. 53
2.6 Advantages and disadvantages of the splitting procedures	p. 53
2.7 Comparison of different splitting procedures	p. 54
2.8 Numerical experiments	p. 58
2.9 Using splitting procedures in connection with other applications	p. 86
2.10 Conclusions and plans for future research	p. 86
3 Treatment of the advection-diffusion phenomena	p. 89
3.1 Treatment of the horizontal advection	p. 90
3.2 Semi-discretization of the advection equation	p. 92
3.3 Time integration of the semi-discretized advection equation	p. 95
3.4 Numerical treatment of the horizontal diffusion	p. 105
3.5 Numerical treatment of the vertical exchange	p. 105
3.6 Applicability to other large-scale models	p. 106
3.7 Concluding remarks and plans for future research	p. 106
4 Treatment of the chemical part: general ideas and major numerical methods	p. 109
4.1 The chemical sub-model	p. 110
4.2 Why is it difficult to handle the chemical sub-model?	p. 111
4.3 Algorithms for the numerical integration of the chemical ODE systems	p. 114
4.4 Numerical results	p. 127
4.5 Treatment of the deposition	p. 134
4.6 Applicability to other large-scale models	p. 134
4.7 Plans for future work	p. 135
5 Error analysis of the partitioning procedures	p. 137
5.1 Statement of the problem	p. 137
5.2 When is [characters not reproducible] small?	p. 139
5.3 An application to air pollution problems	p. 150
5.4 Applicability to other models	p. 153
5.5 Concluding remarks and plans for future work	p. 154
6 Efficient organization of the matrix computations	p. 157
6.1 The horizontal advection-diffusion sub-model	p. 158
6.2 Matrices in the vertical exchange sub-model	p. 161
6.3 Matrices arising in the chemical sub-model	p. 162
6.4 Treatment of the model without splitting	p. 164
6.5 Use of sparse matrix techniques	p. 165
6.6 Utilizing the cache memory for large sparse matrices	p. 187
6.7 Comparisons with another sparse code	p. 197
6.8 Applicability to other models	p. 199
6.9 Concluding remarks	p. 199
7 Parallel computations	p. 201
7.1 The IBM SMP architecture	p. 203
7.2 Running the code on one processor	p. 204
7.3 Parallel runs on one node of the IBM SMP computer	p. 212
7.4 Parallel runs of the code across the nodes	p. 215
7.5 Scalability of the code	p. 216
7.6 When is it most desirable to improve the performance?	p. 217
7.7 Unification of the different versions of the model	p. 219
7.8 OpenMP implementation versus MPI implementation	p. 222
7.9 Parallel computations for general sparse matrices	p. 223
7.10 Applicability to other large-scale models	p. 227
7.11 Concluding remarks and plans for future work	p. 228
8 Studying high pollution levels	p. 233
8.1 Exceedance of some critical levels for ozone	p. 234
8.2 How to reduce the number of "bad" days?	p. 235
8.3 Influence of the biogenic VOC emissions on the ozone pollution levels	p. 237
8.4 Studying the transport of pollutants to a given country	p. 237
8.5 Prediction of the pollution levels for 2010	p. 240
8.6 Some conclusions	p. 243
9 Impact of future climate changes on high pollution levels	p. 245
9.1 Climate changes and air pollution levels	p. 246
9.2 Definition of six scenarios	p. 247
9.3 Validation of the results	p. 252
9.4 Variations of emissions and of meteorological parameters	p. 259
9.5 Results from the climatic scenarios	p. 266
9.6 Conclusions and plans for future work	p. 276
10 Implementation of variational data assimilation	p. 277
10.1 Basic ideas	p. 278
10.2 Calculating the gradient of the functional	p. 279
10.3 Forming the adjoint equations	p. 280
10.4 Algorithmic representation of a data assimilation algorithm	p. 282
10.5 Variational data assimilation for some one-dimensional examples	p. 287
10.6 Numerical results for the one-dimensional transport equation	p. 290
10.7 Treatment of some simple non-linear tests-problems	p. 299
10.8 Concluding remarks	p. 315
11 Discussion of some open questions	p. 317
11.1 Avoiding the use of splitting procedures	p. 317
11.2 Need for reliable error control	p. 319
11.3 Running of air pollution models on fine grids	p. 319
11.4 Transition from regional to urban scale	p. 320
11.5 Static and dynamic local refinement	p. 321
11.6 Need for advanced optimization techniques	p. 322
11.7 Use of data assimilation techniques	p. 322
11.8 Special modules for treatment of particles	p. 324
11.9 Use of computational grids	p. 324
11.10 Applicability of the methods to other large mathematical models	p. 326
Appendix A Colour plots	p. 327
Bibliography	p. 333
Symbol Table	p. 357
Author Index	p. 359
Subject Index	p. 367

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