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.

Use of templates in connection with the treatment of very large modelsPerformance of comprehensive environmental studiesObtaining reliable and robust information about pollution levelsStudying the impact of future climatic changes on high pollution levelsInvestigating trends related to critical levels of pollution

Chapter 1: PDE systems arising in air pollution modelling and justification of the need for high speed computers

Chapter 2: Using splitting techniques in the treatment of air pollution models

Chapter 3: Treatment of the advection-diffusion phenomena

Chapter 4: Treatment of the chemical part: general ideas major numerical methods

Chapter 5: Error analysis of the partitioning procedures

Chapter 6: Efficient organization of the matrix computations

Chapter 7: Parallel computations

Chapter 8: Studying high pollution levels

Chapter 9: Impact of future climate changes on high pollution levels

Chapter 10: Implementation of variational data assimilation

Chapter 11: Discussion of some open questions

Appendix A: Colour plots

Bibliography

Symbol Table

Author Index

Subject Index