Zahari Zlatev – författare

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 models- Performance of comprehensive environmental studies- Obtaining reliable and robust information about pollution levels- Studying the impact of future climatic changes on high pollution levels- Investigating trends related to critical levels of pollution

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 models Performance of comprehensive environmental studies Obtaining reliable and robust information about pollution levels Studying the impact of future climatic changes on high pollution levels Investigating trends related to critical levels of pollution

'Et moi, ...* si j'avait su comment en revenir, One service mathematics has rendered the je n 'y serais point aile.' human race. It has put common sense back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell 0. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'elre of this series.

The long-range transport of air pollutants and the reduction of air pollution to an acceptable level is an important environmental problem. The mathematical models by which this problem can be successfully studied are normally described by partial differential equations. The discretization in space leads to huge systems of ordinary differential equations containing up to several million equations that are to be handled during many time steps. This text shows why the numerical treatment of these problems always causes difficulties, even if high-speed computers are available. Several problems connected with the computer treatment of air pollution models are discussed. This book is an attempt to study these problems systematically. It should be useful for many specialists in air pollution modelling. Some topics are treated rather generally, therefore the book may also be used as a reference source by many scientists and engineers working with large mathematical tools.

The protection of the environment is one of the most important problems facing us today. Reliable and robust strategies for keeping pollution caused by harmful chemicals under safe levels have to be developed and used routinely. Large mathematical models, incorporating all important physical and chemical processes, can solve this task, but the computational burden is huge. The papers presented here describe some of the difficult problems that have to be solved during the development of large scale air pollution models, and different ways to solve them. The work involves combined research by specialists in the fields of environmental modelling, numerical analysis and scientific computing. Further, if models are to run in real time (which may be vital in the case of accidental hazardous releases), then new and efficient methods that harness the potential power of parallel supercomputers must be developed, so that actual computing speed starts to approach the maximum available performance. The articles should be understandable by specialists in the field of air pollution modelling, as well as specialists in the many other related areas.

The protection of the environment is one of the most important problems facing us today. Reliable and robust strategies for keeping pollution caused by harmful chemicals under safe levels have to be developed and used routinely. Large mathematical models, incorporating all important physical and chemical processes, can solve this task, but the computational burden is huge. The papers presented here describe some of the difficult problems that have to be solved during the development of large scale air pollution models, and different ways to solve them. The work involves combined research by specialists in the fields of environmental modelling, numerical analysis and scientific computing. Further, if models are to run in real time (which may be vital in the case of accidental hazardous releases), then new and efficient methods that harness the potential power of parallel supercomputers must be developed, so that actual computing speed starts to approach the maximum available performance. The articles should be understandable by specialists in the field of air pollution modelling, as well as specialists in the many other related areas.

Scientists and engineers are mainly using Richardson extrapolation as a computational tool for increasing the accuracy of various numerical algorithms for the treatment of systems of ordinary and partial differential equations and for improving the computational efficiency of the solution process by the automatic variation of the time-stepsizes. A third issue, the stability of the computations, is very often the most important one and, therefore, it is the major topic studied in all chapters of this book.Clear explanations and many examples make this text an easy-to-follow handbook for applied mathematicians, physicists and engineers working with scientific models based on differential equations. 　 ContentsThe basic properties of Richardson extrapolationRichardson extrapolation for explicit Runge-Kutta methodsLinear multistep and predictor-corrector methodsRichardson extrapolation for some implicit methodsRichardson extrapolation for splitting techniquesRichardson extrapolation for advection problemsRichardson extrapolation for some other problemsGeneral conclusions

Scientists and engineers are mainly using Richardson extrapolation as a computational tool for increasing the accuracy of various numerical algorithms for the treatment of systems of ordinary and partial differential equations and for improving the computational efficiency of the solution process by the automatic variation of the time-stepsizes. A third issue, the stability of the computations, is very often the most important one and, therefore, it is the major topic studied in all chapters of this book.Clear explanations and many examples make this text an easy-to-follow handbook for applied mathematicians, physicists and engineers working with scientific models based on differential equations.

　

ContentsThe basic properties of Richardson extrapolationRichardson extrapolation for explicit Runge-Kutta methodsLinear multistep and predictor-corrector methodsRichardson extrapolation for some implicit methodsRichardson extrapolation for splitting techniquesRichardson extrapolation for advection problemsRichardson extrapolation for some other problemsGeneral conclusions

Scientists and engineers are mainly using Richardson extrapolation as a computational tool for increasing the accuracy of various numerical algorithms for the treatment of systems of ordinary and partial differential equations and for improving the computational efficiency of the solution process by the automatic variation of the time-stepsizes. A third issue, the stability of the computations, is very often the most important one and, therefore, it is the major topic studied in all chapters of this book.Clear explanations and many examples make this text an easy-to-follow handbook for applied mathematicians, physicists and engineers working with scientific models based on differential equations.

　

ContentsThe basic properties of Richardson extrapolationRichardson extrapolation for explicit Runge-Kutta methodsLinear multistep and predictor-corrector methodsRichardson extrapolation for some implicit methodsRichardson extrapolation for splitting techniquesRichardson extrapolation for advection problemsRichardson extrapolation for some other problemsGeneral conclusions

This volume of the Springer Series "Lecture Notes in Computer Science" c- tains refereed papers which were presented at the Fifth International Conf- ence on Numerical Methods and Applications, NMA 2002, held in Borovets, Bulgaria, during 20-24 August 2002. The NMA 2002 Conference was organized by the Central Laboratory for Parallel Processing at the Bulgarian Academy of Sciences in cooperation with SIAM (Society for Industrial and Applied Mat- matics) and GAMM (Gesellschaft fur .. Angewandte Mathematik und Mechanik). Co-organizers of this traditional scienti?c meeting were the Institute of Mat- matics and Informatics at the Bulgarian Academy of Sciences and the Faculty of Mathematics and Informatics at the University of So?a. The conference was devoted to the 70th anniversary of the distinguished Bulgarian mathematician academician Blagovest Sendov - the founder of the Bulgarian school in - merical analysis, the teacher of dozens of Ph. D. students, the author of more than 270 papers, textbooks, and monographs. He is the scholar who has been at the center of the mathematical life in Bulgaria for about 50 years.Over the years he headed the Bulgarian Academy of Sciences, the University of So?a, the Faculty of Mathematics and Informatics of the University of So?a, the C- tral Laboratory for Parallel Processing of the Bulgarian Academy of Sciences, the Department of Mathematical Modeling of the Institute of Mathematics and Informatics, the Union of Bulgarian Mathematicians, etc. He represented with distinctionBulgarianscienceinsuchworld-recognizedorganizationsasIFIP (- ternationalFederationforInformationProcessing),theInternationalAssociation ofUniversities,theInternationalCouncilofScienti?cUnions,theBalkanMat- maticalUnion,etc.

'Et moi, ...* si j'avait su comment en revenir, One service mathematics has rendered the je n 'y serais point aile.' human race. It has put common sense back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell 0. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'elre of this series.

"Models are often the only way of interpreting measurements to in­ vestigate long-range transport, and this is the reason for the emphasis on them in many research programs". B. E. A. Fisher: "A review of the processes and models of long-range transport of air pollutants", Atmospheric Environment, 17(1983), p. 1865. Mathematical models are (potentially, at least) powerful means in the efforts to study transboundary transport of air pollutants, source-receptor relationships and efficient ways of reducing the air pollution to acceptable levels. A mathematical model is a complicated matter, the development of which is based on the use of (i) various mechanisms describing mathematically the physical and chemical properties of the studied phenomena, (ii) different mathematical tools (first and foremost, partial differenti­ al equations), (iii) various numerical methods, (iv) computers (especially, high-speed computers), (v) statistical approaches, (vi) fast and efficient visualization and animation techniques, (vii) fast methods for manipulation with huge sets of data (input data, intermediate data and output data).

"Models are often the only way of interpreting measurements to in­ vestigate long-range transport, and this is the reason for the emphasis on them in many research programs". B. E. A. Fisher: "A review of the processes and models of long-range transport of air pollutants", Atmospheric Environment, 17(1983), p. 1865. Mathematical models are (potentially, at least) powerful means in the efforts to study transboundary transport of air pollutants, source-receptor relationships and efficient ways of reducing the air pollution to acceptable levels. A mathematical model is a complicated matter, the development of which is based on the use of (i) various mechanisms describing mathematically the physical and chemical properties of the studied phenomena, (ii) different mathematical tools (first and foremost, partial differenti­ al equations), (iii) various numerical methods, (iv) computers (especially, high-speed computers), (v) statistical approaches, (vi) fast and efficient visualization and animation techniques, (vii) fast methods for manipulation with huge sets of data (input data, intermediate data and output data).

1. Contents of these proceedings. These proceedings contain most of the papers which were presented at the NATO ARW (Advanced Research Workshop) on "Large Scale Computations in Air Pollution Modelling". The workshop was held, from June 6 to June to, 1998, in Residence Bistritza, a beautiful site near Sofia, the capital of Bulgaria, and at the foot of the mountain Vitosha. 2. Participants in the NATO ARW. Scientists from 23 countries in Europe, North America and Asia attended the meeting and participated actively in the discussions. The total number of participants was 57. The main topic of the discussions was the role of the large mathematical models in resolving difficult problems connected with the protection of our environment. 3. Major topics discussed at the workshop. The protection of our environment is one of the most important problems facing modern society. The importance of this problem has steadily increased during the last two-three decades, and environment protection will become even more important in the next century. Reliable and robust control strategies for keeping the pollution caused by harmful chemical compounds under certain safe levels have to be developed and used in a routine way. Large mathematical models, in which all important physical and chemical processes are adequately described, can successfully be used to solve this task.