The workshop on Modelling and Numerics of Kinetic Dissipative Systems. took place at Hotel Giardino sul Mare in the bay of Lipari, the largest island of the Aeolian arcipelago, Messina, Italy, on May 31th-June 4th 2004. Major aim of the workshop was to stimulate the interaction between young researchers, graduate students and experts coming from diferent felds, as physics, engineering, mathematics and computer simulations, in identifying key ideas, new advances and open questions in the mathematical analysis and the development of numerical methods for dissipative systems, including both kinetic and hydrodynamic descriptions. The papers in this volume originate from the lectures given at the workshop. Many of the authors are renowned experts in the feld, and the articles of this book cover a signi.cant part of the recent progresses on kinetic dissipative systems. The book is divided into three parts, which contain respectively recent results in the kinetic theory of granular gases, kinetic theory of chemically reacting gases, and numerical methods for kinetic systems. Part I is devoted to theoretical aspects of granular gases. Granular gases are dilute systems of inelastically colliding particles. As common for open systems, granular gases reveal a rich variety of self-organized structures such as large scale clusters, vertex .elds, characteristic shock waves and others, which are still far from being completely understood. Applications of such systems range from astrophysics (stellar clouds, planetary rings), to industrial processes (handling of pharmaceuticals) and environment (pollution, erosion processes). Many of the results found in the context of granular gases are independent of the particular interaction of the grains and may be applied to other systems of dissipative particles. To understand the origin of common properties of such systems, which may be very different by their physical nature, remains a challenge for scientists working in these felds. Granular gases can be modelled at different scales, with an increasing level of detail: in hydrodynamical models the gas is described in terms of the density of mass, momentum, and energy; in the kinetic description the gas is described by the evolution of a density function in phase space. Some of the many theoretical problems that such mathematical models pose are covered in this first part. Part II presents recent results on modelling of kinetic systems in which molecules can undergo binary collisions in presence of chemical reactions and/or in presence of quantum effects. The study of these models is of paramount importance for a deep understanding of environmental problems (atmospheric pollution among others), where the e.ects of chemical processes can not be neglected. A correct modelling of kinetic systems is here a frst step in the direction of recovering hydrodynamic descriptions. Other applications of kinetic modelling are also presented in this part ranging from semiconductors to economy. Part III contains several contributions related to the construction of suitable numerical methods and simulations for granular gases. This aspect is particularly important in assessing the behavior of the system. Numerical simulations constitute an essential tool not only for obtaining quantitative prediction, but also as a fundamental bridge between theory and experiment, allowing the theoretician to formulate or corroborate conjectures on the properties of a given model, and the modelist to improve the model itself by the feedback of the comparison between numerics and experiment. The most popular and widely used techniques at the kinetic level are of Monte Carlo type. Other approaches which avoid the presence of .uctuations are based on deterministic discretizations of the collision integral at different level. A particular relevance in this feld is represented by discrete velocity models and spectral methods. As an alternative one can resort to fnite diference or fnite volume methods to solve hydrodynamic models which avoid the costly collision integral. Many of these aspects are illustrated in this last part of the book. We are grateful to all authors for their contributions and to the referees whose comments led in many cases to a marked improvement of the manuscripts. For their essential support, our special thanks go to both National Groups of Mathematical Physics (GNFM) and Scientifc Computing (GNCS) of INdAM (National Institute of High Mathematics of Italy), and to the Research Training Network HYKE fHyperbolic and Kinetic Equations: Asymptotics, Numerics, Applicationsf funded by the European Community as contract HPRN-CT-2002-00282. Our special gratitude goes to Dr. Cristina Milazzo who did a wonderful job in organizing the workshop and in hosting the participants.

### Modelling and Numerics of Kinetic Dissipative Systems

#####
*PARESCHI, Lorenzo;*

##### 2006

#### Abstract

The workshop on Modelling and Numerics of Kinetic Dissipative Systems. took place at Hotel Giardino sul Mare in the bay of Lipari, the largest island of the Aeolian arcipelago, Messina, Italy, on May 31th-June 4th 2004. Major aim of the workshop was to stimulate the interaction between young researchers, graduate students and experts coming from diferent felds, as physics, engineering, mathematics and computer simulations, in identifying key ideas, new advances and open questions in the mathematical analysis and the development of numerical methods for dissipative systems, including both kinetic and hydrodynamic descriptions. The papers in this volume originate from the lectures given at the workshop. Many of the authors are renowned experts in the feld, and the articles of this book cover a signi.cant part of the recent progresses on kinetic dissipative systems. The book is divided into three parts, which contain respectively recent results in the kinetic theory of granular gases, kinetic theory of chemically reacting gases, and numerical methods for kinetic systems. Part I is devoted to theoretical aspects of granular gases. Granular gases are dilute systems of inelastically colliding particles. As common for open systems, granular gases reveal a rich variety of self-organized structures such as large scale clusters, vertex .elds, characteristic shock waves and others, which are still far from being completely understood. Applications of such systems range from astrophysics (stellar clouds, planetary rings), to industrial processes (handling of pharmaceuticals) and environment (pollution, erosion processes). Many of the results found in the context of granular gases are independent of the particular interaction of the grains and may be applied to other systems of dissipative particles. To understand the origin of common properties of such systems, which may be very different by their physical nature, remains a challenge for scientists working in these felds. Granular gases can be modelled at different scales, with an increasing level of detail: in hydrodynamical models the gas is described in terms of the density of mass, momentum, and energy; in the kinetic description the gas is described by the evolution of a density function in phase space. Some of the many theoretical problems that such mathematical models pose are covered in this first part. Part II presents recent results on modelling of kinetic systems in which molecules can undergo binary collisions in presence of chemical reactions and/or in presence of quantum effects. The study of these models is of paramount importance for a deep understanding of environmental problems (atmospheric pollution among others), where the e.ects of chemical processes can not be neglected. A correct modelling of kinetic systems is here a frst step in the direction of recovering hydrodynamic descriptions. Other applications of kinetic modelling are also presented in this part ranging from semiconductors to economy. Part III contains several contributions related to the construction of suitable numerical methods and simulations for granular gases. This aspect is particularly important in assessing the behavior of the system. Numerical simulations constitute an essential tool not only for obtaining quantitative prediction, but also as a fundamental bridge between theory and experiment, allowing the theoretician to formulate or corroborate conjectures on the properties of a given model, and the modelist to improve the model itself by the feedback of the comparison between numerics and experiment. The most popular and widely used techniques at the kinetic level are of Monte Carlo type. Other approaches which avoid the presence of .uctuations are based on deterministic discretizations of the collision integral at different level. A particular relevance in this feld is represented by discrete velocity models and spectral methods. As an alternative one can resort to fnite diference or fnite volume methods to solve hydrodynamic models which avoid the costly collision integral. Many of these aspects are illustrated in this last part of the book. We are grateful to all authors for their contributions and to the referees whose comments led in many cases to a marked improvement of the manuscripts. For their essential support, our special thanks go to both National Groups of Mathematical Physics (GNFM) and Scientifc Computing (GNCS) of INdAM (National Institute of High Mathematics of Italy), and to the Research Training Network HYKE fHyperbolic and Kinetic Equations: Asymptotics, Numerics, Applicationsf funded by the European Community as contract HPRN-CT-2002-00282. Our special gratitude goes to Dr. Cristina Milazzo who did a wonderful job in organizing the workshop and in hosting the participants.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.