- Università di Ferrara
Dipartimento di Matematica
Via Machiavelli 35
44121 Ferrara
Italy - +39-0532-974027
- Main fields of research: Numerical methods for nonlinear PDEs (high order methods for hyperbolic balance laws, stiff ... moreMain fields of research: Numerical methods for nonlinear PDEs (high order methods for hyperbolic balance laws, stiff differential equations, Monte Carlo methods and hybrid methods, spectral methods), Kinetic equations (Boltzmann equation, rarefied gas dynamics, plasma physics, Bose-Einstein condensation), Socio-economic and life sciences modelling (traffic flows, wealth distributions, price formation, carcinogenesis, collective behavior of animals)edit
In recent years kinetic theory has developed in many areas of the physical sciences and engineering, and has extended the borders of its traditional fields of application. New applications in traffic flow engineering, granular media... more
In recent years kinetic theory has developed in many areas of the physical sciences and engineering, and has extended the borders of its traditional fields of application. New applications in traffic flow engineering, granular media modeling, and polymer and phase transition physics have resulted in new numerical algorithms which depart from traditional stochastic Monte--Carlo methods.
This monograph is a self-contained presentation of such recently developed aspects of kinetic theory, as well as a comprehensive account of the fundamentals of the theory. Emphasizing modeling techniques and numerical methods, the book provides a unified treatment of kinetic equations not found in more focused theoretical or applied works.
The book is divided into two parts. Part I is devoted to the most fundamental kinetic model: the Boltzmann equation of rarefied gas dynamics. Additionally, widely used numerical methods for the discretization of the Boltzmann equation are reviewed: the Monte--Carlo method, spectral methods, and finite-difference methods. Part II considers specific applications: plasma kinetic modeling using the Landau--Fokker--Planck equations, traffic flow modeling, granular media modeling, quantum kinetic modeling, and coagulation-fragmentation problems.
Modeling and Computational Methods of Kinetic Equations will be accessible to readers working in different communities where kinetic theory is important: graduate students, researchers and practitioners in mathematical physics, applied mathematics, and various branches of engineering. The work may be used for self-study, as a reference text, or in graduate-level courses in kinetic theory and its applications.
This monograph is a self-contained presentation of such recently developed aspects of kinetic theory, as well as a comprehensive account of the fundamentals of the theory. Emphasizing modeling techniques and numerical methods, the book provides a unified treatment of kinetic equations not found in more focused theoretical or applied works.
The book is divided into two parts. Part I is devoted to the most fundamental kinetic model: the Boltzmann equation of rarefied gas dynamics. Additionally, widely used numerical methods for the discretization of the Boltzmann equation are reviewed: the Monte--Carlo method, spectral methods, and finite-difference methods. Part II considers specific applications: plasma kinetic modeling using the Landau--Fokker--Planck equations, traffic flow modeling, granular media modeling, quantum kinetic modeling, and coagulation-fragmentation problems.
Modeling and Computational Methods of Kinetic Equations will be accessible to readers working in different communities where kinetic theory is important: graduate students, researchers and practitioners in mathematical physics, applied mathematics, and various branches of engineering. The work may be used for self-study, as a reference text, or in graduate-level courses in kinetic theory and its applications.
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Abstract. In this note we describe a novel approach to the numerical solution of the Fokker-Planck-Landau equation in the non-homogeneous case. The method couples, through a time splitting algorithm, a finite volume scheme for the... more
Abstract. In this note we describe a novel approach to the numerical solution of the Fokker-Planck-Landau equation in the non-homogeneous case. The method couples, through a time splitting algorithm, a finite volume scheme for the transport with a fast spectral solver for the efficient solution of the collision operator.
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ABSTRACT Spectral methods based on Fourier series have been recently introduced for the Boltzmann and Landau integral operators. These spectral schemes allow us to obtain spectrally accurate solutions with a reduction of the computational... more
ABSTRACT Spectral methods based on Fourier series have been recently introduced for the Boltzmann and Landau integral operators. These spectral schemes allow us to obtain spectrally accurate solutions with a reduction of the computational cost strictly related to the particular structure of the integral. A reduction from N 2 to Nlog 2 N is readily deducible for the Landau equation, whereas in the Boltzmann case such a reduction does not seem possible. Here we extend the spectral method to the case of the Boltzmann collision integral for one dimensional granular media and discuss the possibility of developing approximate methods that bring the overall computational cost to Nlog 2 N.
For small Knudsen number, simulation of rarefied gas dynamics by the DSMC method becomes computationally intractable because of the large collision rate. To overcome this problem we have developed a hybrid simulation method, combining... more
For small Knudsen number, simulation of rarefied gas dynamics by the DSMC method becomes computationally intractable because of the large collision rate. To overcome this problem we have developed a hybrid simulation method, combining DSMC and a fluid dynamic description in a single seamless method. The molecular distribution function f is represented as a linear combination of a Maxwellian distribution M and a particle distribution g; ie, f = M +(1 − )g. The density, velocity and temperature of M are governed by uid-like equations, while the ...
The study of formations and dynamics of opinions leading to the so-called opinion consensus is one of the most important areas in mathematical modelling of social sciences. Following the Boltzmann-type control approach recently introduced... more
The study of formations and dynamics of opinions leading to the so-called opinion consensus is one of the most important areas in mathematical modelling of social sciences. Following the Boltzmann-type control approach recently introduced by the first two authors, we consider a group of opinion leaders who modify their strategy accordingly to an objective functional with the aim of achieving opinion consensus. The main feature of the Boltzmann-type control is that, owing to an instantaneous binary control formulation, it permits the minimization of the cost functional to be embedded into the microscopic leaders' interactions of the corresponding Boltzmann equation. The related Fokker-Planck asymptotic limits are also derived, which allow one to give explicit expressions of stationary solutions. The results demonstrate the validity of the Boltzmann-type control approach and the capability of the leaders' control to strategically lead the followers' opinion.
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We introduce and discuss a nonlinear kinetic equation of Boltzmann type that describes the influence of knowledge in the evolution of wealth in a system of agents that interact through the binary trades, an equation first introduced by... more
We introduce and discuss a nonlinear kinetic equation of Boltzmann type that describes the influence of knowledge in the evolution of wealth in a system of agents that interact through the binary trades, an equation first introduced by Cordier et al. (2005 J. Stat. Phys. 120, 253-277 (doi:10.1007/S10955-005-5456-0)). The trades, which include both saving propensity and the risks of the market, are here modified in the risk and saving parameters, which now are assumed to depend on the personal degree of knowledge. The numerical simulations show that the presence of knowledge has the potential to produce a class of wealthy agents and to account for a larger proportion of wealth inequality.
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Abstract Some stationary problems for the Broadwell model of the Boltzmann equation are investigated both from analytical and numerical points of view. The knowledge of exact solutions for these stationary problems allows us to discuss an... more
Abstract Some stationary problems for the Broadwell model of the Boltzmann equation are investigated both from analytical and numerical points of view. The knowledge of exact solutions for these stationary problems allows us to discuss an approximation scheme consistent with the nonlinearity of the model itself.
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In the framework of the discrete Boltzmann equation, a suitable space-time discretization of the one-dimensional fourteen discrete velocity model by Cabannes, leads in a bounded domain to the nonlinear Markovian evolution of a probability... more
In the framework of the discrete Boltzmann equation, a suitable space-time discretization of the one-dimensional fourteen discrete velocity model by Cabannes, leads in a bounded domain to the nonlinear Markovian evolution of a probability vector, whose moments represent the macroscopic quantities of the gas. Convergence of the probability vector towards the equilibrium steady state is proven when the walls are at a temperature compatible with the equilibrium itself. A physical application is subsequently dealt with. ...
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In this Note we present methods for the development of fast numerical schemes for the Boltzmann collision integral. These schemes are based on a combination of a Carleman-like representation together with a suitable angular approximation.... more
In this Note we present methods for the development of fast numerical schemes for the Boltzmann collision integral. These schemes are based on a combination of a Carleman-like representation together with a suitable angular approximation. For the hard spheres model in dimension three, we are able to derive spectral methods that can be evaluated through fast algorithms. Estimates for the errors and spectral accuracy are also given. To cite this article: C. Mouhot, L. Pareschi, CR Acad. Sci. Paris, Ser. I 339 (2004).
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We propose a numerical approximation to kinetic equations of Boltzmann type based on the splitting in time between the transport and the collision phases. By means of the representation of the solution to the relaxation part as an... more
We propose a numerical approximation to kinetic equations of Boltzmann type based on the splitting in time between the transport and the collision phases. By means of the representation of the solution to the relaxation part as an arithmetic mean of the convolution iterates of a quadratic positive operator, we construct a stable discretization valid for arbitrary values of the mean free path. Comparison of the present approximation with previous ones is made for Broadwell discrete velocity model.
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SUNTO - In questo articolo vengono studiate alcune propriet~ di regolarit~ della soluzione dell'equazione di Kac senza cutoff. Introducendo opportuni funzionali di Lyapunov si prova che l'equazione senza cutoff converge... more
SUNTO - In questo articolo vengono studiate alcune propriet~ di regolarit~ della soluzione dell'equazione di Kac senza cutoff. Introducendo opportuni funzionali di Lyapunov si prova che l'equazione senza cutoff converge all'equilibrio in vari spazi di Sobolev. ... ABSTRACT - This paper is devoted to the study of some regularity properties of the solution to the non cutoff Kac equation. By introducing suitable Lyapunov hmctionals we prove that the solution to the non cutoff Kac equation converges to equilibrium in various Sobolev spaces.
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... Dans cette Note, nous considérons l'équation de diffusion dégénérée du quatrieme ordre (1.1) qui mod`elise la cellule de Hele-Shaw. ... qui fait passer de l'équation... more
... Dans cette Note, nous considérons l'équation de diffusion dégénérée du quatrieme ordre (1.1) qui mod`elise la cellule de Hele-Shaw. ... qui fait passer de l'équation cinétique a l'équation des films fins est analogue a ceux du type “quasi-elastique” pour les gaz granulaires [11] ou ...
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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... more
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. 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. Part III contains several contributions related to the construction of suitable numerical ...
The construction of efficient numerical schemes for Boltzmann equations represents a real challenge for numerical methods and is of paramount importance in many applications, ranging from rarefied gas dynamics (RGB), plasma physics and... more
The construction of efficient numerical schemes for Boltzmann equations represents a real challenge for numerical methods and is of paramount importance in many applications, ranging from rarefied gas dynamics (RGB), plasma physics and granular media to semiconductors [2; 14; 18; 35]. The main difficulties are essentially due to the structure of the multidimensional integral that describes the interaction among particles. This integration has to be handled carefully since it is at the basis of the macroscopic properties of the ...
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We consider an integro-differential model for evolutionary game theory which describes the evolution of a population adopting mixed strategies. Using a reformulation based on the first moments of the solution, we prove some analytical... more
We consider an integro-differential model for evolutionary game
theory which describes the evolution of a population adopting
mixed strategies. Using a reformulation based on the first
moments of the solution, we prove some analytical properties of
the model and global estimates. The asymptotic behavior and the
stability of solutions in the case of two strategies is analyzed
in details. Numerical schemes for two and three strategies which
are able to capture the correct equilibrium states are also
proposed together with several numerical examples.
theory which describes the evolution of a population adopting
mixed strategies. Using a reformulation based on the first
moments of the solution, we prove some analytical properties of
the model and global estimates. The asymptotic behavior and the
stability of solutions in the case of two strategies is analyzed
in details. Numerical schemes for two and three strategies which
are able to capture the correct equilibrium states are also
proposed together with several numerical examples.
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Abstract In order to embark the development of characteristic based schemes for hyperbolic systems with nonlinear sti source terms we have studied a prototype one-dimensional discrete-velocity Boltzmann equation. We show that the method... more
Abstract In order to embark the development of characteristic based schemes for hyperbolic systems with nonlinear sti source terms we have studied a prototype one-dimensional discrete-velocity Boltzmann equation. We show that the method can be evaluated at the cost of an explicit scheme and that yield accurate solutions even when the relaxation time is much less than the time step. Numerical experiments con rm that the quality of the results obtained is comparable with those of earlier approaches.
In this paper, we discuss the large-time behavior of solution of a simple kinetic model of Boltzmann–Maxwell type, such that the temperature is time decreasing and/or time increasing. We show that, under the combined effects of the... more
In this paper, we discuss the large-time behavior of solution of a simple kinetic model of Boltzmann–Maxwell type, such that the temperature is time decreasing and/or time increasing. We show that, under the combined effects of the nonlinearity and of the time-monotonicity of the temperature, the kinetic model has non trivial quasi-stationary states with power law tails. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution. The same idea is applied to investigate the large-time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and/or decreasing of the mean wealth. In this last case, the large-time behavior of the solution shows a Pareto power law tail. Numerical results confirm the previous analysis.
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Abstract In this paper we introduce a new smoothed scheme derived from the spectral Fourier method for the homogeneous Boltzmann equation recently introduced in [14, 15]. More precisely we show that using suitable smoothing filters the... more
Abstract In this paper we introduce a new smoothed scheme derived from the spectral Fourier method for the homogeneous Boltzmann equation recently introduced in [14, 15]. More precisely we show that using suitable smoothing filters the method can be designed in such a way that the spectral solution remains positive in time and preserves the total mass. Several numerical examples are given to illustrate the previous analysis.