On the convergence toward nonequilibrium stationary states in thermostatted kinetic models

A differential equation‐based framework is suitable for the modeling of nonequilibrium complex systems if its solution is able to reach, as time goes to infinity, the related nonequilibrium steady states. The thermostatted kinetic theory framework has been recently proposed for the modeling of complex systems subjected to an external force field. The present paper is devoted to the mathematical proof of the convergence of the solutions of the thermostatted kinetic framework towards the related nonequilibrium stationary states. The proof of the main result is gained by employing the Fourier transform and distribution theory arguments.     

A thermostatted kinetic theory model for a hybrid multisource system with storage

This paper deals with the modeling of a hybrid energy multisource network composed by a non-renewable energy source and a renewable energy source. The mathematical model is derived within the framework of the thermostatted kinetic theory where the external force field coupled to the thermostat term mimics the construction of the energy storage. The parameters of the mathematical model are set in order to promote the use of the renewable energy source thus improving the quality of the provided energy. A computational analysis is performed to show the emerging phenomena that the model is able to capture. Specifically the computational analysis is mainly addressed to a sensitivity analysis on the switching-source parameters and the transition-energy parameters. Moreover the construction of the energy storage is analyzed by performing a sensitivity analysis on the magnitude of the external force field. Discussions and future research perspectives are postponed to the last section of the paper.     

Onset of nonlinearity in thermostatted active particles models for complex systems

This paper is concerned with the derivation of a new discrete general framework of the kinetic theory, suitable for the modeling of complex systems under the action of an external force field and constrained to kept constant the mass or density, and the kinetic or activation energy. The resulting model relies on the interactions of single individuals within the population and is expressed by means of nonlinear ordinary or partial integro-differential equations. The global in time existence and uniqueness of the solution to the relative Cauchy problem are proved for which the density and the energy of the solution are preserved. A critical analysis, proposed in the last part of the paper, outlines suitable applications and research perspectives.     

Modelling pedestrian dynamics into a metro station by thermostatted kinetic theory methods

This paper deals with the modelling of pedestrian dynamics at the entry of a metro station by means of the thermostatted kinetic theory framework. Specifically, the model depicts the time evolution of the pedestrian dynamics at the turnstiles under no panic conditions. The modelling of the microscopic interactions is based on the stochastic game theory and reflects the decision dynamics of the turnstiles pursued by pedestrians. A qualitative analysis is addressed to the equilibrium solutions by means of the classical stability theory of perturbations. Numerical simulations aim at showing the emerging behaviours captured by the model. In particular the model validation is obtained by performing a sensitivity analysis on the parameters and on the initial conditions. Further refinements and research perspective, including the modelling under panic conditions, are discussed in the last section of the paper.     

High-order computational methods for option valuation under multifactor models

Many of the different numerical techniques in the partial differential equations framework for solving option pricing problems have employed only standard second-order discretization schemes. A higher-order discretization has the advantage of producing low size matrix systems for computing sufficiently accurate option prices and this paper proposes new computational schemes yielding high-order convergence rates for the solution of multi-factor option problems. These new schemes employ Galerkin finite element discretizations with quadratic basis functions for the approximation of the spatial derivatives in the pricing equations for stochastic volatility and two-asset option problems and time integration of the resulting semi-discrete systems requires the computation of a single matrix exponential. The computations indicate that this combination of high-order finite elements and exponential time integration leads to efficient algorithms for multi-factor problems. Highly accurate European prices are obtained with relatively coarse meshes and high-order convergence rates are also observed for options with the American early exercise feature. Various numerical examples are provided for illustrating the accuracy of the option prices for Heston’s and Bates stochastic volatility models and for two-asset problems under Merton’s jump-diffusion model.     

Implicit kinetic schemes for scalar conservation laws

Based on kinetic formulation for scalar conservation laws, we present implicit kinetic schemes. For time stepping these schemes require resolution of linear systems of algebraic equations. The scheme is conservative at steady states. We prove that if time marching procedure converges to some steady state solution, then the implicit kinetic scheme converges to some entropy steady state solution. We give sufficient condition of the convergence of time marching procedure. For scalar conservation laws with a stiff source term we construct a stiff numerical scheme with discontinuous artificial viscosity coefficients that ensure the scheme to be equilibrium conserving. We couple the developed implicit approach with the stiff space discretization, thus providing improved stability and equilibrium conservation property in the resulting scheme. Numerical results demonstrate high computational capabilities (stability for large CFL numbers, fast convergence, accuracy) of the developed implicit approach. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 26–43, 2002     

High-order compact finite difference scheme for option pricing in stochastic volatility models

We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility models. The scheme is fourth order accurate in space and second order accurate in time. Under some restrictions, theoretical results like unconditional stability in the sense of von Neumann are presented. Where the analysis becomes too involved we validate our findings by a numerical study. Numerical experiments for the European option pricing problem are presented. We observe fourth order convergence for non-smooth payoff.     

Cosmological inflation models in the kinetic approximation

We consider the construction of cosmological inflation models with an approximate linear dependence of the kinetic energy of the scalar field on the state parameter. We compare the obtained solutions with known cosmological models and calculate the main parameters of cosmological perturbations.     

A kinetic method for strictly nonlinear conservation laws

Ideas from kinetic theory are used to construct a new solution method for nonlinear conservation laws of the formu ₁+f(u)_x=0. We choose a class of distribution functionsG=G(t, x, ), which are compactly supported with respect to the artificial velocity. This can be done in an optimal way, i.e. so that the-integral of the solution of the linear kinetic equationG _t+G_x=0 solves the nonlinear conservation law exactly.Introducing a time step and variousx-discretisations one easily obtains a variety of numerical schemes. Among them are interesting new methods as well as known upstream schemes, which get a new interpretation and the possibility to incorporate boundary value problems this way.     

A kinetic decomposition for singular limits of non-local conservation laws

We consider a non-local regularization of nonlinear hyperbolic conservation laws in several space variables. The regularization is motivated by the theory of phase dynamics and is based on a convolution operator. We formulate the initial value problem and begin by deriving a priori estimates which are independent of the regularization parameter. Following Hwang and Tzavaras we establish a kinetic decomposition associated with the problem under consideration, and we conclude that the sequence of solutions generated by the non-local model converges to a weak solution of the corresponding hyperbolic problem. Depending on the scaling introduced in the non-local dispersive term, this weak limit is either a classical Kruzkov solution satisfying all entropy inequalities or, more interestingly, a nonclassical entropy solution in the sense defined by LeFloch, that is, a weak solution satisfying a single entropy inequality and containing undercompressive shock waves possibly selected by a kinetic relation. Finally, we illustrate our analytical conclusions with numerical experiments in one spatial variable.     

On a kinetic theory approach to modelling degradation phenomena in conservation sciences

This paper deals with the modelling of degradation phenomena for works of art under the action applied by external agents. The analysis is based on a suitable development of the methods of the kinetic theory for active particles. The model consists in an evolution equation for the probability distribution of the degradation stage. The interpretation of empirical data provides the identification of the parameters of the model and a quantitative prediction of degradation events.     

Hybrid kinetic/fluid models for nonequilibrium systems

Our purpose is to derive a model describing the evolution of particles at various scales following their kinetic energy. Fast particles will be described through a collisional kinetic equation of Boltzmann-BGK type. This equation will be coupled with a fluid model (Euler equations) that describes the evolution of slower particles. The main interest of this approach is to reduce the cost of numerical simulations. To cite this article: N. Crouseilles et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Kinetic decomposition for kinetic models of BGK type

In this paper, we consider kinetic models of BGK type which describe the scalar conservation law at the microscopic scale. We use new technique developed in Comm. Partial Differential Equations 27 (2002) 1229 in order to get the convergence. First, we obtain the approximate transport equation for the given kinetic models of BGK type. Then using the averaging lemma, we obtain the convergence. This paper shows how to relate the given kinetic model with the averaging lemma to get the convergence.     

Existence of stationary solutions in kinetic models with Gaussian thermostats

The thermostatted kinetic framework has been recently proposed in [C. Bianca, Nonlinear Analysis: Real World Applications 13 (2012) 2593‐2608] for the modeling of complex systems in the applied sciences under the action of an external force field that moves out of equilibrium the system. The framework consists in an integro‐differential equation with quadratic nonlinearity coupled with the Gaussian isokinetic thermostat. This paper is concerned with the existence of stationary solutions proof. The main result is gained by fixed point and measure theory arguments. Copyright © 2013 John Wiley & Sons, Ltd.     

Large deviations for generalized compound Poisson risk models and its bankruptcy moments

We extend the classical compound Poisson risk model to the case where the premium income process, based on a Poisson process, is no longer a linear function. For this more realistic risk model, Lundberg type limiting results on the finite time ruin probabilities are derived. Asymptotic behaviour of the tail probabilities of the claim surplus process is also investigated.     

Stochastic approach and functional models in the kinetic theory of gases

Theoretical and Mathematical Physics -     

Finding all steady state solutions of chemical kinetic models

Ordinary differential equations are used frequently by theoreticians to model kinetic process in chemistry and biology. These systems can have stable and unstable steady states and oscillations. This paper presents an algorithm to find all steady state solutions to a restricted class of ODE models, for which the right-hand sides are linear combinations of rational functions of variables and parameters. The algorithm converts the steady state equations into a system of polynomial equations and uses a globally convergent homotopy method to find all the roots of the system of polynomials. All steady state solutions of the original ODEs are guaranteed to be present as roots of the polynomial equations. The conversion may generate some spurious roots that do not correspond to steady state solutions. The stability properties of the steady states are not revealed. This paper explains the algorithms used and gives results for a cell cycle modeling problem.