In this survey we consider the development and the mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity.\ud \ud Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic preserving methods and the construction of hybrid schemes

We discuss implicit-explicit (IMEX) Runge-Kutta methods which are particularly adapted to stiff kinetic equations of Boltzmann type. We consider both the case of easy invertible collision operators and the challenging case of Boltzmann collision operators. We give sufficient conditions in order that such methods are asymptotic preserving and asymptotically accurate. Their monotonicity properties are also studied. In the case of the Boltzmann operator the methods are based on the introduction of a penalization technique for the collision integral. This reformulation of the collision operator permits us to construct penalized IMEX schemes which work uniformly for a wide range of relaxation times avoiding the expensive implicit resolution of the collision operator. Finally, we show some numerical results which confirm the theoretical analysis. Introduction.The numerical solution of Boltzmann-type equations close to fluid regimes represents a real challenge for numerical methods. In these regimes, in fact, the intermolecular collision rate grows exponentially and the collisional time becomes very small. On the other hand, the actual time scale for evolution is the fluid dynamic time scale, which can be much larger than the collisional time. A nondimensional measure of the importance of collisions is given by the Knudsen number which is large in the rarefied regions and small in the fluid ones. Standard computational approaches lose their efficiency due to the necessity of using very small time steps in deterministic schemes or, equivalently, a large number of collisions in probabilistic approaches. Unfortunately the use of implicit solvers originates a prohibitive computational cost due to the high dimensionality and the nonlinearity of the collision integral.Several authors have tackled the above problem for the Boltzmann equation in the recent past (see [2,18,15,20,19] and the references therein). In summary, the possible approaches that permit us to overcome such a difficulty can be subdivided into two main classes: domain decomposition strategies and asymptotic preserving (AP) schemes. The first class of methods permits us to avoid the problem of very small Knudsen number by identifying the regions where it is possible to use the reduced fluid model and the regions where the full kinetic model must be solved. The literature in this direction has a long history and we recall here references [6, 33] and some recent works by Degond and coauthors [11,12]. A closely related research approach

We introduce a class of exponential Runge-Kutta integration methods for kinetic equations. The methods are based on a decomposition of the collision operator into an equilibrium and a non equilibrium part and are exact for relaxation operators of BGK type. For Boltzmann type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes. We give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving. Such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times. The methods also offer favorable properties such as nonnegativity of the solution and entropy inequality. For this reason, as we will show, the methods are suitable both for deterministic as well as probabilistic numerical techniques.

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This paper collects the efforts done in our previous works [8], [11], [10] to build a robust multiscale kinetic-fluid solver. Our scope is to efficiently solve fluid dynamic problems which present non equilibrium localized regions that can move, merge, appear or disappear in time.The main ingredients of the present work are the followings ones: a fluid model is solved in the whole domain together with a localized kinetic upscaling term that corrects the fluid model wherever it is necessary; this multiscale description of the flow is obtained by using a micromacro decomposition of the distribution function [10]; the dynamic transition between fluid and kinetic descriptions is obtained by using a time and space dependent transition function; to efficiently define the breakdown conditions of fluid models we propose a new criterion based on the distribution function itself. Several numerical examples are presented to validate the method and measure its computational efficiency.

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SUMMARYIn this work we propose a new approach for the numerical simulation of kinetic equations through Monte Carlo schemes. We introduce a new technique that permits to reduce the variance of particle methods through a matching with a set of suitable macroscopic moment equations. In order to guarantee that the moment equations provide the correct solutions, they are coupled to the kinetic equation through a nonequilibrium term. The basic idea, on which the method relies, consists in guiding the particle positions and velocities through moment equations so that the concurrent solution of the moment and kinetic models furnishes the same macroscopic quantities.

Kinetic equations play a major rule in modeling large systems of interacting particles. Recently the legacy of classical kinetic theory found novel applications in socio-economic and life sciences, where processes characterized by large groups of agents exhibit spontaneous emergence of social structures. Well-known examples are the formation of clusters in opinion dynamics, the appearance of inequalities in wealth distributions, flocking and milling behaviors in swarming models, synchronization phenomena in biological systems and lane formation in pedestrian traffic. The construction of kinetic models describing the above processes, however, has to face the difficulty of the lack of fundamental principles since physical forces are replaced by empirical social forces. These empirical forces are typically constructed with the aim to reproduce qualitatively the observed system behaviors, like the emergence of social structures, and are at best known in terms of statistical information of the modeling parameters. For this reason the presence of random inputs characterizing the parameters uncertainty should be considered as an essential feature in the modeling process. In this survey we introduce several examples of such kinetic models, that are mathematically described by nonlinear Vlasov and Fokker-Planck equations, and present different numerical approaches for uncertainty quantification which preserve the main features of the kinetic solution.