We consider implicit-explicit (IMEX) Runge Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strong-stabilitypreserving (SSP) scheme, and the implicit part is treated by an L-stable diagonally implicit Runge Kutta (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by finite difference discretization with Weighted Essentially Non Oscillatory (WENO) reconstruction. After a brief description of the mathematical properties of the schemes, several applications will be presented.

In this paper, we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading. We show that the kinetic model admits non trivial quasi-stationary states with power law tails of Pareto type. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution of wealth among individuals. For this equation the stationary state can be easily derived and shows a Pareto power law tail. Numerical results confirm the previous analysis.

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

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In this paper we show that the use of spectral Galerkin methods for the approximation of the Boltzmann equation in the velocity space permits one to obtain spectrally accurate numerical solutions at a reduced computational cost. We prove that the spectral algorithm preserves the total mass and approximates with infinite-order accuracy momentum and energy. Consistency of the method is also proved, and a stability result for a smoothed positive scheme is given. We demonstrate that the Fourier coefficients associated with the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly. Numerical examples for homogeneous test problems in two and three dimensions confirm the advantages of the method.AMS subject classifications. 65L60, 65R20, 76P05, 82C40 PII. S0036142998343300 1. Introduction. This paper is the first of two papers devoted to the development of numerical schemes for the accurate computation of the Boltzmann equation and related kinetic models. Here we will mainly concentrate on the approximation of the collision operator and hence of the velocity space. In a companion paper we will consider the problem of space and time discretizations showing how to develop under resolved numerical schemes that work with uniform accuracy with respect to the Knudsen number [34].The numerical solution of the Boltzmann equation represents a real challenge for numerical methods. This is essentially due to the nonlinearity, to the large number of variables (t for the time, two three-dimensional vectors x for space, and v for velocity), and to the five-fold integral that defines the collision operator. Furthermore, this integration has to be handled carefully since it is at the basis of the macroscopic properties of the Boltzmann equation.For example, if we denote by n the number of parameters which characterize the density with respect to the velocity variables, the computational cost of a conventional deterministic method for the evaluation of the collisional integral is much larger than n 2 . Moreover, we have to multiply this cost times the number of points in the physical space.As a consequence most numerical computations are based on probabilistic Monte Carlo techniques at different levels. Examples are the direct simulation Monte Carlo method (DSMC) by Bird [3] and the modified Monte Carlo method by Nanbu [30]. For a detailed description of such methods we refer to [3,12,24,37].Probabilistic methods present different advantages. First, the computational cost is strongly reduced and can be considered approximately of the order of the number

We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such regime the system relaxes towards a convection-diffusion equation. The first objective of the paper is to show that traditional partitioned IMEX R-K schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system. Of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step. The main goal of the paper is to present an approach, based on IMEX R-K schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the convection-diffusion equation, in which the diffusion is treated implicitly. This is achieved by an original reformulation of the problem, and subsequent application of IMEX R-K schemes to it. An analysis on such schemes to the reformulated problem shows that the schemes reduce to IMEX R-K schemes for the limit equation, under the same conditions derived for hyperbolic relaxation [8]. Several numerical examples including neutron transport equations confirm the theoretical analysis.

Abstract. The development of accurate and fast numerical schemes for the five-fold Boltzmann collision integral represents a challenging problem in scientific computing. For a particular class of interactions, including the so-called hard spheres model in dimension three, we are able to derive spectral methods that can be evaluated through fast algorithms. These algorithms are based on a suitable representation and approximation of the collision operator. Explicit expressions for the errors in the schemes are given and spectral accuracy is proved. Parallelization properties and adaptivity of the algorithms are also discussed.

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