### Numerical approximation of kinetic equations for granular gases

 Clusters formation
A granular gas is a conglomeration of discrete solid, particles characterized by a loss of energy whenever the particles interact. Unlike conventional gases granular materials will tend to cluster and clump due to the dissipative nature of the collisions between grains which may be expressed by a decay of its granular temperature. Applications include the handling and storage of cereals, granular chemicals, sand, coal, pharmaceuticals, and more.
Planetary rings and interstellar dust are examples of celestial granular systems, and many pollutants qualify as granular systems of environmental importance [6]. Some catastrophic events, such as snow avalanches, rock and land slides, or the collapse of silos are clearly instances of granular dynamics.
In the sequel we report some images and a simple animation for the homogeneous cooling process of a one-dimensional granular gas computed by spectral methods [1,2]. The different figures refers to different values of the quasi-elastic limit parameter n. On the x-axis we have the time t, the y-axis refers to the distribution function f(v,t). For n=1 we have the Boltzmann equation for a granular gas whereas as n goes to infinity we obtain a friction equation. The initial data is the sum of two one-dimensional gaussian and n is in the range 1-100. The change of the cooling process appears evident. Multidimensional computations can be found in [4], we refer also to [5] for a general introduction to kinetic theory of granular gases.

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References
1. L.Pareschi, On the fast evaluation of kinetic equations for driven granular media, Numerical Analysis and Advanced Applications - Proceedings of ENUMATH 2001, the 4th conference on numerical mathematics and advanced applications, Ischia, July 2001, Springer-Italia, (2003).
2. G.Naldi, L.Pareschi, G.Toscani, Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit, Mathematical Models and Numerical Analysis, 37, (2003), pp. 73-90.
3. L.Pareschi, G.Toscani, Modelling and numerical methods for granular gases, Chapter 9, Modeling and computational methods for kinetic equations, Series: Modeling and Simulation in Science, Engineering and Technology, Birkhauser (2004), pp.259-286.
4. F.Filbet, L.Pareschi, G.Toscani, Accurate numerical methods for the collisional motion of (heated) granular flows, J. Comp. Phys., 202, (2005), pp.216-235.
5. Lorenzo Pareschi, Giuseppe Toscani, Giovanni Russo, Modeling and numerics of kinetic dissipative systems, Nova Science Publishers, New York (2006 ).
6. Ferrari, E. ; Pareschi, L. Modelling and numerical methods for the dynamics of impurities in a gas. Internat. J. Numer. Methods Fluids 57 (2008), no. 6, 693--713.