Numerics of the inelastic Boltzmann equation
Seminar Room 1, Newton Institute
AbstractIn the present lecture we give an overview of the analytical and numerical properties of the spatially homogeneous granular Boltzmann equation. The presentation is based on the three articles. In the first article , we consider the uniformly heated spatially homogeneous granular Boltzmann equation. A new stochastic numerical algorithm for this problem is presented and tested using analytical relaxation of the temperature. The tails of the steady state distribution, which are overpopulated for the steady state solutions of the granular Boltzmann equation, are computed using this algorithm and the results are compared with the available analytical information. In the second paper , we deal again with the same equation and consider the DSMC-error due to splitting technique using the time step Delta_t. This equation provides the possibility to see the Delta_t error of the first order when using standard splitting technique and of the second order for Strang?s splitting method. In contrast, a new direct simulation method recently introduced in  is "exact", i.e. the error is a function of the number of particles n and of the number of independent ensembles N_rep. Thus this method can be seen as a "correct" generalization of the classical DSMC for the inelastic Boltzmann equation. Finally, in , we consider an inelastic gas in a host medium. The numerical algorithm is tested by the use of the analytical relaxation of the momentum and of the temperature.
 I. M. Gamba, S. Rjasanow, and W. Wagner. Direct simulation of the uniformly heated granular Boltzmann equation. Math. Comput. Modelling , 42(5-6):683?700, 2005.
 S. Rjasanow and W. Wagner. Time splitting error in DSMC schemes for the inelastic Boltzmann equation. SIAM J. Numerical Anal., 45(1):54? 67, 2007.
 M. Bisi, S. Rjasanow and G. Spiga. A numerical study of the granular gas in a host medium. Preprint NI10051 of the Isaac Newton Institute for Mathematical Sciences, 2010.