Simulation of Rare Events by the Stochastic Weighted Particle Method for the Boltzmann Equation

An extension of the stochastic weighted particle method for the numerical treatment of the Boltzmann equation is presented. A new procedure for modelling the inflow boundary condition is introduced and its performance is tested in a two-dimensional example with strong density gradients. A gain factor in computing time of several orders of magnitude is achieved in specific situations

Stochastic weighted particle method -- Theory and numerical examples

In the present paper we give a theoretical background of the Stochastic Weighted Particle Method (SWPM) for the classical Boltzmann equation. This numerical method was developed for problems with big deviation in magnitude of values of interest. We describe the corresponding algorithms, give a brief summary of the convergence theory and illustrate the new possibilities by numerical tests

A temperature time counter scheme for the Boltzmann equation

This paper gives a rigorous derivation of a new stochastic particle method for the Boltzmann equation and illustrates its numerical efficiency. Using estimates based on the local temperature of the simulation cells, any truncation error related to large velocities is avoided. Moreover, time steps between collisions are larger than in the standard direct simulation method. This fact and an efficient modelling procedure for the index distribution of the collision partners lead to a considerable reduction of computational effort in certain applications

Time splitting error in DSMC schemes for the inelastic Boltzmann equation

The paper is concerned with the numerical treatment of the uniformly heated inelastic Boltzmann equation by the direct simulation Monte Carlo (DSMC) method. This technique is presently the most widely used numerical method in kinetic theory. We consider three modifications of the DSMC method and study them with respect to their efficiency and convergence properties. Convergence is investigated both with respect to the number of particles and to the time step. The main issue of interest is the time step discretization error due to various splitting strategies. A scheme based on the Strang-splitting strategy is shown to be of second order with respect to time step, while there is only first order for the commonly used Euler-splitting scheme. On the other hand, a no-splitting scheme based on appropriate Markov jump processes does not produce any time step error. It is established in numerical examples that the no-splitting scheme is about two orders of magnitude more efficient than the Euler-splitting scheme. The Strang-splitting scheme reaches almost the same level of efficiency compared to the no-splitting scheme, since the deterministic time step error vanishes sufficiently fast

On time counting procedures in the DSMC method for rarefied gases

The DSMC (direct simulation Monte Carlo) method for rarefied gas dynamics is studied. The behaviour of the underlying stochastic particle system is determined by three main components - the time step between subsequent collisions, the random mechanism for the choice of the collision partners, and the random mechanism for calculating the result of the collision. The purpose of this paper is to illustrate the interplay between these various components and to propose some new modifications of the DSMC method. Different time counting procedures are derived and their influence on the other parts of the algorithm is investigated. Various modifications of the DSMC method are compared with respect to different criteria such as efficiency, systematic error, and statistical fluctuations

On estimates of the Boltzmann collision operator with cutoff

We present new estimates of the Boltzmann collision operator in weighted Lebesgue and Bessel potential spaces. The main focus is put on hard potentials under the assumption that the angular part of the collision kernel fulfills some weighted integrability condition. In addition, the proofs for some previously known \mathbb{L}_{p^{-}} estimates have been considerably shortened and carried out by elementary methods. For a class of metric spaces, the collision integral is seen to be a continous operator into the same space. Furthermore, we give a new pointwise lower bound as well as asymptotic estimates for the loss term without requiring that the entropy is finite

Reduction of the number of particles in the stochastic weighted particle method for the Boltzman equation

Different ideas for reducing the number of particles in the stochastic weighted particle method for the Boltzmann equation are described and discussed. The corresponding error bounds are obtained and numerical tests for the spatially homogeneous Boltzmann equation presented. It is shown that if an appropriate reduction procedure is used then any effect on the accuracy of the numerical scheme is negligible

Direct simulation of the uniformly heated granular Boltzmann equation

In the present paper we give an overview of the analytical properties of the steady state solution of the spatially homogeneous uniformly heated granular Boltzmann equation. The asymptotic properties of this distribution (so called tails) are formulated for different models of interaction. 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 are computed using this algorithm and the results are compared with the available analytical information