Journal ArticleDOI
A convergence proof for Nanbu's simulation method for the full Boltzmann equation
Hans Babovsky,Reinhard Illner +1 more
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In this paper, it was shown that the discrete measures given by the Nanbu simulation method converge with respect to the weak topology of measures to solutions of the Boltzmann equation.Abstract:
It is shown that the discrete measures given by the Nanbu simulation method converge with respect to the weak topology of measures to solutions of the Boltzmann equation. The main conditions for this result are that the Cauchy problem for the Boltzmann equation has a sufficiently smooth solution and that the discretization parameters (cell size, timestep and test particle number) satisfy suitable constraints.read more
Citations
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Journal ArticleDOI
A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation
TL;DR: In this article, the limit of the random empirical measures associated with the Bird algorithm is shown to be a deterministic measure-valued function satisfying an equation close (in a certain sense) to the Boltzmann equation.
Journal ArticleDOI
Direct Simulation Monte Carlo: Recent Advances and Applications
TL;DR: In this article, the principles of and procedures for implementing direct simulation Monte Carlo (DSMC) are described and guidelines to inherent and external errors common in DSMC applications are provided.
Book ChapterDOI
Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models
Sylvie Méléard,Sylvie Méléard +1 more
Journal ArticleDOI
Probability theory of electron-molecule, ion-molecule, molecule-molecule, and Coulomb collisions for particle modeling of materials processing plasmas and cases
TL;DR: In this paper, the kinetic theory basis of the particle simulation method is first described and state-of-the-art probabilistic treatments of collisions are described for electron-molecule, ion-molescule, molecule-molcule, and Coulomb collisions.
Journal ArticleDOI
Stochastic particle approximations for generalized Boltzmann models and convergence estimates
Carl Graham,Sylvie Méléard +1 more
TL;DR: In this paper, the Markov process corresponding to a generalized mollified Boltzmann equation with general motion between collisions and nonlinear bounded jump (collision) operator is given, and the nonlinear martingale problem is solved.
References
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Book
Convergence of Probability Measures
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Journal ArticleDOI
The Boltzmann equation. I. Uniqueness and local existence
Shmuel Kaniel,Marvin Shinbrot +1 more
TL;DR: In this paper, an abstract form of the spatially nonhomogeneous Boltzmann equation is derived which includes the usual, more concrete form for any kind of potential, hard or soft, with finite cutoff.
Journal ArticleDOI
The Boltzmann equation: global existence for a rare gas in an infinite vacuum
Reinhard Illner,Marvin Shinbrot +1 more
TL;DR: In this paper, it was shown that the Boltzmann equation can be solved globally in time under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough.
Journal ArticleDOI
On the Cauchy Problem of the Boltzmann Equation with a Soft Potential
Seiji Ukai,Kiyoshi Asano +1 more
TL;DR: In this article, the probability density of a gas in the space K is defined in terms of the probability that a gas particle will collide with another gas particle at time t in a given position x and velocity f. The operator Q describes binary collisions of gas particles and the function q is determined by the corresponding interaction potential.