7,925 research outputs found

    Convergence of the stochastic weighted particle method for the Boltzmann equation

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    This paper studies convergence of the stochastic weighted particle method for the Boltzmann equation. First the method is extended by introducing new stochastic reduction procedures, in order to control the number of simulation particles. Then, under rather general conditions, convergence to the solution of the Boltzmann equation is proved. Finally, numerical experiments are performed illustrating both convergence and considerable variance reduction, for the specific problem of calculating tails of the velocity distribution

    Stochastic weighted particle method -- Theory and numerical examples

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    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 direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization

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    A deterministic method is proposed for solving the Boltzmann equation. The method employs a Galerkin discretization of the velocity space and adopts, as trial and test functions, the collocation basis functions based on weights and roots of a Gauss-Hermite quadrature. This is defined by means of half- and/or full-range Hermite polynomials depending whether or not the distribution function presents a discontinuity in the velocity space. The resulting semi-discrete Boltzmann equation is in the form of a system of hyperbolic partial differential equations whose solution can be obtained by standard numerical approaches. The spectral rate of convergence of the results in the velocity space is shown by solving the spatially uniform homogeneous relaxation to equilibrium of Maxwell molecules. As an application, the two-dimensional cavity flow of a gas composed by hard-sphere molecules is studied for different Knudsen and Mach numbers. Although computationally demanding, the proposed method turns out to be an effective tool for studying low-speed slightly rarefied gas flows

    A new approach to quantitative propagation of chaos for drift, diffusion and jump processes

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    This paper is devoted the the study of the mean field limit for many-particle systems undergoing jump, drift or diffusion processes, as well as combinations of them. The main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles, as the number of particles goes to infinity. To this end we introduce a general functional framework which reduces this question to the one of proving a purely functional estimate on some abstract generator operators (consistency estimate) together with fine stability estimates on the flow of the limiting nonlinear equation (stability estimates). Then we apply this method to a Boltzmann collision jump process (for Maxwell molecules), to a McKean-Vlasov drift-diffusion process and to an inelastic Boltzmann collision jump process with (stochastic) thermal bath. To our knowledge, our approach yields the first such quantitative results for a combination of jump and diffusion processes.Comment: v2 (55 pages): many improvements on the presentation, v3: correction of a few typos, to appear In Probability Theory and Related Field

    Binary interaction algorithms for the simulation of flocking and swarming dynamics

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    Microscopic models of flocking and swarming takes in account large numbers of interacting individ- uals. Numerical resolution of large flocks implies huge computational costs. Typically for NN interacting individuals we have a cost of O(N2)O(N^2). We tackle the problem numerically by considering approximated binary interaction dynamics described by kinetic equations and simulating such equations by suitable stochastic methods. This approach permits to compute approximate solutions as functions of a small scaling parameter ε\varepsilon at a reduced complexity of O(N) operations. Several numerical results show the efficiency of the algorithms proposed
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