The Finite Pointset Method for hypersonic flows in the rarefied gas regime

Neunzert, Helmut; Struckmeier, Jens

doi:10.1007/978-1-4612-0375-9_7

Cited by 2 publications

(2 citation statements)

References 14 publications

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“…The technique is very similar to particle methods for Euler equations which can be found for example in [3, 41. Concerning the general concept of particle methods we refer to [6] where the reader may find a much more elaborated particle method for the full Boltzmann equation. The general idea of a particle method for a nonlinear transport equation is the following: The kinetic distribution function G(t, .x, 5 ) is considered as the density of a continuous measure pt on the phase space R x R which varies with time t according to the given transport equation.…”

Section: A Particle Methods For Shallow Water Wavesmentioning

confidence: 99%

On a kinetic model for shallow water waves

Struckmeier¹

1995

Math Methods in App Sciences

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The system of shallow water waves is one of the classical examples for non-linear, two-dimensional conservation laws. The paper investigates a simple kinetic equation depending on a parameter E which leads for E -+ 0 to the system of shallow water waves. The corresponding 'equilibrium' distribution function has a compact support which depends on the eigenvalues of the hyperbolic system. It is shown that this kind of kinetic approach is restricted to a special class of non-linear conservation laws. The kinetic model is used to develop a simple particle method for the numerical solution of shallow water waves. The particle method can be implemented in a straightforward way and produces in test examples sufficiently accurate results.

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Section: A Particle Methods For Shallow Water Wavesmentioning

confidence: 99%

On a kinetic model for shallow water waves

Struckmeier¹

1995

Math Methods in App Sciences

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“…Low discrepancy sequences were introduced instead of sequences of random numbers in some parts of the Nanbu-Babovsky procedure, later called finite pointset method (cf. [152], [150]). Further studies concerning low discrepancy sequences in the context of the Boltzmann equation were performed in [118], [119], [120].…”

Section: Further Referencesmentioning

confidence: 99%

Stochastic Numerics for the Boltzmann Equation

Rjasanow¹,

Wagner²

2005

Springer Series in Computational Mathematics

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The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.Cover design: design&production, Heidelberg Typeset by the authors using a Springer L A T E X macro package Printed on acid-free paper 46/3142sz-5 4 3 2 1 0 PrefaceStochastic numerical methods play an important role in large scale computations in the applied sciences. Such algorithms are convenient, since inherent stochastic components of complex phenomena can easily be incorporated. However, even if the real phenomenon is described by a deterministic equation, the high dimensionality often makes deterministic numerical methods intractable. A stochastic procedure, called direct simulation Monte Carlo (DSMC) method, has been developed in the physics and engineering community since the sixties. This method turned out to be a powerful tool for numerical studies of complex rarefied gas flows. It was successfully applied to problems ranging from aerospace engineering to material processing and nanotechnology. In many situations, DSMC can be considered as a stochastic algorithm for solving some macroscopic kinetic equation. An important example is the classical Boltzmann equation, which describes the time evolution of large systems of gas molecules in the rarefied regime, when the mean free path (distance between subsequent collisions of molecules) is not negligible compared to the characteristic length scale of the problem. This means that either the mean free path is big (space-shuttle design, vacuum technology), or the characteristic length is small (micro-device engineering). As the dimensionality of this nonlinear integro-differential equation is high (time, position, velocity), its numerical treatment is a typical application field of Monte Carlo algorithms.Intensive mathematical research on stochastic algorithms for the Boltzmann equation started in the eighties, when techniques for studying the convergence of interacting particle systems became available. Since that time much progress has been made in the justification and further development of these numerical methods.The purpose of this book is twofold. The first goal is to give a mathematical description of various classical DSMC procedures, using the theory of Markov processes (in particular, stochastic interacting particle systems) as a unifying framework. The second goal is a systematic treatment of an extension of DSMC, called stochastic weighted particle method (SWPM). This VI Preface method has been developed by the authors during the last decade. SWPM includes several new features, which are introduced for the purpose of variance reduction (rare event simulation). Rigorous results concerning the approximation of solutions to the Boltzmann equation by particle systems are given, covering both DSMC and SWPM. Thorough numerical experiments are performed, illustrating the behavior of systematic...

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The Finite Pointset Method for hypersonic flows in the rarefied gas regime

Cited by 2 publications

References 14 publications

On a kinetic model for shallow water waves

On a kinetic model for shallow water waves

Stochastic Numerics for the Boltzmann Equation

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