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Thu, 06 Jul 2000 00:00:00 +0200Thu, 06 Jul 2000 00:00:00 +0200Nonstandard Hydrodynamics for the Boltzmann Equation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/552
The Boltzmann equation solutions are considered for the small Knudsen number. The main attention is devoted to certain deviations from the classical Navier-Stokes description. The equations for the quasistationary slow flows are derived. These equations do not contain the Knudsen number and provide in this sense a limiting description of hydrodynamical variables. Two well-known special cases are also indicated. In the isothermal case the equations are equivalent to the incompressible Navier-Stokes equations, in stationary case they coincide with the equations of slow non-isothermal flows. It is shown that the derived equations possess all principal properties of the Boltzmann equation on contrast to the Burnett equations. In one dimension the equations reduce to the nonlinear diffusion equations, being exactly solvable for Maxwell molecules. Multidimensional stationary heat-transfer problems are also discussed. It is shown that one can expect an essential difference between the Boltzmann equaiton solution in the limit of the continuous media and the corresponding solution of the Navier-Stokes equations.A.V. Bobylevpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/552Wed, 07 Jun 2000 00:00:00 +0200Quasistationary Solutions of the Boltzmann Equation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/554
Equations of quasistationary hydrodynamics are derived from the Boltzmann equation by using the modified Hilbert approach. The physical and mathematical meaning of quasistationary solutions are discussed in detail.A.V. Bobylevpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/554Wed, 07 Jun 2000 00:00:00 +0200Partikelapproximation von Wahrscheinlichkeitsmaßen mit minimalem Lipschitzabstand
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/542
J. Mohringpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/542Tue, 06 Jun 2000 00:00:00 +0200Diffusion Approximation and Hyperbolic Automorphisms of the Torus
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/543
In this article a diffusion equation is obtained as a limit of a reversible kinetic equation with an ad hoc scaling. The diffusion is produced by the collisions of the particles with the boundary. These particles are assumed to be reflected according to a reversible law having convenient mixing properties. Optimal convergence results are obtained in a very simple manner. This is made possible because the model, based on Arnold" s cat map can be handled with Fourier series instead of the symbolic dynamics associated to a Markow partition.C. Bardos; F. Golse; J.-F. Colonnapreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/543Tue, 06 Jun 2000 00:00:00 +0200On the Computation of Stress in Stationary Loaded Journal Bearings
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/540
In this paper we deal with the problem of computing the stresses in stationary loaded bearings. A method to obtain the pressure in the lubrication fluid, which is given as a solution of Reynolds" differential equation, is presented. Furthermore, using the theory of plain stress, the stresses in the bearing shell are described by derivatives of biharmonic functions. A spline interpolation method for computing these functions is developed and an estimate for the error on the boundaries is presented. Finally the described methods are tested theoretically as well as with real examples.T. Grünholzpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/540Wed, 31 May 2000 00:00:00 +0200Factorization Theory for Stable, Discrete-Time Inner Functions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/530
We develop a factorization theory for stable inner functions relative to the unit circle.Paul A. Fuhrmann; Jörg Hoffmannpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/530Mon, 03 Apr 2000 00:00:00 +0200On Balanced Realizations of Bounded Real and Positive Real Functions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/531
Based on normalized coprime factorizations with respect to indefinite metrics and the construction of suitable characteristic functions, the Ober balanced canonical forms for the classes of bounded real and positive real are derived. This uses a matrix representation of the shift realization with respect to a basis related to sets of orthogonal polynomials.Paul A. Fuhrmann; Jörg Hoffmannpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/531Mon, 03 Apr 2000 00:00:00 +0200Lifetime Estimation in the Car Industry
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/533
Whenever new parts of a car have been developed, the manufacturer needs an estimation of the lifetime of this new part. On one hand the construction must not be too weak, so that the part holds long enough to satisfy the customer, but on the other hand, if the construction is too excessive, the part gets too heavy.; One is interested in methods that only need few measured data from the specimen itself, but use data about the material, because constructing and testing of specimen is expensive.Michael Hackpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/533Mon, 03 Apr 2000 00:00:00 +0200Generation of Random Variates Using Asymptotic Expansions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/535
Monte-Carlo methods are widely used numerical tools in various fields of application, like rarefied gas dynamics, vacuum technology, stellar dynamics or nuclear physics. A central part in all applications is the generation of random variates according to a given probability law. Fundamental techniques to generate non-uniform random variates are the inversion principle or the acceptance-rejection method. Both procedures can be quite time-consuming if the given probability law has a complicated structure.; In this paper we consider probability laws depending on a small parameter and investigate the use of asmptotic expansions to generate random variates. The results given in the paper are restrictedto first order expansions. We show error estimates for the discrepancy as well as for the bounded Lipschitz distance of the asymptotic expansion. Furthermore the integration error for some special classes of functions is given. The efficiency of the method is proved by a numerical example from rarefied gas flows.Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/535Mon, 03 Apr 2000 00:00:00 +0200Tutorial on Asymptotic Analysis I
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/536
This text summarizes parts of the exercises of the tutorialon 'Asymptotic Analysis' held in the winter term 1993/94 atthe University of Kaiserslautern.Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/536Mon, 03 Apr 2000 00:00:00 +0200Scale-Space Properties of Nonlinear Diffusion Filtering with a Diffusion Tensor
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/539
In spite of its lack of theoretical justification, nonlinear diffusion filtering has become a powerful image enhancement tool in the recent years. The goal of the present paper is to provide a mathematical foundation for nonlinear diffusion filtering as a scale-space transformation which is flexible enough to simplify images without loosing the capability of enhancing edges. By stuying the Lyapunow functional, it is shown that nonlinear diffusion reduces Lp norms and central moments and increases the entropy of images. The proposed anisotropic class utilizes a diffusion tensor which may be adapted to the image structure. It permits existence, uniqueness and regularity results, the solution depends continuously on the initial image, and it fulfills an extremum principle. All considerations include linear and certain nonlinear isotropic models and apply to m-dimensional vector-valued images. The results are juxtaposed to linear and morphological scale-spaces.Joachim Weickertpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/539Mon, 03 Apr 2000 00:00:00 +0200Boltzmann Simulation by Particle Methods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/541
Particle methods to simulate rarefied gas flows have found an increasing interest in Computational Fluid Dynamics during the last decade, see for example [1], [2], [3] and [4]. The general goal is to develop numerical schemes which are reliable enough to substitute real windtunnel experiments, needed for example in space research, by computer experiments. In order to achieve this goal one needs numerical methods solving the Boltzmann equation including all important physical effects. In general this means 3D computations for a chemically reacting rarefied gas. With codes of this kind at hand, Boltzmann simulation becomes a powerful tool in studying rarefied gas phenomena.Helmut Neunzert; Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/541Mon, 03 Apr 2000 00:00:00 +0200Convergence of Alternating Domain Decomposition Schemes for Kinetic and Aerodynamic Equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/546
A domain decomposition scheme linking linearized kinetic and aerodynamic equations is considered. Convergence of the alternating scheme is shown. Moreover the physical correctness of the obtained coupled solutions is discussed.Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/546Mon, 03 Apr 2000 00:00:00 +0200A Numerical Method for Computing Asymptotic States and Outgoing Distributions for Kinetic Linear Half-Space Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/547
Linear half-space problems can be used to solve domain decomposition problems between Boltzmann and aerodynamic equations. A new fast numerical method computing the asymptotic states and outgoing distributions for a linearized BGK half-space problem is presented. Relations with the so-called variational methods are discussed. In particular, we stress the connection between these methods and Chapman-Enskog type expansions.F. Golse; Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/547Mon, 03 Apr 2000 00:00:00 +0200Domain Decomposition for Kinetic Problems with Nonequilibrium States
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/548
A nonequilibrium situation governed by kinetic equations with strongly contrasted Knudsen numbers in different subdomains is discussed. We consider a domain decomposition problem for Boltzmann- and Euler equations, establish the correct coupling conditions and prove the validity of the obtained coupled solution. Moreover numerical examples comparing different types of coupling conditions are presented.Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/548Mon, 03 Apr 2000 00:00:00 +0200On the Connection of the Formulae for Entropy and Stationary Distribution
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/550
As it is well known in statistical physics the stationary distribution can be obtained by maximizing entropy. We show how one can reconstruct the formula for entropy knowing the formula for the stationary distribution. A general case is discussed and some concrete physical examples are considered.Y. Arkhipov; Axel Klar; V. Vedenyapinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/550Mon, 03 Apr 2000 00:00:00 +0200Computation of Nonlinear Functionals in Particle Methods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/551
We consider the numerical computation of nonlinear functionals of distribution functions approximated by point measures. Two methods are described and estimates for the speed of convergence as the number of points tends to infinity are given. Moreover numerical results for the entropy functional are presented.Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/551Mon, 03 Apr 2000 00:00:00 +0200Implicit and Iterative Methods for the Boltzmann Equation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/553
The paper presents some approximation methods for the Boltzmann equation. In the first part fully implicit discretization techniques for the spatially homogeneous Boltzmann equation are investigated. The implicit equation is solved using an iteration process. It is shown that the iteration converges to the correct solution for the moments of the distribution function as long as the mass conservation is strictly fulfilled. For a simple model Boltzmann equation some unexpected features of the implicit scheme and the corresponding iteration process are clarified. In the second part a new iteration algorithm is proposed which should be used for the stationary Boltzmann equation. The realization of the method is very similar to the standard splitting algorithms except some new stochastic elements.A.V. Bobylev; Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/553Mon, 03 Apr 2000 00:00:00 +0200A Nonlinear Ray Theory
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/529
A proof of the famous Huygens" method of wavefront construction is reviewed and it is shown that the method is embedded in the geometrical optics theory for the calculation of the intensity of the wave based on high frequency approximation. It is then shown that Huygens" method can be extended in a natural way to the construction of a weakly nonlinear wavefront. This is an elegant nonlinear ray theory based on an approximation published by the author in 1975 which was inspired by the work of Gubkin. In this theory, the wave amplitude correction is incorporated in the eikonal equation itself and this leads to a sytem of ray equations coupled to the transport equation. The theory shows that the nonlinear rays stretch due to the wave amplitude, as in the work of Choquet-Bruhat (1969), followed by Hunter, Majda, Keller and Rosales, but in addition the wavefront rotates due to a non-uniform distribution of the amplitude on the wavefront. Thus the amplitude of the wave modifies the rays and the wavefront geometry, which in turn affects the growth and decay of the amplitude. Our theory also shows that a compression nonlinear wavefront may develop a kink but an expansion one always remains smooth. In the end, an exact solution showing the resolution of a linear caustic due to nonlinearity has been presented. The theory incorporates all features of Whitham" s geometrical shock dynamics.P. Prasadpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/529Sat, 01 Jan 1994 00:00:00 +0100Image Processing Using a Wavelet Algorithm for Nonlinear Diffusion
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/532
The edge enhancement property of a nonlinear diffusion equation with a suitable expression for the diffusivity is an important feature for image processing. We present an algorithm to solve this equation in a wavelet basis and discuss its one dimensional version in some detail. Sample calculations demonstrate principle effects and treat in particular the case of highly noise perturbed signals. The results are discussed with respect to performance, efficiency, choice of parameters and are illustrated by a large number of figures. Finally, a comparison with a Fourier method and a finite volume method is performed.Joachim Weickert; Jochen Fröhlichpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/532Sat, 01 Jan 1994 00:00:00 +0100Particle Methods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/544
Helmut Neunzertpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/544Sat, 01 Jan 1994 00:00:00 +0100Von der neuen Rolle der Mathematik; Vom Nutzen der Mathematik; Mathematik und Computersimulation: Modelle, Algorithmen, Bilder
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/545
Helmut Neunzertpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/545Sat, 01 Jan 1994 00:00:00 +0100