## Fachbereich Mathematik

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#### Year of publication

- 1994 (34) (remove)

#### Keywords

- Boltzmann Equation (2)
- Numerical Simulation (2)
- Domain Decomposition (1)
- Particle Methods (1)
- Perona-Malik filter (1)
- Random number generation (1)
- Rarefied Gas Dynamics (1)
- anisotropic diffusion (1)
- asymptotic expansions (1)
- finite volume methods (1)
- image processing (1)
- inversion method (1)
- pseudospectral methods (1)
- scale-space (1)
- wavelets (1)
- well-posedness (1)

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.

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 Nonlinear Ray Theory
(1994)

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.

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.

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.

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.

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.