## Fachbereich Mathematik

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#### Keywords

- asymptotic analysis (2)
- kinetic equations (2)
- low Mach number limit (2)
- CFL type conditions (1)
- Chorin's projection scheme (1)
- Discrete velocity models (1)
- Evolution Equations (1)
- Hybrid Codes (1)
- Particle Methods (1)
- Smoothed Particle Hydrodynamics (1)
- adaptive grid generation (1)
- asymptotic preserving numerical scheme (1)
- domain decomposition (1)
- drift-diffusion limit (1)
- fixpoint theorem (1)
- fluid dynamic equations (1)
- incompressible Euler equation (1)
- incompressible Navier-Stokes equations (1)
- kinetic models (1)
- kinetic semiconductor equations (1)
- lattice Boltzmann method (1)
- linear transport equation (1)
- numerical methods for stiff equations (1)
- particle methods (1)
- second order upwind discretization (1)
- slope limiter (1)
- stability uniformly in the mean free path (1)
- stationary solutions (1)
- vehicular traffic (1)

A way to derive consistently kinetic models for vehicular traffic from microscopic follow the leader models is presented. The obtained class of kinetic equations is investigated. Explicit examples for kinetic models are developed with a particular emphasis on obtaining models, that give realistic results. For space homogeneous traffic flow situations numerical examples are given including stationary distributions and fundamental diagrams.

In this paper the kinetic model for vehicular traffic developed in [3,4] is considered and theoretical results for the space homogeneous kinetic equation are presented. Existence and uniqueness results for the time dependent equation are stated. An investigation of the stationary equation leads to a boundary value problem for an ordinary differential equation. Existence of the solution and some properties are proved. A numerical investigation of the stationary equation is included.

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.

An asymptotic-induced scheme for kinetic semiconductor equations with the diffusion scaling is developed. The scheme is based on the asymptotic analysis of the kinetic semiconductor equation. It works uniformly for all ranges of mean free paths. The velocity discretization is done using quadrature points equivalent to a moment expansion method. Numerical results for different physical situations are presented.

In this paper we present a domain decomposition approach for the coupling of Boltzmann and Euler equations. Particle methods are used for both equations. This leads to a simple implementation of the coupling procedure and to natural interface conditions between the two domains. Adaptive time and space discretizations and a direct coupling procedure leads to considerable gains in CPU time compared to a solution of the full Boltzmann equation. Several test cases involving a large range of Knudsen numbers are numerically investigated.

An asymptotic-induced scheme for nonstationary transport equations with thediffusion scaling is developed. The scheme works uniformly for all ranges ofmean free paths. It is based on the asymptotic analysis of the diffusion limit ofthe transport equation. A theoretical investigation of the behaviour of thescheme in the diffusion limit is given and an approximation property is proven.Moreover, numerical results for different physical situations are shown and atheuniform convergence of the scheme is established numerically.

This paper deals with domain decomposition methods for kinetic and drift diffusion semiconductor equations. In particular accurate coupling conditions at the interface between the kinetic and drift diffusion domain are given. The cases of slight and strong nonequilibrium situations at the interface are considered and some numerical examples are shown.

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.