## 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)

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.

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.

This survey contains a description of different types of mathematical models used for the simulation of vehicular traffic. It includes models based on ordinary differential equations, fluid dynamic equations and on equations of kinetic type. Connections between the different types of models are mentioned. Particular emphasis is put on kinetic models and on simulation methods for these models.

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.

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.

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.

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.