## Fachbereich Mathematik

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#### Schlagworte

- A Kinetic Model for Vehicular Traffic Derived from a Stochastic Microscopic Model (1995)
- 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.

- An adaptive domain decomposition procedure for Boltzmann and Euler equations (1998)
- 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.

- Particle Methods: Theory and Applications (1995)
- In the present paper a review on particle methods and their applications to evolution equations is given. In particular, particle methods for Euler- and Boltzmann equations are considered.

- Enskog-like kinetic models for vehicular traffic (1996)
- In the present paper a general criticism of kinetic equations for vehicular traffic is given. The necessity of introducing an Enskog-type correction into these equations is shown. An Enskog-line kinetic traffic flow equation is presented and fluid dynamic equations are derived. This derivation yields new coefficients for the standard fluid dynamic equations of vehicular traffic. Numerical simulations for inhomogeneous traffic flow situations are shown together with a comparison between kinetic and fluid dynamic models.

- Uniform Stability of a Finite Difference Scheme for Transport Equations in Diffusive Regimes (2000)
- An asymptotic preserving numerical scheme (with respect to diffusion scalings) for a linear transport equation is investigated. The scheme is adopted from a class of recently developped schemes. Stability is proven uniformly in the mean free path under a CFL type condition turning into a parabolic CFL condition in the diffusion limit.

- Transition from Kinetic Theory to Macroscopic Fluid Equations: A Problem fo Domain Decomposition and a Source for New Algorithms (1999)
- In the paper we discuss the transition from kinetic theory to macroscopic fluid equations, where the macroscopic equations are defined as aymptotic limits of a kinetic equation. This relation can be used to derive computationally efficient domain decomposition schemes for the simulaion of rarefied gas flows close to the continuum limit. Moreover, we present some basic ideas for the derivation of kinetic induced numerical schemes for macroscopic equations, namely kinetic schemes for general conservation laws as well as Lattice-Boltzmann methods for the incompressible Navier-Stokes equations.

- Mathematical Models for Vehicular Traffic (1995)
- 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.

- Convergence of Alternating Domain Decomposition Schemes for Kinetic and Aerodynamic Equations (1994)
- 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.

- Domain Decomposition for Kinetic Problems with Nonequilibrium States (1994)
- 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.

- Computation of Nonlinear Functionals in Particle Methods (1994)
- 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.