## Fachbereich Mathematik

- 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.

- Discretizations for the Incompressible Navier-Stokes Equations based on the Lattice Boltzmann Method (1999)
- A discrete velocity model with spatial and velocity discretization based on a lattice Boltzmann method is considered in the low Mach number limit. A uniform numerical scheme for this model is investigated. In the limit, the scheme reduces to a finite difference scheme for the incompressible Navier-Stokes equation which is a projection method with a second order spatial discretization on a regular grid. The discretization is analyzed and the method is compared to Chorin's original spatial discretization. Numerical results supporting the analytical statements are presented.

- 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.

- An Asymptotic-Induced Scheme for Nonstationary Transport Equations in the Diffusive Limit (1997)
- 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.

- 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.

- A Numerical Method for Computing Asymptotic States and Outgoing Distributions for Kinetic Linear Half-Space Problems (1994)
- 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.

- 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.

- On the Connection of the Formulae for Entropy and Stationary Distribution (1994)
- 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.

- 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.