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Mon, 10 Dec 2001 00:00:00 +0100Mon, 10 Dec 2001 00:00:00 +0100A limiter based on kinetic theory
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1271
Mapundi K. Banda; Michael Junk; Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1271Mon, 10 Dec 2001 00:00:00 +0100Uniform Stability of a Finite Difference Scheme for Transport Equations in Diffusive Regimes
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1114
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.Axel Klar; Andreas Unterreiterpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1114Wed, 16 Aug 2000 00:00:00 +0200Discretizations for the Incompressible Navier-Stokes Equations based on the Lattice Boltzmann Method
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/652
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.Michael Junk; Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/652Sun, 25 Jun 2000 08:20:00 +0200An adaptive domain decomposition procedure for Boltzmann and Euler equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/625
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.Sudarshan Tiwari; Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/625Sun, 25 Jun 2000 00:00:00 +0200An Asymptotic-Induced Scheme for Nonstationary Transport Equations in the Diffusive Limit
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/610
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.Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/610Sat, 24 Jun 2000 00:00:00 +0200Convergence of Alternating Domain Decomposition Schemes for Kinetic and Aerodynamic Equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/546
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.Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/546Mon, 03 Apr 2000 00:00:00 +0200A Numerical Method for Computing Asymptotic States and Outgoing Distributions for Kinetic Linear Half-Space Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/547
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.F. Golse; Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/547Mon, 03 Apr 2000 00:00:00 +0200Domain Decomposition for Kinetic Problems with Nonequilibrium States
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/548
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.Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/548Mon, 03 Apr 2000 00:00:00 +0200On the Connection of the Formulae for Entropy and Stationary Distribution
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/550
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.Y. Arkhipov; Axel Klar; V. Vedenyapinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/550Mon, 03 Apr 2000 00:00:00 +0200Computation of Nonlinear Functionals in Particle Methods
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/551
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.Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/551Mon, 03 Apr 2000 00:00:00 +0200Asymptotic-Induced Domain Decomposition Methods for Kinetic and Drift Diffusion Semiconductor Equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/568
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.Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/568Mon, 03 Apr 2000 00:00:00 +0200A Kinetic Model for Vehicular Traffic Derived from a Stochastic Microscopic Model
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/569
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.R. Wegener; Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/569Mon, 03 Apr 2000 00:00:00 +0200Particle Methods: Theory and Applications
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/586
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.Helmut Neunzert; Axel Klar; Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/586Mon, 03 Apr 2000 00:00:00 +0200Mathematical Models for Vehicular Traffic
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/591
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.Axel Klar; R. D. Kühne; R. Wegenerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/591Mon, 03 Apr 2000 00:00:00 +0200Enskog-like kinetic models for vehicular traffic
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/596
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.Axel Klar; R. Wegenerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/596Mon, 03 Apr 2000 00:00:00 +0200Particle Methods for Evolution Equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/602
Michael Junk; Axel Klar; Jens Struckmeier; Sudarshan Tiwaripreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/602Mon, 03 Apr 2000 00:00:00 +0200A Numerical Method for Kinetic Semiconductor Equations in the Drift Diffusion limit
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/623
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.Axel Klarpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/623Mon, 03 Apr 2000 00:00:00 +0200A kinetic model for vehicular traffic: Existence of stationary solutions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/627
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.Reinhard Illner; Axel Klar; H. Lange; Andreas Unterreiter; Raimund Wegenerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/627Mon, 03 Apr 2000 00:00:00 +0200Transition from Kinetic Theory to Macroscopic Fluid Equations: A Problem fo Domain Decomposition and a Source for New Algorithms
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/636
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.Axel Klar; Helmut Neunzert; Jens Struckmeierpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/636Mon, 03 Apr 2000 00:00:00 +0200A hierarchy of models for multilane vehicular traffic PART I: Modeling
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/744
In the present paper multilane models for vehicular traffic are considered. A microscopic multilane model based on reaction thresholds is developed. Based on this model an Enskog like kinetic model is developed. In particular, care is taken to incorporate the correlations between the vehicles. From the kinetic model a fluid dynamic model is derived. The macroscopic coefficients are deduced from the underlying kinetic model. Numerical simulations are presented for all three levels of description in [10]. Moreover, a comparison of the results is given there.Axel Klar; Raimund Wegenerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/744Mon, 03 Apr 2000 00:00:00 +0200A hierarchy of models for multilane vehicular traffic PART II: Numerical and stochastic investigations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/745
In this paper the work presented in [6] is continued. The present paper contains detailed numerical investigations of the models developed there. A numerical method to treat the kinetic equations obtained in [6] are presented and results of the simulations are shown. Moreover, the stochastic correlation model used in [6] is described and investigated in more detail.Axel Klar; Raimund Wegenerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/745Mon, 03 Apr 2000 00:00:00 +0200Boundary Layers and Domain Decomposition for Radiative Heat Transfer and Diffusion Equations: Applications to Glass Manufacturing Processes
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/746
In this paper domain decomposition methods for radiative transfer problems including conductive heat transfer are treated. The paper focuses on semi-transparent materials, like glass, and the associated conditions at the interface between the materials. Using asymptotic analysis we derive conditions for the coupling of the radiative transfer equations and a diffusion approximation. Several test cases are treated and a problem appearing in glass manufacturing processes is computed. The results clearly show the advantages of a domain decomposition approach. Accuracy equivalent to the solution of the global radiative transfer solution is achieved, whereas computation time is strongly reduced.Axel Klar; Norbert Siedwopreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/746Mon, 03 Apr 2000 00:00:00 +0200