Matthias Kunik

Kinetic schemes for the ultra-relativistic Euler equations

We present a kinetic numerical scheme for the relativistic Euler equations, which describe the flow of a perfect fluid in terms of the particle density n, the spatial part of the four-velocity u and the pressure p. The kinetic approach is very simple in the ultra-relativistic limit, but may also be applied to more general cases. The basic ingredients of the kinetic scheme are the phase-density in equilibrium and the free flight. The phase-density generalizes the nonrelativistic Maxwellian for a gas in local equilibrium. The free flight is given by solutions of a collision free kinetic transport equation. The scheme presented here is an explicit method and unconditionally stable. We establish that the conservation laws of mass, momentum and energy as well as the entropy inequality are everywhere exactly satisfied by the solution of the kinetic scheme. For that reason we obtain weak admissible Euler solutions including arbitrarily complicated shock interactions. In the numerical case studies the results obtained from the kinetic scheme are compared with the first order upwind and centered schemes.

Kinetic schemes for the relativistic gas dynamics

Summary.A kinetic solution for the relativistic Euler equations is presented. This solution describes the flow of a perfect gas in terms of the particle density n, the spatial part of the four-velocity u and the inverse temperature β. In this paper we present a general framework for the kinetic scheme of relativistic Euler equations which covers the whole range from the non-relativistic limit to the ultra-relativistic limit. The main components of the kinetic scheme are described now. (i) There are periods of free flight of duration τM, where the gas particles move according to the free kinetic transport equation. (ii) At the maximization times tn=nτM, the beginning of each of these free-flight periods, the gas particles are in local equilibrium, which is described by Jüttners relativistic generalization of the classical Maxwellian phase density. (iii) At each new maximization time tn>0 we evaluate the so called continuity conditions, which guarantee that the kinetic scheme satisfies the conservation laws and the entropy inequality. These continuity conditions determine the new initial data at tn. iv If in addition adiabatic boundary conditions are prescribed, we can incorporate a natural reflection method into the kinetic scheme in order to solve the initial and boundary value problem. In the limit τM→0 we obtain the weak solutions of Euler’s equations including arbitrary shock interactions. We also present a numerical shock reflection test which confirms the validity of our kinetic approach.

On the approximation of the Fokker-Planck equation by moment systems

The aim of this paper is to show that moment approximations of kinetic equations based on a maximum-entropy approach can suffer from severe drawbacks if the kinetic velocity space is unbounded. As example, we study the Fokker-Planck equation where explicit expressions for the moments of solutions to Riemann problems can be derived. The quality of the closure relation obtained from the maximum-entropy approach as well as the Hermite/Grad approach is studied in the case of five moments. It turns out that the maximum-entropy closure is even singular in equilibrium states while the Hermite/Grad closure behaves reasonably. In particular, the admissible moments may lead to arbitrarily large speeds of propagation, even for initial data arbitrary close to global eqilibrium.

Cold, thermal and oscillator closure of the atomic chain

We consider a simple microscopic model for a solid body and study the problematic nature of micro-macro transitions. The microscopic model describes the solid body by a many-particle system that develops according to Newtons equations of motion. We discuss various Riemannian initial value problems that lead to the propagation of waves. The initial value problems are solved directly from the microscopic equations of motion. Additionally, these equations serve to establish macroscopic field equations. The macroscopic field equations consist of conservation laws, which follow rigorously from the microscopic equations, and of closure relations which are completely determined by the distributions of the microscopic motion. In particular, we consider three kinds of closure relations which correspond to three different kinds of equilibrium. It turns out that closure relations cannot be given appropriately without relating them to the initial conditions, and that closure relations might change during the temporal development of the initial data, because the body undergoes several transitions between different states of local equilibrium. In those examples that we have considered, the macroscopic variables of mass density and temperature do not constitute a unique kind of microscopic motion in equilibrium.

Reflections of Eulerian Shock Waves at Moving Adiabatic Boundaries

This study solves the initial and boundary value problem for the Euler equations of gases. The boundaries are allowed to move and are assumed to be adiabatic. In addition we shall discuss that isothermal walls are not possibly within the Euler theory. We do not formulate the boundary conditions in terms of the macroscopic basic variables as the mass density, velocity and temperature. Instead we consider the underlying kinetic picture which exhibits the interaction of the gas atoms with the boundaries. Hereby the advantage is offered to formulate the boundary conditions in a very suggestive manner. This procedure becomes possible, because we approach the solution of the Euler equations by the following limit: We rely on the moment representation of the macroscopic basic variables, and in order to obtain the temporal development of the phase density, we decompose a given macroscopic time interval into periods of free flight of the gas atoms. These periods of duration TME are interrupted by a maximization of entropy, thus introducing a simulation of the interatomic interaction. In [2] we have shown that the Euler equations may be established in the limit TME -> 0. 232 W. Dreyer and M. Kunik

A REDUCTION OF THE BOLTZMANN-PEIERLS EQUATION

This paper deals with the solutions of initial value problems of the Boltzmann-Peierls equation (BPE). This integro-differential equation describes the evolution of heat in crystalline solids at very low temperatures. The BPE describes the evolution of the phase density of a phonon gas. The corresponding entropy density is given by the entropy density of a Bose-gas. We derive a reduced three-dimensional kinetic equation which has a much simpler structure than the original BPE. By using special coordinates in the one-dimensional case, we can perform a further reduction of the kinetic equation. By assuming one-dimensionality on the initial phase density one can show that this property is preserved for all later times. We derive kinetic schemes for the kinetic equation as well as for the derived moment systems. Several numerical test cases are presented to validate the theory.

KINETIC SCHEMES AND INITIAL BOUNDARY VALUE PROBLEMS FOR THE EULER SYSTEM

ABSTRACT We study kinetic solutions, including shocks, of initial and boundary value problems for the Euler equations of gases. In particular we consider moving adiabatic boundaries, which may be driven either by a given path or because they are subjected to forces. In the latter case we consider a gas contained in a cylinder which is closed by a piston. Here the boundary represents the piston that suffers forces by the incoming and outgoing gas particles. Moreover, we will study periodic boundary conditions. A kinetic scheme consists of three ingredients: (i) There are periods of free flight of duration τ M , where the gas particles move according to the free transport of equation. (ii) It is assumed that the distribution of the gas particles at the beginning of each of these periods is given by a Maxwellian. (iii) The interaction of gas particles with a boundary is described by a so called extension law, which determines the phase density at the boundary, and provides additionally continuity conditions for the fields at the boundary in order to achieve convergence. The Euler equations result in the limit τ M → 0. We prove rigorous results for these kinetic schemes concerning (i) regularity, (ii) weak conservation laws, (iii) entropy inequality and (iv) continuity conditions for the fields at the boundaries. The study is supplemented by some numerical examples. This approach is by no mean restricted to the Euler equations or to adiabatic boundaries, but it holds also for other hyperbolic systems, namely those that rely on a kinetic formulation.

Anwendungen der Funktionentheorie

Einige Anregungen zur Entstehung dieses Abschnittes stammen aus den Lehrbuchern von Weinberg [42, Chapter 1.1], Konigsberger [27] und Zeitler [45]. Wir erinnern uns an das PoincarescheM odell der hyperbolischen Geometrie im Einheitskreis das wir im Abschnitt 5.2 eingefuhrt haben.

Grundlagen der Funktionentheorie

Die Funktionentheorie beschaftigt sich mit den analytischen Eigenschaften komplex differenzierbarer Funktionen, die auf offenen Teilmengen der komplexen Zahlenebene ℂ definiert sind. Man nennt sie auch holomorphe Funktionen. Erst die Funktionentheorie ermoglicht ein tieferes Verstandnis grundlegender reellwertiger Funktionen, wenn diese auf die komplexe Zahlenebene fortgesetzt werden.

On Energy Conditions for Electromagnetic Diffraction by Apertures

The diffraction of light is considered for a plane screen with an open bounded aperture. The corresponding solution behind the screen is given explicitly in terms of the Fourier transforms of the tangential components of the electric boundary field in the aperture. All components of the electric as well as the magnetic field vector are considered. We introduce solutions with global finite energy behind the screen and describe them in terms of two boundary potential functions. This new approach leads to a decoupling of the vectorial boundary equations in the aperture in the case of global finite energy. For the physically admissible solutions, that is, the solutions with local finite energy, we derive a characterisation in terms of the electric boundary fields.