We consider a self-gravitating collisionless gas as described by the Vlasov-Poisson or Einstein-Vlasov system or a self-gravitating fluid ball as described by the Euler-Poisson or Einstein-Euler system. We give a simple proof for the finite extension of spherically symmetric equilibria, which covers all these models simultaneously. In the Vlasov case the equilibria are characterized by a local growth condition on the microscopic equation of state, i.e., on the dependence of the particle distribution on the particle energy, at the cut-off energy E 0 , and in the Euler case by the corresponding growth condition on the equation of state p = P (ρ) at ρ = 0. These purely local conditions are slight generalizations to known such conditions.

Numerical evidence is given that spherically symmetric perturbations of stable spherically symmetric steady states of the gravitational Vlasov-Poisson system lead to solutions which oscillate in time. The oscillations can be periodic in time or damped. Along one-parameter families of polytropic steady states we establish an Eddington-Ritter type relation which relates the period of the oscillation to the central density of the steady state. The numerically obtained periods are used to estimate possible periods for typical elliptical galaxies.

This paper considers the parallelization of two Particle-in-Cell (PIC) codes which simulate the time evolution of galaxies and globular clusters in the Newtonian or the general relativistic framework. The corresponding models are known as the Vlasov-Poisson or the Einstein-Vlasov system, and the latter is designed in particular to study the formation of black holes and spacetime singularities. We start with a stepby-step shared-memory parallelization of the Vlasov-Poisson code using POSIX Threads and finally develop message passing codes using MPI. The parallel codes have been investigated on three modern supercomputer systems using up to 4096 cores, and speedups above 1300 have been reached. The speedup obtained through parallelization has already helped in finding new numerical results, such as oscillating solutions of the Vlasov-Poisson system.

We consider spherically symmetric steady states of the Vlasov-Poisson system, which describe equilibrium configurations of galaxies or globular clusters. If the microscopic equation of state, i.e., the dependence of the steady state on the particle energy (and angular momentum) is fixed, a one-parameter family of such states is obtained. In the polytropic case the mass of the state along such a one-parameter family is a monotone function of its radius. We prove that for the King, Woolley-Dickens, and related models this mass-radius relation takes the form of a spiral.

We derive a new discretisation method for first order PDEs of arbitrary spatial dimension, which is based upon a meshfree spatial approximation. This spatial approximation is similar to the SPH (smoothed particle hydrodynamics) technique and is a typical kernel-based method. It differs, however, significantly from the SPH method since it employs an Eulerian and not a Lagrangian approach. We prove stability and convergence for the resulting semi-discrete scheme under certain smoothness assumptions on the defining function of the PDE. The approximation order depends on the underlying kernel and the smoothness of the solution. Hence, we also review an easy way of constructing smooth kernels yielding arbitrary convergence orders. Finally, we give a numerical example by testing our method in the case of a one-dimensional Burgers equation.