## Berichte der Arbeitsgruppe Technomathematik (AGTM Report)

- 168
- Particle Methods for Evolution Equations (1996)

- 212
- 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.

- 211
- A Finite Difference Interpretation of the Lattice Boltzmann Method (1999)
- Compared to conventional techniques in computational fluid dynamics, the lattice Boltzmann method (LBM) seems to be a completely different approach to solve the incompressible Navier-Stokes equations. The aim of this article is to correct this impression by showing the close relation of LBM to two standard methods: relaxation schemes and explicit finite difference discretizations. As a side effect, new starting points for a discretization of the incompressible Navier-Stokes equations are obtained.

- 218
- A new perspective on kinetic schemes (1999)
- Compared to standard numerical methods for hyperbolic systems of conservation laws, Kinetic Schemes model propagation of information by particles instead of waves. In this article, the wave and the particle concept are shown to be closely related. Moreover, a general approach to the construction of Kinetic Schemes for hyperbolic conservation laws is given which summarizes several approaches discussed by other authors. The approach also demonstrates why Kinetic Schemes are particularly well suited for scalar conservation laws and why extensions to general systems are less natural.

- 229
- On the approximation of kinetic equations by moment systems (2000)
- The aim of this article 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 arbitrary large speeds of propagation, even for initial data arbitrary close to global eqilibrium.

- 220
- Exponentially exact hyperbolic systems (2000)
- Starting with general hyperbolic systems of conservation laws, a special sub - class is extracted in which classical solutions can be expressed in terms of a linear transport equation. A characterizing property of this sub - class which contains, for example, all linear systems and non - linear scalar equations, is the existence of so called exponentially exact entropies.

- 226
- Consistency analysis of mesh-free methods for conservation laws (2000)
- Based on general partitions of unity and standard numerical flux functions, a class of mesh-free methods for conservation laws is derived. A Lax-Wendroff type consistency analysis is carried out for the general case of moving partition functions. The analysis leads to a set of conditions which are checked for the finite volume particle method FVPM. As a by-product, classical finite volume schemes are recovered in the approach for special choices of the partition of unity.

- 246
- Maximum Entropy Moment Systems and Galilean Invariance (2001)
- In this article, we investigate the maximum entropy moment closure in gas dynamics. We show that the usual choice of polynomial weight functions may lead to hyperbolic systems with an unpleasant state space: equilibrium states are boundary points with possibly singular fluxes. In order to avoid singularities, the necessary arises to find weight functions which growing sub-quadratically at infinity. Unfortunately, this requirement leads to a conflict with Galilean invariance of the moment systems because we can show that rotational and translational invariant, finite dimensional function spaces necessarily consist of polynomials.

- 241
- A limiter based on kinetic theory (2001)