## Fachbereich Mathematik

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- Particle Methods (2)
- Boltzmann Equation (1)
- Domain Decomposition (1)
- Evolution Equations (1)
- Hybrid Codes (1)
- Numerical Simulation (1)
- Smoothed Particle Hydrodynamics (1)

We discuss how kinetic and aerodynamic descriptions of a gas can be matched at some prescribed boundary. The boundary (matching) conditions arise from requirement that the relevant moments (p,u,...) of the particle density function be continuous at the boundary, and from the requirement that the closure relation, by which the aerodynamic equations (holding on one side of the boundary) arise from the kinetic equation (holding on the other side), be satisfied at the boundary. We do a case study involving the Knudsen gas equation on one side and a system involving the Burgers equation on the other side in section 2, and a discussion for the coupling of the full Boltzmann equation with the compressible Navier-Stokes equations in section 3.

This paper contains the basic ideas and practical aspects for numerical methods for solving the Boltzmann Equation. The main field of application considered is the reentry of a Space Shuttle in the transition from free molecular flow to continuum flow. The method used will be called Finite Pointset Method (FPM) approximating the solution by finite sets of particles in a rigorously defined way. Convergence results are cited while practical aspects of the algorithm are emphasized. Ideas for the transition to the Navier Stokes domain are shortly discussed.

Particle methods to simulate rarefied gas flows have found an increasing interest in Computational Fluid Dynamics during the last decade, see for example [1], [2], [3] and [4]. The general goal is to develop numerical schemes which are reliable enough to substitute real windtunnel experiments, needed for example in space research, by computer experiments. In order to achieve this goal one needs numerical methods solving the Boltzmann equation including all important physical effects. In general this means 3D computations for a chemically reacting rarefied gas. With codes of this kind at hand, Boltzmann simulation becomes a powerful tool in studying rarefied gas phenomena.

We have presented here a two-dimensional kinetical scheme for equations governing the motion of a compressible flow of an ideal gas (air) based on the Kaniel method. The basic flux functions are computed analytically and have been used in the organization of the flux computation. The algorithm is implemented and tested for the 1D shock and 2D shock-obstacle interaction problems.