## 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)

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.

This report contains the following three papers about computations of rarefied gas flows:; ; a) Rarefied gas flow around a disc with different angles of attack, published in the proceedings of the 17th RGD Symposium, Aachen, 1990.; ; b) Hypersonic flow calculations around a 3D-deltawing at low Knudsen numbers, published in the proceedings of the 17th RGD Symposium,; Aachen, 1990.; ; c) Rarefied gas flow around a 3D-deltawing, published in the proceedings of the Workshop on Hypersonic Flows for Reentry Problems,; Part 1, Antibes, France, January 22-25, 1990.; ; All computations are part of the HERMES Research and Development Program.

Patterns are considered as normalized measures and distances between them are defined as distances of the corresponding measures using metrics in measure spaces. This idea can be applied for pattern recognition if smeared patterns have to be compared with given ideal patterns. Different metrics are sensitive to different characteristics of the patterns - this is demonstrated in discussing examples. Particular attention is paid to a problem of Quality Control for an artificial fabric, where the distance to uniformity is defined and evaluated; the results are now used in industry.

Particle Methods
(1994)

In these lectures we will mainly treat a billard game. Our particles will be hard spheres. Not always: We will also touch cases, where particles have interior energies due to rotation or vibration, which they exchange in a collision, and we will talk about chemical reactions happening during a collision. But many essential aspects occur already in the billard case which will be therefore paradigmatic. I do not know enough about semiconductors to handle collisions there - the Boltzmann case is certainly different but may give some idea even for the other cases.

As an alternative to the commonly used Monte Carlo Simulation methods for solving the Boltzmann equation we have developed a new code with certain important improvements. We present results of calculations on the reentry phase of a space shuttle. One aim was to test physical models of internal energies and of gas-surface interactions.

Industrial mathematics has many faces; but its essential feature is the cooperation of partners - from industry and from universities - with quite different interest (business versus academic carreer), normally working on different time scales. They measure success in a different way (selling rate against citing index), they have different hierarchies of values and are very often distrusting each other. Industry doubts that mathematicians are willing and/or able to produce something real practical and useful (and the mathematicians should not be too much surprised about this attitude, they very often doubt themselves) - mathematicians are afraid to loose their competence (their ideal of scientific truth, to say it more idealistically), to sell their souls.