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- Fachbereich Mathematik (28)
- Fraunhofer (ITWM) (1)

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.

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.

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.

The paper presents a parallelization technique for the finite pointset method, a numerical method for rarefied gas flows.; First we give a short introduction to the Boltzmann equation, which describes the behaviour of rarefied gas flows. The basic ideas of the finite pointset method are presented and a strategy to parallelize the algorithm will be explained. It is shown that a static processor partition leads to an insufficient load-balance of the processors. Therefore an optimized parallelization technique based on an adaptive processor partition will be introduced, which improves the efficiency of the simulation code over the whole region of interesting flow situation. Finally we present a comparison of the CPU-times between a parallel computer and a vector computer.

The paper presents theoretical and numerical investigations on simulation methods for the Boltzmann equation with axisymmetric geometry. The main task is to reduce the computational effort by taking advantage of the symmetry in the solution of the Boltzmann equation.; The reduction automatically leads to the concept of weighting functions for the radial space coordinate and therefore to a modified Boltzmann equation. Consequently the classical simulation methods have to be modified according to the new equation.; The numerical results shown in this paper - rarefied gas flows around a body with axisymmetric geometry - were done in the framework of the European space project HERMES.

The system of shallow water waves is one of the classical examples for nonlinear, twodimensional conservation laws. The paper investigates a simple kinetic equation depending on a parameter e which leads for e to 0 to the system of shallow water waves. The corresponding equilibrium distribution function has a compact support which depends on the eigenvalues of the hyperbolic system. It is shown that this kind of kinetic approach is restricted to a special class of nonlinear conservation laws. The kinetic model is used to develop a simple particle method for the numerical solution of shallow water waves. The particle method can be implemented in a straightforward way and produces in test examples sufficiently accurate results.

The paper presents a fast implementation of a constructive method to generate a special class of low-discrepancy sequences which are based on Van Neumann-Kakutani tranformations. Such sequences can be used in various simulation codes where it is necessary to generate a certain number of uniformly distributed random numbers on the unit interval.; From a theoretical point of view the uniformity of a sequence is measured in terms of the discrepancy which is a special distance between a finite set of points and the uniform distribution on the unit interval.; Numerical results are given on the cost efficiency of different generators on different hardware architectures as well as on the corresponding uniformity of the sequences. As an example for the efficient use of low-discrepancy sequences in a complex simulation code results are presented for the simulation of a hypersonic rarefied gas flow.

Simulation methods like DSMC are an efficient tool to compute rarefied gas flows. Using supercomputers it is possible to include various real gas effects like vibrational energies or chemical reactions in a gas mixture. Nevertheless it is still necessary to improve the accuracy of the current simulation methods in order to reduce the computational effort. To support this task the paper presents a comparison of the classical DSMC method with the so called finite Pointset Method. This new approach was developed during several years in the framework of the European space project HERMES. The comparison given in the paper is based on two different testcases: a spatially homogeneous relaxation problem and a 2-dimensional axisymmetric flow problem at high Mach numbers.

Monte-Carlo methods are widely used numerical tools in various fields of application, like rarefied gas dynamics, vacuum technology, stellar dynamics or nuclear physics. A central part in all applications is the generation of random variates according to a given probability law. Fundamental techniques to generate non-uniform random variates are the inversion principle or the acceptance-rejection method. Both procedures can be quite time-consuming if the given probability law has a complicated structure.; In this paper we consider probability laws depending on a small parameter and investigate the use of asmptotic expansions to generate random variates. The results given in the paper are restrictedto first order expansions. We show error estimates for the discrepancy as well as for the bounded Lipschitz distance of the asymptotic expansion. Furthermore the integration error for some special classes of functions is given. The efficiency of the method is proved by a numerical example from rarefied gas flows.

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.

The paper presents some approximation methods for the Boltzmann equation. In the first part fully implicit discretization techniques for the spatially homogeneous Boltzmann equation are investigated. The implicit equation is solved using an iteration process. It is shown that the iteration converges to the correct solution for the moments of the distribution function as long as the mass conservation is strictly fulfilled. For a simple model Boltzmann equation some unexpected features of the implicit scheme and the corresponding iteration process are clarified. In the second part a new iteration algorithm is proposed which should be used for the stationary Boltzmann equation. The realization of the method is very similar to the standard splitting algorithms except some new stochastic elements.

Second Order Scheme for the Spatially Homogeneous Boltzmann Equation with Maxwellian Molecules
(1995)

In the standard approach, particle methods for the Boltzmann equation are obtained using an explicit time discretization of the spatially homogeneous Boltzmann equation. This kind of discretization leads to a restriction of the discretization parameter as well as on the differential cross section in the case of the general Boltzmann equation. Recently, it was shown, how to construct an implicit particle scheme for the Boltzmann equation with Maxwellian molecules. The present paper combines both approaches using a linear combination of explicit and implicit discretizations. It is shown that the new method leads to a second order particle method, when using an equiweighting of explicit and implicit discretization.

Numerical Simulation of the Stationary One-Dimensional Boltzmann Equation by Particle Methods
(1995)

The paper presents a numerical simulation technique - based on the well-known particle methods - for the stationary, one-dimensional Boltzmann equation for Maxwellian molecules. In contrast to the standard splitting methods, where one works with the instationary equation, the current approach simulates the direct solution of the stationary problem. The model problem investigated is the heat transfer between two parallel plates in the rarefied gas regime. An iteration process is introduced which leads to the stationary solution of the exact - space discretized - Boltzmann equation, in the sense of weak convergence.

The paper presents numerical results on the simulation of boundary value problems for the Boltzmann equation in one and two dimensions. In the one-dimensional case, we use prescribed fluxes at the left and diffusive conditions on the right end of a slab to study the resulting steady state solution. Moreover, we compute the numerical density function in velocity space and compare the result with the Chapman-Enskog distribution obtained in the limit for continuous media. The aim of the two-dimensional simulations is to investigate the possibility of a symmetry break in the numerical solution.

The paper presents some new estimates on the gain term of the Boltzmann collision operator. For Maxwellian molecules, it is shown that the L -norm of the gain term can be bounded in terms of the L1 and L -norm of the density function f. In the case of more general collision kernels, like the hard-sphere interaction potential, the gain term is estimated pointwise by the L -norm of the density function and the loss term of the Boltzmann collision operator.

The paper presents some adaptive load balance techniques for the simulation of rarefied gas flows on parallel computers. It is shown that a static load balance is insufficient to obtain a scalable parallel efficiency. Hence, two adaptive techniques are investigated which are based on simple algorithms. Numerical results show that using heuristic techniques one can achieve a sufficiently high efficiency over a wide range of different hardware platforms.