## Berichte der Arbeitsgruppe Technomathematik (AGTM Report)

### Filtern

#### Erscheinungsjahr

- 1993 (11) (entfernen)

#### Volltext vorhanden

- ja (11) (entfernen)

#### Schlagworte

- Low-discrepancy sequences (1)
- Monte Carlo method (1)
- Van Neumann-Kakutani transformation (1)
- discrete measure (1)
- distribution (1)
- low discrepancy (1)
- particle method (1)

99

Exact Solutions of Discrete Kinetic Models and Stationary Problems for the Plane Broadwell Model
(1993)

92

This paper is concerned with the development of a self-adaptive spatial descretization for PDEs using a wavelet basis. A Petrov-Galerkin method [LPT91] is used to reduce the determination of the unknown at the new time step to the computation of scalar products. These have to be discretized in an appropriate way. We investigate this point in detail and devise an algorithm that has linear operation count with respect to the number of unknowns. It is tested with spline wavelets and Meyer wavelets retaining the latter for their better localisation at finite precision. The algorithm is then applied to the one dimensional thermodiffusive equations. We show that the adaption strategy merits to be modified in order to take into account the particular and very strong nonlinearity of this problem. Finally, a supplementary Fourier discretization permits the computation of two dimensional flame fronts.

89

90

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.

96

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.

94

This paper considers the numerical solution of a transmission boundary-value problem for the time-harmonic Maxwell equations with the help of a special finite volume discretization. Applying this technique to several three-dimensional test problems, we obtain large, sparse, complex linear systems, which are solved by using BiCG, CGS, BiCGSTAB resp., GMRES. We combine these methods with suitably chosen preconditioning matrices and compare the speed of convergence.

87

The paper presents the shuffle algorithm proposed by Baganoff, which can be implemented in simulation methods for the Boltzmann equation to simplify the binary collision process. It is shown that the shuffle algborithm is a discrete approximation of an isotropic collision law. The transition probability as well as the scattering cross section of the shuffle algorithm are opposed to the corresponding quantities of a hard-sphere model. The discrepancy between measures on a sphere is introduced in order to quantify the approximation error by using the shuffle algorithm.

100

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.

93

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.