## Berichte der Arbeitsgruppe Technomathematik (AGTM Report)

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#### Year of publication

- 1999 (28) (remove)

#### Document Type

- Preprint (28) (remove)

#### Keywords

- incompressible Navier-Stokes equations (2)
- lattice Boltzmann method (2)
- low Mach number limit (2)
- Chorin's projection scheme (1)
- Collocation Method plus (1)
- Differential Cross-Sections (1)
- Discrete velocity models (1)
- Experimental Data (1)
- Fredholm integral equation of the second kind (1)
- GPS-satellite-to-satellite tracking (1)

- 219
- On an Integral Equation Model for Slender Bodies in Low Reynolds-Number Flows (1999)
- The interation of particular slender bodies with low Reynolds-number flows is in the limit 'slenderness to 0' described by a linear Fredholm integral equation of the second kind. The integral operator of this equation has a denumerable set of polynomial eigenfunctions whose corresponding eigenvalues are non-positive and of logarithmic growth. A theorem similiar to a classical result of Plemelj-Privalov for integral operators with Cauchy kernels is proven. In contrast to Cauchy kernel operators, the integral operator maps no Hölder space into itself. A spectral analysis of the integral operator restricted to an appropriate class of analytic functions is performed. The spectral properties of this restricted integral operator suggest a collocation-like method to solve the integral equation numerically. For this numerical scheme, convergence is proven and several computations are presented.

- 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.

- 217
- Wavelet Analysis of the Geomagnetic Field Within Source Regions of Ionospheric and Magnetospheric Currents (1999)
- A multiscale method is introduced using spherical (vector) wavelets for the computation of the earth's magnetic field within source regions of ionospheric and magnetospheric currents. The considerations are essentially based on two geomathematical keystones, namely (i) the Mie representation of solenoidal vector fields in terms of toroidal and poloidal parts and (ii) the Helmholtz decomposition of spherical (tangential) vector fields. Vector wavelets are shown to provide adequate tools for multiscale geomagnetic modelling in form of a multiresolution analysis, thereby completely circumventing the numerical obstacles caused by vector spherical harmonics. The applicability and efficiency of the multiresolution technique is tested with real satellite data.

- 216
- Least-squares Geopotential Approximation by Windowed Fourier and Wavelet Transform (1999)
- Two possible substitutes of the Fourier transform in geopotential determination are windowed Fourier transform (WFT) and wavelet transform (WT). In this paper we introduce harmonic WFT and WT and show how it can be used to give information about the geopotential simultaneously in the space domain and the frequency (angular momentum) domain. The counterparts of the inverse Fourier transform are derived, which allow us to reconstruct the geopotential from its WFT and WT, respectively. Moreover, we derive a necessary and sufficient condition that an otherwise arbitrary function of space and frequency has to satisfy to be the WFT or WT of a potential. Finally, least - squares approximation and minimum norm (i.e. least - energy) representation, which will play a particular role in geodetic applications of both WFT and WT, are discussed in more detail.

- 215
- Moment contractivity and stability exponents of nonlinear stochastic dynamical systems (1999)
- Nonlinear stochastic dynamical systems as ordinary stochastic differential equations and stochastic difference methods are in the center of this presentation in view of the asymptotical behaviour of their moments. We study the exponential p-th mean growth behaviour of their solutions as integration time tends to infinity. For this purpose, the concepts of nonlinear contractivity and stability exponents for moments are introduced as generalizations of well-known moment Lyapunov exponents of linear systems. Under appropriate monotonicity assumptions we gain uniform estimates of these exponents from above and below. Eventually, these concepts are generalized to describe the exponential growth behaviour along certain Lyapunov-type functionals.

- 214
- On Moment-Dissipative Stochastic Dynamical (1999)
- Nonlinear dissipativity, asymptotical stability, and contractivity of (ordinary) stochastic differential equations (SDEs) with some dissipative structure and their discretizations are studied in terms of their moments in the spirit of Pliss (1977). For this purpose, we introduce the notions and discuss related concepts of dissipativity, growth- bounded and monotone coefficient systems, asymptotical stability and contractivity in wide and narrow sense, nonlinear A-stability, AN-stability, B-stability and BN-stability for stochastic dynamical systems - more or less as stochastic counterparts to deterministic concepts. The test class of in a broad sense interpreted dissipative SDEs as natural analogon to dissipative deterministic differential systems is suggested for stochastic-numerical methods. Then, in particular, a kind of mean square calculus is developed, although most of ideas and analysis can be carried over to general "stochastic Lp-case" (p * 1). By this natural restriction, the new stochastic concepts are theoretically meaningful, as in deterministic analysis. Since the choice of step sizes then plays no essential role in related proofs, we even obtain nonlinear A-stability, AN-stability, B-stability and BN-stability in the mean square sense for this implicit method with respect to appropriate test classes of moment-dissipative SDEs.

- 213
- On a Recursive Approximation of Singularity Perturbed Parabolic Equations (extended version) (1999)
- The asymptotic analysis of IBVPs for the singularly perturbed parabolic PDE ... in the limit epsilon to zero motivate investigations of certain recursively defined approximative series ("ping-pong expansions"). The recursion formulae rely on operators assigning to a boundary condition at the left or the right boundary a solution of the parabolic PDE. Sufficient conditions for uniform convergence of ping-pong expansions are derived and a detailed analysis for the model problem ... is given.

- 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.

- 210
- The Stationary Current-Voltage Characteristics of the Quantum Drift Diffusion Model (1999)
- This paper is concerned with numerical algorithms for the bipolar quantum drift diffusion model. For the thermal equilibrium case a quasi-gradient method minimizing the energy functional is introduced and strong convergence is proven. The computation of current - voltage characteristics is performed by means of an extended emph{Gummel - iteration}. It is shown that the involved fixed point mapping is a contraction for small applied voltages. In this case the model equations are uniquely solvable and convergence of the proposed iteration scheme follows. Numerical simulations of a one dimensional resonant tunneling diode are presented. The computed current - voltage characteristics are in good qualitative agreement with experimental measurements. The appearance of negative differential resistances is verified for the first time in a Quantum Drift Diffusion model.