Oberseminar Numerik (Junk)

We consider car-following models for a single-line trac ow on a circular road. The model is characterized by a choice of a particular optimal velocity function which is related to each individual driver. The aim is to investigate oscillatory solutions to the model. Nevertheless, these solutions may not be physical. The analysis shows that a trajectory becomes non physical since an event which can be interpreted as a collision with the preceding car. The natural action of a driver at that moment would be to overtake the slower car. A model of the overtaking is proposed. We will present an equivalent formulation of the overtaking maneuver: Introducing new state variables, we formulate our model as a Filippov system, i.e., ordinary dierential equations with discontinuous righthand sides. As a case study, we investigate typical oscillatory patterns of three cars. The objective is to continue the patterns with respect to a parameter.

We consider the Analytic Singular Value Decomposition, ASVD, of matrix valued functions. ASVD is smooth up to isolated parameter values at which either a multiple singular value or a zero singular value turns up on the path. These exceptional points are called non-generic. Note that ASVD-computations require information on all singular values on the path and hence the algorithms were not able to cope with large sparse input data. Moreover, we investigated a pa- thfollowing of just one simple singular value and the corresponding left/right singular vector. A breakdown of the continuation is related to non-generic points on the path. We apply Singularity Theory to analyze and classify these non-generic points. Our analysis will include the questions concerning structural stability.

The presentation reports on the numerical simulation of a complex shock vortex interaction. In a schlieren animation we observe the diffraction of a shock wave in air down a step and the formation of a larve vortex in the wake. The shock is reflected in the bottom to pass back through the vortex. During this interaction a chain of secondary vortices develops, which can only be observed numerically using high resolution schemes.

The numerical results presented in this talk are obtained using a new uniformly third order FV scheme based on logarithmic reconstruction. The talk presents joint work with Robert Artebrant at Lund University.

Let an mxn matrix A depend smoothly on a real parameter t \in ‹a,b›. The problem is to construct a smooth path of SVD's:

A(t) = U(t) S(t) V(t)T,

If A is a real analytic matrix function on ‹a,b›, the factorization is called Analytic Singular Value Decomposition (ASVD). For large sparse matrices A=A(t), our aim is to construct ASVD numerically. In particular, we will consider branches of p, p<n, selected singular values and the corresponding left/right singular vectors. We apply a predictor-corrector algorithm with an adaptive step size control.

The algorithm may get stuck at isolated points. It is possible to investigate these points as classical singularities of a mapping and apply techniques of the dimensional reduction and bifurcation analysis. This analysis can be exploited to improve the performance of the algorithm in a neighborhood of the singular point.

We consider a simple microscopic follow - the - leader model of N identical cars on a circular road. We extend the model in order to define overtaking of cars. Mathematically, the resulting model is a kind of a Filippov system.

The main objective is to study a long time behavior of the model. We observed several oscillatory patterns which apparently exhibit spatialtemporal symmetries. We will suggest classification and identification techniques.

The Adaptive Cross Approximation (ACA) method is an innovative technique used to construct a low rank approximation of a given matrix. In combination with clustering procedure, the ACA has proven to be an efficient tool when applied to boundary element matrices. Convergence of this method was established for Nystrom, callocation and, most recently, for Galerkin discretizations of BEM. The main advantage of the ACA is that it uses only a small portion of the matrix entries to construct the approximation.

The aim of this presentation is to explain the method and show its application examples in function interpolation and in Galerkin BEM for the Lame system.

Für elliptische und damit verwandte Randwertprobleme in allgemeinen Gebieten, welche nicht an irgendein Standardgitter angepasst sind, haben sich die so genannten Immersed Interface und ähnliche Methoden als effektiv herausgestellt. Eine der Stärken dieser Verfahren liegt in der sehr einfachen und schnellen Gittergenerierung, was insbesondere bei zeitabhängigen Prozessen einen wichtigen Vorteil darstellen kann. Die Einbettungsidee, welche allen Immersed Interface Methoden zu Grunde liegt, erlaubt es, verschiedene schnelle Löser auf "einfachen" Gebieten, wie z.B. Würfeln, einzusetzen.

Eines dieser Verfahren ist die so genannte Explicit Jump Immersed Interface Methode (EJIIM). In diesem Vortrag wird die Erweiterung von EJIIM für zweidimensionale Stokes Gleichungen eingeführt und erste numerische Ergebnisse präsentiert.