## 510 Mathematik

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Die vorliegende Arbeit beschäftigt sich mit der numerischen Behandlung Differential-Algebraischer Gleichungen (DAE" s). DAE" s treten beispielsweise bei der Modellierung der Dynamik mechanischer System, der Schaltkreissimulation sowie der chemischen Reaktionskinetik auf. Es werden Rosenbrock-Wanner ähnliche Verfahren zu deren Lösung hergeleitet und an technischen Modellen (Fahrzeugachse und Verstärker) getestet.

The numerical solution of implicit ordinary differential equations arising in vehicle dynamic
(1988)

Two Rosenbrock-Wanner type methods for the numerical treatment of differential-algebraic equations are presented. Both methods possess a stepsize control and an index-1 monitor. The first method DAE34 is of order (3)4 and uses a full semi-implicit Rosenbrock-Wanner scheme. The second method RKF4DA is derived from the Runge-Kutta-Fehlberg 4(5)-pair, where a semi-implicit Rosenbrock-Wanner method is embedded, in order to solve the nonlinear equations. The performance of both methods is discussed in artificial test problems and in technical applications.

Zur Perzentilberechnung
(1990)

Ein mathematisches Modell zur schiffahrtsbezogenen Wasserstandsvorhersage am Beispiel des Rheins
(1996)

River alarm systems are designed for the forecasting of water stages during floods or low flow conditions or the prediction of the transport of pollution plumes. The basic model equations are introduced and a Method of Lines approach for their numerical solution is discussed. The approach includes adaptive space-mesh strategies and a Rosenbrock-Wanner scheme for the time integration. It fits into a PC environment and fulfills the requirements on an implementation within river alarm systems.

Hydrological processes, like the water flow or transport of soluble substances in rivers, are usually described by partial differential equations (PDEs). The numerical solution of these equations offer various fields of important applications, e.g., the prediction of water stages during floods or low water periods (optimal load capacity of the ships) and the prediction of the fate of pollution plumes in case of an accident. The aim is to develop numerical schemes that are applicable to mostly all those simulations and well-tested with respect to their stability and efficiency. The method of lines (MOL)-approach was found to be a suitable numerical technique to solve the whole class of the discussed problems. The subject of this paper is the comparison of different numerical schemes for the space discretizations in the MOL-approach, in particular the ENO-discretizations which are found to be very useful even when sharp gradients occur.

Wissenschaftliches Rechnen
(1999)

Wissenschaftliches Rechnen
(1999)

A Method of Lines Flux-Difference Splitting Finite Volume Approach for 1D and 2D River Flow Problems
(2001)

The reliable forecasting of water levels is a very important issue. The modelling approach for water level forecasting at the Middle and Lower River Rhine is based on hydrodynamic river flow models coupled with precipitation-runoff models. The hydrodynamic model is defined by a numerical solution of the one-dimensional (1d) shallow water equations. If flood plains or flood risk maps are important, a two-dimensional (2d) model is required.
In this paper the usage of the Alcrudo-Garcia-Navarro scheme (Alcrudo and Garcia-Navarro, 1993) for 1d and 2d problems within the method of lines framework is described. The scheme is slightly modified, in order to allow a more accurate solution of problems with strong variations in the bottom topography. The problem of drying and re-wetting of mesh-cells is not yet fully sufficiently solved. Numerical results for some test problems and an application to a natural river will be presented.

The system for operational water level forecast and prediction of (fortunately not daily) pollutant transport for the river Rhine is in daily use. This model is based on the Saint-Venant or one-dimensional shallow water equations.
The model is augmented by additional terms and equations to model the effect of dead zones and the transport of soluble components.
The next step is to move towards two-dimensional models. An important problem that arises is that the domain of the fluid is not fixed by the given data, but depends on the water level and is therefore part of the solution of the model. Even worse, depending on the topography of the river bed, even the topology of the fluid domain may change, as islands may appear at low water and get flooded at high water situations.

Integrated modelling approaches for whole river catchments require the coupling of different types of models. As an example, river flow and forecast models in one- and two-space dimensions are discussed. Usually, these models are based on the hyperbolic shallow water equations and require special discretizations like ENO or Godunov-type methods.
The basic coupling mechanisms like coupling via source terms, via boundary conditions, via state variables and simulator coupling are introduced by examples. Their properties with respect to performance and accuracy requirements and implementation issues are presented.
If coupling conditions are considered, additional algebraic equations arise. By the method of lines approach it is possible to translate the partial differential equations and the algebraic equations into a large system of differential algebraic equations (DAEs). The DAEs can efficiently be solved if the special structure of the Jacobian of the coupled model components is taken into account.

For many practical problems an efficient solution of the one-dimensional shallow water equations (Saint-Venant equations) is important, especially when large networks of rivers, channels or pipes are considered. In order to test and develop numerical methods four test problems are formulated. These tests include the well known dam break and hydraulic jump problems and two steady state problems with varying channel bottom, channel width and friction.