## 510 Mathematik

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Zur Perzentilberechnung
(1990)

Wissenschaftliches Rechnen
(1999)

Wissenschaftliches Rechnen
(1999)

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.

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.

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.

Simulating free-surface and pressurised flow is important to many fields of application, especially in network approaches. Modelling equations to describe flow behaviour arising in these problems are often expressed by one-dimensional formulations of the hyperbolic shallow water equations. One established approach to realise their numerical computation is the method of lines based on semi-discretisation in space (Steinebach and Rentrop, An adaptive method of lines approach for modeling flow and transport in rivers. In: Vande Wouwer, Saucez, Schiesser (eds) Adaptive method of lines, pp 181–205. Chapman & Hall/CRC, Boca Raton, London, New York, Washington, DC, 2001; Steinebach and Weiner, Appl Numer Math 62:1567–1578, 2012; Steinebach et al., Modeling and numerical simulation of pipe flow problems in water supply systems. In: Martin, Klamroth, et al. (eds) Mathematical optimization of water networks. International series of numerical mathematics, vol 162, pp 3–15. Springer, Basel, 2012). It leads to index-one DAE systems as algebraic constraints are required to realise coupling and boundary conditions of single reaches.Linearly implicit ROW schemes proved to be effective to solve these DAE systems (Steinebach and Rentrop, An adaptive method of lines approach for modeling flow and transport in rivers. In: Vande Wouwer, Saucez, Schiesser (eds) Adaptive method of lines, pp 181–205. Chapman & Hall/CRC, Boca Raton, London, New York, Washington, DC, 2001). However, under certain conditions an extended partial explicit time-integration of the shallow water equations could be worthwhile to save computational effort. To restrict implicit solution by ROW schemes to stiff components while using explicit solution by RK methods for remaining terms, we adapt ROW method ROS34PRW (Rang, J Comput Appl Math 262:105–114, 2014) to an AMF and IMEX combining approach (Hundsdorfer and Verwer, Numerical solution of time-dependent advection-diffusion-reaction equations. Springer, Berlin, Heidelberg, New York, 2003). Applied to first test problems regarding open channel flow, efficiency is analysed with respect to flow behaviour. Results prove to be advantageous especially concerning dynamical flow.

We derive rates of convergence for limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume–Emery–Griffith model. The theorems consist of scaling limits for the total spin. The model depends on the inverse temperature β and the interaction strength K. The rates of convergence results are obtained as (β,K) converges along appropriate sequences (βn,Kn) to points belonging to various subsets of the phase diagram which include a curve of second-order points and a tricritical point. We apply Stein’s method for normal and non-normal approximation avoiding the use of transforms and supplying bounds, such as those of Berry–Esseen quality, on approximation error.

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.
The space discretization of the partial differential equations is based on a finite volume approach with central WENO interpolation and local Lax–Friedrich fluxes (Kurganov and Levy, 2000) [7]. For time-integration new linearly-implicit two-step peer methods of orders three and four are developed. These methods are especially adapted to the usage within the method of lines framework. They show a good performance compared to the well-established methods like ode15s, radau5 or rodasp.

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.

In this chapter the simulation of a water supply system on a mesoscale abstraction level is considered. The water network consists of storage tanks, pipes, pumps and valves. It is operated by the characteristics of the water supplier, the consumer and the pumps. For all network elements the modeling equations are given. They include mass and momentum conservation for pressurized pipe flow. For their numerical solution the method of lines is proposed. The discretization in space is based on a finite volume approach together with a local Lax-Friedrich splitting and central WENO reconstruction. Boundary and coupling conditions are implemented as algebraic equations. This leads to a system of differential-algebraic equations in time which is solved by a special Rosenbrock method. The paper ends with some typical simulation results of the network.

The simulation of fluid flows is of importance to many fields of application, especially in industry and infrastructure. The modelling equations applied describe a coupled system of non-linear, hyperbolic partial differential equations given by one-dimensional shallow water equations that enable the consistent implementation of free surface flows in open channels as well as pressurised flows in closed pipes. The numerical realisation of these equations is complicated and challenging to date due to their characteristic properties that are able to cause discontinuous solutions.

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.

The lattice Boltzmann method is a modern approach to simulate fluid flow. In its original formulation, it is restricted to regular grids, second-order discretizations, and a unity CFL number. This paper describes our new off-lattice Boltzmann solver NATriuM, an extensible and parallel C++ code to perform lattice Boltzmann simulations on irregular grids. NATriuM also allows high-order spatial discretizations and non-unity CFL numbers to be used. We demonstrate how these features can efficiently decrease the number of grid points required in a simulation and thus reduce the computational time, compared to the standard lattice Boltzmann method. We detail the implementation of a recently proposed semi-Lagrangian lattice Boltzmann method and prove its efficiency in comparisons to other state-of-the-art off-lattice Boltzmann schemes.