## 510 Mathematik

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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.

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.

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.

In this paper an overview on modelling techniques and numerical methods applied to problems in water network simulation is given. The considered applications cover river alarm systems (Rentrop and Steinebach, Surv Math Ind 6:245–265, 1997), water level forecast methods (Steinebach and Wilke, J CIWEM 14(1):39–44, 2000) up to sewer and water supply networks (Steinebach et al., Mathematical Optimization of Water Networks Martin. Springer, Basel, 2012).
The hyperbolic modelling equations are derived from mass and momentum conservation laws. A typical example are the well known Saint-Venant equations. For their numerical solution a conservative semi-discretisation in space by finite differences is proposed. A new well-balanced space discretisation scheme is presented which improves the local Lax-Friedrichs approach applied so far. Higher order discretisations are achieved by WENO methods (Kurganov and Levy, SIAM J Sci Comput 22(4):1461–1488, 2000).
Together with appropriate boundary and coupling conditions this method of lines approach leads to an index-one DAE system. Efficient solution of the DAE system is the topic of Jax and Steinebach (ROW methods adapted to network simulation for fluid flow, in preparation).

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.

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.

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.

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

Zur Perzentilberechnung
(1990)

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.

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.