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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.
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).
A new type of Rosenbrock-Wanner (ROW) methods for solving semi-explicit DAEs of index-1 is introduced. The scheme considers arbitrary approximations to Jacobian entries resulting for the differential part and thus corresponds to a first attempt of applying W methods to DAEs. Besides, it is a generalized class covering many ROW-type methods known from literature. Order conditions are derived by a consistent approach that combines theories of ROW methods with exact Jacobian for DAEs (Roche, 1988) and W methods with arbitrary Jacobian for ODEs (Steihaug and Wolfbrandt, 1979). In this context, rooted trees based on Butcher’s theory that include a new type of vertices are used to describe non-exact differentials of the numerical solution. Resulting conditions up to order four are given explicitly, including new conditions for realizing schemes of higher order. Numerical tests emphasize the relevance of satisfying these conditions when solving DAEs together with approximations to Jacobian entries of the differential part.
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.
Since being introduced in the sixties and seventies, semi-implicit RosenbrockWanner (ROW) methods have become an important tool for the timeintegration of ODE and DAE problems. Over the years, these methods have been further developed in order to save computational effort by regarding approximations with respect to the given Jacobian [5], reduce effects of order reduction by introducing additional conditions [2, 4] or use advantages of partial explicit integration by considering underlying Runge-Kutta formulations [1]. As a consequence, there is a large number of different ROW-type schemes with characteristic properties for solving various problem formulations given in literature today.
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.
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.
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.
Die im Folgenden dargestellten wichtigsten Ergebnisse des Teilprojektes 5 "Mathematische Beschreibung der relevanten physikalischen Prozesse und numerische Simulation von Wasseraufbereitung und -verteilung" beziehen sich auf die Arbeitspakete 2 "Daten und Methoden zum Modellaufbau, zur Zustandsschätzung, Prognose und Bewertung" und 3 "Physikalische Modelle und Numerische Verfahren".
Neue technologische Entwicklungen basieren immer mehr auf einer
zunehmenden Mathematisierung, gerade in den Ingenieurwissenschaften.
Nicht erst seit PISA ist jedoch zu beobachten, dass sich das
belastbare mathematische Grundwissen vieler Studienanfänger in den letzten Jahren verringert hat.
Im vorliegenden Beitrag wird dieses Spannungsfeld, in dem sich die Ingenieurmathematik befindet, aus Sicht von Fachhochschuldozenten beschrieben. Ausgehend von den Ausbildungszielen der Ingenieurmathematik werden Anforderungen an die Schulmathematik abgeleitet.
Diese Anforderungen werden beispielhaft für die Einführung und den Umgang mit den mathematischen Objekten Zahlen, Terme, Gleichungen und Funktionen konkretisiert.
Ziel ist eine Sensibilisierung von Mathematiklehrerinnen und -lehrern, um ihre Schulabsolventinnen und -absolventen besser für ein zukünftiges ingenieurwissenschaftliches Studium zu rüsten.
Von Fluiden durchströmte Rohr- und Kanalnetzwerke spielen in vielen technischen Anwendungen eine zentrale Rolle. Die beschreibenden hyperbolischen Modellgleichungen basieren auf Erhaltungsgesetzen von Masse, Impuls und Energie. Dazu können Konvektions-Diffusions-Reaktionsgleichungen kommen, falls die Fluide Inhaltsstoffe transportieren und deren chemisch-biologische Reaktionen betrachtet werden. Für die verschiedenen Modellgleichungen wird ein einheitlicher numerischer Lösungsansatz vorgeschlagen. Die Ortsdiskretisierung erfolgt mit dem Kurganov-Levi Verfahren. Damit können Stoßwellen aufgelöst werden, ohne auf die Eigenstruktur der hyperbolischen Systeme zurück zu greifen. Je nach Anwendungsgebiet können dann unterschiedliche Verfahren zur Lösung der entstehenden Systeme gewöhnlicher oder differential-algebraischer Gleichungssysteme eingesetzt werden. Anhand von Testproblemen mit unstetigem Lösungsverlauf wird die Eignung der gewählten Diskretisierungsansätze demonstriert.
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 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.
