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Rosenbrock–Wanner methods for systems of stiff ordinary differential equations are well known since the seventies. They have been continuously developed and are efficient for differential-algebraic equations of index-1, as well. Their disadvantage that the Jacobian matrix has to be updated in every time step becomes more and more obsolete when automatic differentiation is used. Especially the family of Rodas methods has proven to be a standard in the Julia package DifferentialEquations. However, the fifth-order Rodas5 method undergoes order reduction for certain problem classes. Therefore, the goal of this paper is to compute a new set of coefficients for Rodas5 such that this order reduction is reduced. The procedure is similar to the derivation of the methods Rodas4P and Rodas4P2. In addition, it is possible to provide new dense output formulas for Rodas5 and the new method Rodas5P. Numerical tests show that for higher accuracy requirements Rodas5P always belongs to the best methods within the Rodas family.
In this contribution, we perform computer simulations to expedite the development of hydrogen storages based on metal hydride. These simulations enable in-depth analysis of the processes within the systems which otherwise could not be achieved. That is, because the determination of crucial process properties require measurement instruments in the setup which are currently not available. Therefore, we investigate the reliability of reaction values that are determined by a design of experiments.
Specifically, we first explain our model setup in detail. We define the mathematical terms to obtain insights into the thermal processes and reaction kinetics. We then compare the simulated results to measurements of a 5-gram sample consisting of iron-titanium-manganese (FeTiMn) to obtain the values with the highest agreement with the experimental data. In addition, we improve the model by replacing the commonly used Van’t-Hoff equation by a mathematical expression of the pressure-composition-isotherms (PCI) to calculate the equilibrium pressure.
Finally, the parameters’ accuracy is checked in yet another with an existing metal hydride system. The simulated results demonstrate high concordance with experimental data, which advocate the usage of approximated kinetic reaction properties by a design of experiments for further design studies. Furthermore, we are able to determine process parameters like the entropy and enthalpy.
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 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.
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.
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".
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.