510 Mathematik
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This book discusses the development of the Rosenbrock—Wanner methods from the origins of the idea to current research with the stable and efficient numerical solution and differential-algebraic systems of equations, still in focus. The reader gets a comprehensive insight into the classical methods as well as into the development and properties of novel W-methods, two-step and exponential Rosenbrock methods. In addition, descriptive applications from the fields of water and hydrogen network simulation and visual computing are presented. (Verlagsangaben)
In this paper, an analysis of the error ellipsoid in the space of solutions of stationary gas transport problems is carried out. For this purpose, a Principal Component Analysis of the solution set has been performed. The presence of unstable directions is shown associated with the marginal fulfillment of the resistivity conditions for the equations of compressors and other control elements in gas networks. Practically, the instabilities occur when multiple compressors or regulators try to control pressures or flows in the same part of the network. Such problems can occur, in particular, when the compressors or regulators reach their working limits. Possible ways of resolving instabilities are considered.
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.
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(2020)
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.
This textbook contains and explains essential mathematical formulas within an economic context. A broad range of aids and supportive examples will help readers to understand the formulas and their practical applications. This mathematical formulary is presented in a practice-oriented, clear, and understandable manner, as it is needed for meaningful and relevant application in global business, as well as in the academic setting and economic practice.
The lattice Boltzmann method (LBM) is an efficient simulation technique for computational fluid mechanics and beyond. It is based on a simple stream-and-collide algorithm on Cartesian grids, which is easily compatible with modern machine learning architectures. While it is becoming increasingly clear that deep learning can provide a decisive stimulus for classical simulation techniques, recent studies have not addressed possible connections between machine learning and LBM. Here, we introduce Lettuce, a PyTorch-based LBM code with a threefold aim. Lettuce enables GPU accelerated calculations with minimal source code, facilitates rapid prototyping of LBM models, and enables integrating LBM simulations with PyTorch's deep learning and automatic differentiation facility. As a proof of concept for combining machine learning with the LBM, a neural collision model is developed, trained on a doubly periodic shear layer and then transferred to a different flow, a decaying turbulence. We also exemplify the added benefit of PyTorch's automatic differentiation framework in flow control and optimization. To this end, the spectrum of a forced isotropic turbulence is maintained without further constraining the velocity field.
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.