## 510 Mathematik

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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.

Novel methods for contingency analysis of gas transport networks are presented. They are motivated by the transition of our energy system where hydrogen plays a growing role. The novel methods are based on a specific method for topological reduction and so-called supernodes. Stationary Euler equations with advanced compressor thermodynamics and a gas law allowing for gas compositions with up to 100% hydrogen are used. Several measures and plots support an intuitive comparison and analysis of the results. In particular, it is shown that the newly developed methods can estimate locations and magnitudes of additional capacities (injection, buffering, storage etc.) with a reasonable performance for networks of relevant composition and size.

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
(2022)

Solving transport network problems can be complicated by non-linear effects. In the particular case of gas transport networks, the most complex non-linear elements are compressors and their drives. They are described by a system of equations, composed of a piecewise linear ‘free’ model for the control logic and a non-linear ‘advanced’ model for calibrated characteristics of the compressor. For all element equations, certain stability criteria must be fulfilled, providing the absence of folds in associated system mapping. In this paper, we consider a transformation (warping) of a system from the space of calibration parameters to the space of transport variables, satisfying these criteria. The algorithm drastically improves stability of the network solver. Numerous tests on realistic networks show that nearly 100% convergence rate of the solver is achieved with this approach.

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)

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.

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.

Off-lattice Boltzmann methods increase the flexibility and applicability of lattice Boltzmann methods by decoupling the discretizations of time, space, and particle velocities. However, the velocity sets that are mostly used in off-lattice Boltzmann simulations were originally tailored to on-lattice Boltzmann methods. In this contribution, we show how the accuracy and efficiency of weakly and fully compressible semi-Lagrangian off-lattice Boltzmann simulations is increased by velocity sets derived from cubature rules, i.e. multivariate quadratures, which have not been produced by the Gauß-product rule. In particular, simulations of 2D shock-vortex interactions indicate that the cubature-derived degree-nine D2Q19 velocity set is capable to replace the Gauß-product rule-derived D2Q25. Likewise, the degree-five velocity sets D3Q13 and D3Q21, as well as a degree-seven D3V27 velocity set were successfully tested for 3D Taylor–Green vortex flows to challenge and surpass the quality of the customary D3Q27 velocity set. In compressible 3D Taylor–Green vortex flows with Mach numbers on-lattice simulations with velocity sets D3Q103 and D3V107 showed only limited stability, while the off-lattice degree-nine D3Q45 velocity set accurately reproduced the kinetic energy provided by literature.

Solving diﬀerential-algebraic equations (DAEs) eﬃciently by means of appropriate numerical schemes for time-integration is an ongoing topic in applied mathematics. In this context, especially when considering large systems that occur with respect to many ﬁelds of practical application eﬀective computation becomes relevant. In particular, corresponding examples are given when having to simulate network structures that consider transport of ﬂuid and gas or electrical circuits. Due to the stiﬀness properties of DAEs, time-integration of such problems generally demands for implicit strategies. Among the schemes that prove to be an adequate choice are linearly implicit Rung-Kutta methods in the form of Rosenbrock-Wanner (ROW) schemes. Compared to fully implicit methods, they are easy to implement and avoid the solution of non-linear equations by including Jacobian information within their formulation. However, Jacobian calculations are a costly operation. Hence, necessity of having to compute the exact Jacobian with every successful time-step proves to be a considerable drawback. To overcome this drawback, a ROW-type method is introduced that allows for non-exact Jacobian entries when solving semi-explicit DAEs of index one. The resulting scheme thus enables to exploit several strategies for saving computational eﬀort. Examples include using partial explicit integration of non-stiﬀ components, utilizing more advantageous sparse Jacobian structures or making use of time-lagged Jacobian information. In fact, due to the property of allowing for non-exact Jacobian expressions, the given scheme can be interpreted as a generalized ROW-type method for DAEs. This is because it covers many diﬀerent ROW-type schemes known from literature. To derive the order conditions of the ROW-type method introduced, a theory is developed that allows to identify occurring diﬀerentials and coeﬃcients graphically by means of rooted trees. Rooted trees for describing numerical methods were originally introduced by J.C. Butcher. They signiﬁcantly simplify the determination and deﬁnition of relevant characteristics because they allow for applying straightforward procedures. In fact, the theory presented combines strategies used to represent ROW-type methods with exact Jacobian for DAEs and ROW-type methods with non-exact Jacobian for ODEs. For this purpose, new types of vertices are considered in order to describe occurring non-exact elementary diﬀerentials completely. The resulting theory thus automatically comprises relevant approaches known from literature. As a consequence, it allows to recognize order conditions of familiar methods covered and to identify new conditions. With the theory developed, new sets of coeﬃcients are derived that allow to realize the ROW-type method introduced up to orders two and three. Some of them are constructed based on methods known from literature that satisfy additional conditions for the purpose of avoiding eﬀects of order reduction. It is shown that these methods can be improved by means of the new order conditions derived without having to increase the number of internal stages. Convergence of the resulting methods is analyzed with respect to several academic test problems. Results verify the theory determined and the order conditions found as only schemes satisfying the order conditions predicted preserve their order when using non-exact Jacobian expressions.

Network aggregation
(2020)

Demand forecast
(2020)

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
(2018)

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.

Solving differential-algebraic equations (DAEs) efficiently is an ongoing topic in applied mathematics. Applications are given with respect to many fields of practical interest, such as multiphysics problems or network simulations. Due to the stiffness properties of DAEs, linearly implicit Runge-Kutta methods in the form of Rosenbrock-Wanner (ROW) schemes are an appropriate choice for effecitive numerical time-integration. Compared to fully implicit schemes, they are easy to implement and avoid having to solve non-linear equations by including Jacobian information in their formulation explicity. But, especially when having to solve large coupled systems, computing the Jacobian is costly and proves to be a considerable drawback. Inspired by the works of Steihaug and Wolfbrandt [4], we introduce concepts to realize linearly-implicit Runge-Kutta methods for DAEs in the form of so-called W-methods. These schemes allow for arbitrary approximations to given Jacobian entries and, thus, for versatile strategies to reduce computational effort significantly when solving semi-explicit DAE problems of index-1. An approach extending Roche’s procedure [3] will be presented that enables to derive order conditions of the resulting methods by an algebraic theory using rooted trees, a strategy originally introduced by Butcher regarding Runge-Kutta schemes [1,2]. Besides, suitable sets of coefficients for implementing embedded schemes and their potential of increasing efficincy when solving DAEs will be demonstrated.