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