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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.
Further development on globally convergent algorithms for solution of stationary network problems is presented. The algorithms make use of global non-degeneracy of Jacobi matrix of the system, composed of Kirchhoff's flow conservation conditions and transport element equations. This property is achieved under certain monotonicity conditions on element equations and guarantees an existence of a unique solution of the problem as well as convergence to this solution from an arbitrary starting point. In application to gas transport networks, these algorithms are supported by a proper modeling of gas compressors, based on individually calibrated physical characteristics. This paper extends the modeling of compressors by hierarchical methods of topological reduction, combining the working diagrams for parallel and sequential connections of compressors. Estimations are also made for application of topological reduction methods beyond the compressor stations in generic network problems. Efficiency of the methods is tested by numerical experiments on realistic networks.
The formulation of transport network problems is represented as a translation between two domain specific languages: from a network description language, used by network simulation community, to a problem description language, understood by generic non-linear solvers. A universal algorithm for this translation is developed, an estimation of its computational complexity given, and an efficient application of the algorithm demonstrated on a number of realistic examples. Typically, for a large gas transport network with about 10K elements the translation and solution of non-linear system together require less than 1 sec on the common hardware. The translation procedure incorporates several preprocessing filters, in particular, topological cleaning filters, which accelerate the solution procedure by factor 8.
Network aggregation
(2020)
Demand forecast
(2020)
