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After introducing the waveform relaxation (WR) concept, the authors briefly describe how WR was implemented in the simulator SISAL. They analyze a refinement of WR called waveform relaxation Newton (WRN). They reveal the interrelation of both methods and introduce an improved implementation technique of the WRN algorithm. Numerical results are reported.

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.

The paper describes methods for calculating chemical equilibria based on a constrained Gibbs free energy minimization. The methods allow the treatment of multicomponent systems with multiple phases, including gaseous phases, condensed phases, and stoichiometric phases. A special aspect is the detection and treatment of miscibility gaps. The underlying mathematical problem is described in detail together with the algorithmic approach for its solution. Results are presented for some test cases, including the computation of phase diagrams for ternary systems.

The paper presents the topological reduction method applied to gas transport networks, using contraction of series, parallel and tree-like subgraphs. The contraction operations are implemented for pipe elements, described by quadratic friction law. This allows significant reduction of the graphs and acceleration of solution procedure for stationary network problems. The algorithm has been tested on several realistic network examples. The possible extensions of the method to different friction laws and other elements are discussed.