TY - INPR A1 - Stolwijk, Jeroen J. A1 - Mehrmann, Volker T1 - Error Analysis and Model Adaptivity for Flows in Gas Networks N2 - In the simulation and optimization of gas flow in a pipeline network, a hierarchy of models is used that employs different formulations of the Euler equations. While the optimization is performed on piecewise linear models, the flow simulation is based on the simulation of one to three dimensional Euler equations including the temperature distributions. To decide which model class in the hierarchy is adequate to achieve a desired accuracy, this paper presents an error and perturbation analysis for a two level model hierarchy including the isothermal Euler equations in semilinear form and the stationary Euler equations in purely algebraic form. The focus of the work is on the effect of data uncertainty, discretization and rounding errors in the numerical simulation of these models and their interaction. Two simple discretization schemes for the semilinear model are compared with respect to their conditioning and temporal stepsizes are determined for which a well-conditioned problem is obtained. The results are based on new componentwise relative condition numbers for the solution of nonlinear systems of equations. Moreover, the model error between the semilinear and the algebraic model is computed, the maximum pipeline length is determined for which the algebraic model can be used safely, and a condition is derived for which the isothermal model is adequate. KW - gas network KW - isothermal Euler equations KW - error analysis KW - condition number KW - data uncertainty Y1 - 2017 ER - TY - INPR A1 - Mehrmann, Volker A1 - Stolwijk, Jeroen J. T1 - Error Analysis for the Euler Equations in Purely Algebraic Form N2 - The presented work contains both a theoretical and a statistical error analysis for the Euler equations in purely algebraic form, also called the Weymouth equations or the temperature dependent algebraic model. These equations are obtained by performing several simplifications of the full Euler equations, which model the gas flow through a pipeline. The theoretical analysis is executed by first calculating the backward error and then the individual relative condition numbers. This error analysis results in a statement about the maximum pipeline length such that the algebraic model can be used safely. The statistical analysis is performed using both a Monte Carlo Simulation and the Univariate Reduced Quadrature Method and is used to illustrate and confirm the obtained theoretical results. KW - error analysis KW - measurement error KW - condition number KW - backward error KW - statistical analysis Y1 - 2015 ER - TY - JOUR A1 - Domschke, Pia A1 - Dua, Aseem A1 - Stolwijk, Jeroen J. A1 - Lang, Jens A1 - Mehrmann, Volker T1 - Adaptive Refinement Strategies for the Simulation of Gas Flow in Networks using a Model Hierarchy N2 - A model hierarchy that is based on the one-dimensional isothermal Euler equations of fluid dynamics is used for the simulation and optimisation of gas flow through a pipeline network. Adaptive refinement strategies have the aim of bringing the simulation error below a prescribed tolerance while keeping the computational costs low. While spatial and temporal stepsize adaptivity is well studied in the literature, model adaptivity is a new field of research. The problem of finding an optimal refinement strategy that combines these three types of adaptivity is a generalisation of the unbounded knapsack problem. A refinement strategy that is currently used in gas flow simulation software is compared to two novel greedy-like strategies. Both a theoretical experiment and a realistic gas flow simulation show that the novel strategies significantly outperform the current refinement strategy with respect to the computational cost incurred. KW - gas supply networks KW - model hierarchy KW - error estimators KW - model adaptivity KW - refinement strategies Y1 - 2017 U6 - https://doi.org/10.1553/etna_vol48s97 VL - Electronic Transactions on Numerical Analysis IS - Vol. 48 SP - 97 EP - 113 ER - TY - JOUR A1 - Mehrmann, Volker A1 - Schmidt, Martin A1 - Stolwijk, Jeroen J. T1 - Model and Discretization Error Adaptivity within Stationary Gas Transport Optimization JF - Vietnam Journal of Mathematics N2 - The minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and isothermal gas flows. The error control is realized by switching to more detailed models or finer discretizations if necessary to guarantee that a prescribed model and discretization error tolerance is satisfied in the end. We prove convergence of the adaptively controlled optimization method and illustrate the new approach with numerical examples. KW - Gas network optimization KW - Isothermal stationary Euler equations KW - Model hierarchy KW - Adaptive error control KW - Marking strategy Y1 - 2017 IS - 46(4) SP - 779 EP - 801 ER -