# American Institute of Mathematical Sciences

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December  2018, 13(4): 641-661. doi: 10.3934/nhm.2018029

## Optimal model switching for gas flow in pipe networks

 1 Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Lehrstuhl Angewandte Mathematik Ⅱ, Cauerstr. 11, 91058 Erlangen, Germany 2 Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany

* Corresponding author

Received  January 2018 Revised  August 2018 Published  November 2018

We consider model adaptivity for gas flow in pipeline networks. For each instant in time and for each pipe in the network a model for the gas flow is to be selected from a hierarchy of models in order to maximize a performance index that balances model accuracy and computational cost for a simulation of the entire network. This combinatorial problem involving partial differential equations is posed as an optimal switching control problem for abstract semilinear evolutions. We provide a theoretical and numerical framework for solving this problem using a two stage gradient descent approach based on switching time and mode insertion gradients. A numerical study demonstrates the practicability of the approach.

Citation: Fabian Rüffler, Volker Mehrmann, Falk M. Hante. Optimal model switching for gas flow in pipe networks. Networks & Heterogeneous Media, 2018, 13 (4) : 641-661. doi: 10.3934/nhm.2018029
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A gas network with a supply node $N_1$ and two costumer nodes $N_2$ and $N_3$
Snapshot of the fully simulated solution showing density (solid, blue) and flux (dashed, red, scaled by $0.05$). On the outer pipes 1 to 5 we see a lot of fluctuation due to the oscillatory boundary flows. The pipes 6 to 9 of the inner circle, however, remain nearly constant
(A): resulting optimized switching sequence showing, for each time step from $t_0 = 0\ \text{s}$ to $T = 1800\ \text{s}$ and each edge $e_1, \ldots, e_{10}$, if the solution is calculated with the fine model (white) or frozen (black). (B), (C): filtered results with two different filters. (D): $L^2$-error relative to maximum values of the solution $\bar{z}$ corresponding to freezing edges $6$ to $10$ completely
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