Abstract
In this chapter we consider the solution of the model equations of water supply networks and continuous optimal control tasks. We begin with the description of our simulation tool in Sect. 2.1, in particular the numerical treatment of the water hammer equations. This includes the description of the implemented discretization scheme together with a stability and convergence analysis. As we will see, the applied scheme perfectly matches with the properties of the water hammer equations and thus builds a useful foundation for the solution of the entire model equations as well as optimal control tasks.
In Sect. 2.2 we consider the computation of sensitivity information, which is necessary for the application of gradient-based optimization techniques. Here, we follow a first-discretize approach to derive adjoint equations. Due to the special structure of the considered problems, very efficient algorithms can be applied.
Finally, Sect. 2.3 deals with the problem of singularities in the model equations of water supply networks. Here, a physically motivated regularization approach is applied and also extended to be applicable in an adjoint calculus.
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Kolb, O., Lang, J. (2012). Simulation and Continuous Optimization. In: Martin, A., et al. Mathematical Optimization of Water Networks. International Series of Numerical Mathematics, vol 162. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0436-3_2
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DOI: https://doi.org/10.1007/978-3-0348-0436-3_2
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