# TU Darmstadt ULB TUbiblio

# Second-Order Implicit Methods for Conservation Laws with Applications in Water Supply Networks

## Abstract

In this thesis, we develop and analyse numerical methods for the simulation of water transport processes in networks. In this context, the possibility of combining such a method with adjoint-based optimization algorithms is of special importance. These algorithms are used in a simulation-based assistance system which computes energy-optimized operation plans of drinking water supply networks. In the first part, we develop and analyse suitable numerical methods to solve the so-called water hammer equations which describe the flow of water through pressurized pipes. From a mathematical point of view, the challenges are the hyperbolic character of this one-dimensional system on the one hand, and a possibly stiff source term modelling the friction effects on the other hand. For the time integration, we use so-called strong stability preserving (SSP) singly-diagonal implicit Runge-Kutta (SDIRK) methods. Such methods are advantageous with respect to their numerical implementation and further, they preserve the nonlinear stability which is an important property in the context of hyperbolic partial differential equations. Concerning hyperbolic equations, there are two important characteristic features which numerical methods need to display – being conservative and handling discontinuities and shocks. For this reason, we use Finite Volume and Discontinuous Galerkin methods for the spatial discretization. For the fully discrete schemes, which are combinations of the schemes mentioned above, we derive important properties: well-balancedness with respect to the water hammer equations and a discrete maximum principle. As a result, the numerical methods are able to exactly approximate the stationary state of the water hammer equations, which can be used to prove asymptotic stability. Further, the numerical solution which is computed by the methods lies in a certain range, which depends on the initial condition. All theoretical results are additionally verified by numerical tests. The results presented here were achieved within a project that aims to develop a simulation-based assistance system for drinking water supply. We therefore describe the structure of the entire system in the second part of the thesis. In particular, we take a closer look at the incorporated optimization module and the model equations for all network components. The assistance system is capable of successfully reducing the energy consumption of the whole network, which we demonstrate by two examples based on real data provided by our project partners.

Item Type: Ph.D. Thesis
Erschienen: 2018
Creators: Wagner, Lisa
Title: Second-Order Implicit Methods for Conservation Laws with Applications in Water Supply Networks
Language: English
Abstract:

In this thesis, we develop and analyse numerical methods for the simulation of water transport processes in networks. In this context, the possibility of combining such a method with adjoint-based optimization algorithms is of special importance. These algorithms are used in a simulation-based assistance system which computes energy-optimized operation plans of drinking water supply networks. In the first part, we develop and analyse suitable numerical methods to solve the so-called water hammer equations which describe the flow of water through pressurized pipes. From a mathematical point of view, the challenges are the hyperbolic character of this one-dimensional system on the one hand, and a possibly stiff source term modelling the friction effects on the other hand. For the time integration, we use so-called strong stability preserving (SSP) singly-diagonal implicit Runge-Kutta (SDIRK) methods. Such methods are advantageous with respect to their numerical implementation and further, they preserve the nonlinear stability which is an important property in the context of hyperbolic partial differential equations. Concerning hyperbolic equations, there are two important characteristic features which numerical methods need to display – being conservative and handling discontinuities and shocks. For this reason, we use Finite Volume and Discontinuous Galerkin methods for the spatial discretization. For the fully discrete schemes, which are combinations of the schemes mentioned above, we derive important properties: well-balancedness with respect to the water hammer equations and a discrete maximum principle. As a result, the numerical methods are able to exactly approximate the stationary state of the water hammer equations, which can be used to prove asymptotic stability. Further, the numerical solution which is computed by the methods lies in a certain range, which depends on the initial condition. All theoretical results are additionally verified by numerical tests. The results presented here were achieved within a project that aims to develop a simulation-based assistance system for drinking water supply. We therefore describe the structure of the entire system in the second part of the thesis. In particular, we take a closer look at the incorporated optimization module and the model equations for all network components. The assistance system is capable of successfully reducing the energy consumption of the whole network, which we demonstrate by two examples based on real data provided by our project partners.

Divisions: 04 Department of Mathematics
04 Department of Mathematics > Numerical Analysis and Scientific Computing
Date Deposited: 18 Mar 2018 20:55 View Item