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

Wagner L (2018)


Publication Language: English

Publication Type: Thesis

Publication year: 2018

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 energyoptimized
operation plans of drinking water supply networks.
In the first part, we develop and analyse suitable numerical methods to solve the socalled
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.

How to cite

APA:

Wagner, L. (2018). Second-Order Implicit Methods for Conservation Laws with Applications in Water Supply Networks (Dissertation).

MLA:

Wagner, Lisa. Second-Order Implicit Methods for Conservation Laws with Applications in Water Supply Networks. Dissertation, 2018.

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