PhD Thesis

Wavelet-based Numerical Methods Adaptive Modelling Of Shallow Water Flows

PhD Awarded 2015

PhD Research - Wavelet-based Numerical Methods Adaptive Modelling Of Shallow Water Flows

Welcome! In my doctoral research, I explored a novel approach to modelling shallow water flows using wavelet theory. As you know, numerical models based on shallow water flow equations are essential tools for water resource engineers. They allow us to predict flow conditions in real-time, eliminating the need for expensive and time-consuming field measurements.

These models traditionally rely on classical discretization techniques, such as the finite difference method (FDM), finite volume method (FVM), or finite element method (FEM). However, the workhorse for shallow water flow simulations is often the finite volume method, particularly the Godunov-type methods. While these methods offer accuracy and conservation properties, they are limited to first-order accuracy and require dense meshes for complex problems.

Adaptive techniques have been introduced to maintain accuracy with fewer mesh cells. However, they can be cumbersome due to challenges like selecting error sensors, managing multiple user-defined parameters, and transferring data between different resolution scales.

High-order finite volume methods, on the other hand, utilize reconstruction procedures to recover point values from cell averages. This approach can compromise the locality, a key advantage of FVM methods. Discontinuous Galerkin (DG) methods offer an alternative by enabling high-order accuracy within a Godunov-type framework while preserving locality and conservation properties. However, DG methods come with a trade-off – they are computationally expensive due to a very restrictive CFL condition. This is where adaptive techniques, such as h-adaptation, p-adaptation, or a combination of both, become valuable for improving efficiency.

Recognizing the limitations of existing methods, my research aimed to develop new adaptive multi-resolution schemes. To achieve this, I incorporated the power of wavelet theory, specifically Haar wavelets and their generalization called multiwavelets, into the design of Godunov-type and DG methods, respectively.

In the following sections, we'll delve deeper into the background of this research, explore the concept of wavelets and multiwavelets, and then showcase the proposed adaptive techniques and the results obtained through various test cases.

Intrigued to learn more about the intricacies of wavelet-based adaptive modelling? The full dissertation is available for download, along with detailed explanations of the methodologies and the test cases employed.

QUALIFICATION DETAILS:

TITLE: Wavelet-based Numerical Methods Adaptive Modelling Of Shallow Water Flows

AUTHOR: Dilshad Abduljabbar Haleem [M.Sc.]

ISSN:

AWARDING BODY: Department of Civil and Structural Engineering, Faculty of Engineering, The University of Sheffield

CURRENT INSTITUTION: The University of Duhok

DATE AWARDED: 2015

Full Text Link: [PLEASE CLICK TO VIEW THE FULL TEXT OF MY PhD]

SUPERVISOR: Dr Georges Kesserwani, 2nd: Prof. Dr Harm Askes and 3rd: Dr Chris Keylock

SPONSOR: The University of Duhok and The University of Sheffield

QUALIFICATION NAME: Degree of Doctor of Philosophy in Civil and Structural Engineering

QUALIFICATION LEVEL: PhD

REPOSITORY LINK:

Citation to this work:

Haleem, DA, (2015), Wavelet-based Numerical Methods Adaptive Modelling Of Shallow Water Flows, PhD Thesis, University of Sheffield, Department of Civil and Structural Engineering, Faculty of Engineering [Access: https://etheses.whiterose.ac.uk/11759/2/D_haleem_thesis_civil_Eng.pdf]

Abstract:

Mesh adaptation techniques are commonly coupled with the numerical schemes in an attempt to improve the modelling efficiency and capturing of the different physical scales which are involved in the shallow water flow problems. This work designs an adaptive technique that avails from the wavelets theory for transforming the local single resolution information into multiresolution information in which these data information became accessible. The adaptivity of wavelets was first comprehensively tested via using an arbitrary function in which the spatial resolution adaptivity was achieved from the local solution itself, and it was based on a single user-prescribed parameter.

Secondly, the adaptive technique was combined with two standard numerical modelling schemes (i.e. finite volume and discontinuous Galerkin schemes) to produce two wavelet-based adaptive schemes. These schemes are designed for modelling one dimensional shallow water flows and are referred as the Haar wavelets finite volume (HWFV) and multiwavelet discontinuous Galerkin (MWDG) schemes. Both adaptive schemes were systematically tested using hydraulic test cases. The results demonstrated that the proposed adaptive technique could serve as a lucid foundation on which to construct holistic and smart adaptive schemes for simulating real shallow water flow.

Keywords:

Wavelet-based Modeling, Shallow Water Flows, Adaptive Discretization, Godunov-type Methods, Discontinuous Galerkin Methods

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