phd thesis numerical methods

TopSCHOLAR®

• < Previous

Home > GRADUATE > THESES > 2053

Masters Theses & Specialist Projects

Analysis and implementation of numerical methods for solving ordinary differential equations.

Muhammad Sohel Rana , Western Kentucky University Follow

Publication Date

Advisor(s) - committee chair.

Dr. Mark Robinson (Director), Dr. Ferhan Atici and Dr. Ngoc Nguyen

Degree Program

Department of Mathematics

Degree Type

Master of Science

Numerical methods to solve initial value problems of differential equations progressed quite a bit in the last century. We give a brief summary of how useful numerical methods are for ordinary differential equations of first and higher order. In this thesis both computational and theoretical discussion of the application of numerical methods on differential equations takes place. The thesis consists of an investigation of various categories of numerical methods for the solution of ordinary differential equations including the numerical solution of ordinary differential equations from a number of practical fields such as equations arising in population dynamics and astrophysics. It includes discussion what are the advantages and disadvantages of implicit methods over explicit methods, the accuracy and stability of methods and how the order of various methods can be approximated numerically. Also, semidiscretization of some partial differential equations and stiff systems which may arise from these semidiscretizations are examined.

• Disciplines

Numerical Analysis and Computation | Ordinary Differential Equations and Applied Dynamics | Partial Differential Equations

Recommended Citation

Rana, Muhammad Sohel, "Analysis and Implementation of Numerical Methods for Solving Ordinary Differential Equations" (2017). Masters Theses & Specialist Projects. Paper 2053. https://digitalcommons.wku.edu/theses/2053

Since November 27, 2017

Included in

Numerical Analysis and Computation Commons , Ordinary Differential Equations and Applied Dynamics Commons , Partial Differential Equations Commons

Advanced Search

• Notify me via email or RSS
• Colleges, Departments, Units
• WKU Journals/Peer-Reviewed Series
• Conferences and Events

Author Corner

• Author Submissions
• TopSCHOLAR copyright form
• WKU Libraries
• WKU Homepage
• WKU Libraries OA Hall of Fame
• Kentucky Research Commons
• Digital Commons Repositories

Home | About | FAQ | My Account | Accessibility Statement

Privacy Copyright

Academia.edu no longer supports Internet Explorer.

To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to  upgrade your browser .

Enter the email address you signed up with and we'll email you a reset link.

• We're Hiring!
• Help Center

Numerical Methods for Hyperbolic PDE

Various Numerical techniques for solving the Hyperbolic Partial Differential Equations(PDE) in one space dimension are discussed. The advection-diffusion equation with constant coefficient is chosen as a model problem to introduce, analyze and compare numerical techniques used. These numerical techniques can then be generalized to nonlinear equations and even systems of equations. Starting with the Upwind and the Lax-Wendroff schemes for the advection equation. These techniques are based on the two-level finite difference approximations. Next we investigate dissipation and dispersion of a numerical scheme. An alternative description of many numerical methods is based on the advection-diffusion equation. The Godunov method(Finite Volume) is generalization of the upwind scheme to nonlinear equations. The results of a numerical experiment are presented, and their consistency, stability, convergence and accuracy are discussed and compared.

Related Papers

Hagos Hailu

In this PhD thesis, we construct numerical methods to solve problems described by advectiondiffusion and convective Cahn-Hilliard equations. The advection-diffusion equation models a variety of physical phenomena in fluid dynamics, heat transfer and mass transfer or alternatively describing a stochastically-changing system. The convective Cahn-Hilliard equation is an equation of mathematical physics which describes several physical phenomena such as spinodal decomposition of phase separating systems in the presence of an external field and phase transition in binary liquid mixtures (Golovin et al., 2001; Podolny et al., 2005). In chapter 1, we define some concepts that are required to study some properties of numerical methods. In chapter 2, three numerical methods have been used to solve two problems described by 1D advection-diffusion equation with specified initial and boundary conditions. The methods used are the third order upwind scheme (Dehghan, 2005), fourth order scheme (De...

Advection-diffusion equation with constant and variable coefficients has a wide range of practical and industrial applications. Due to the importance of advection-diffusion equation the present paper, solves and analyzes these problems using a new finite difference equation as well as a numerical scheme. The developed scheme is based on a mathematical combination between Siemieniuch and Gradwell approximation for time and Dehghan's approximation for spatial variable. In the proposed scheme a special discretization for the spatial variable is made in such away that when applying the finite difference equation at any time level (j + 1) two nodes from both ends of the domain are left. After that the unknowns at the two nodes adjacent to the boundaries are obtained from the interpolation technique. The results are compared with some available analytical solutions and show a good agreement.

International Journal for Numerical Methods in Fluids

Jose Alberto Cuminato

Longe I Oluwaseun

Advances in Water Resources

Roger A Falconer - Cardiff University

Abstract This paper discusses and compares the spatial accuracy of the QUICK finite difference scheme and the third-order convection, second-order diffusion (TCSD) scheme for the severe case with pure advection only. It is shown that the QUICK scheme is generally second-order accurate in space and that a general explicit finite difference representation of various upwind difference schemes is numerically unstable for the severest transport case. Various modified forms of the implicit TCSD scheme are presented with their numerical stability properties being studied and analysed. These modified schemes have been applied to an idealised one-dimensional test basin for the cases of pure advection and of advection and diffusion using three different initial-boundary conditions, including: (a) a sharp front concentration gradient; (b) a Gaussian concentration distribution; and (c) a plug source. A two-dimensional version of the modified TCSD scheme has also been formulated based upon the standard ADI technique and has been applied to three two-dimensional test cases, including pure advection for a circular column and a Gaussian distribution, and advection with diffusion for a Gaussian distribution. Both uniform and rotational flow fields were used for these two-dimensional tests. The scheme has been shown to be attractive, with details of the comparisons between these modified schemes and other similar higher order schemes together with the corresponding analytical solutions also being included in the paper.

Clifford Pinto

Mehrdad Manteghian

The Several numerical techniques have been developed and compared for solving the one- dimensional advection-diffusion equation with constant coefficients. These techniques are based on the finite difference methods (FDM). By changing the values of temporal and spatial weighted parameters, solutions are obtained for both explicit and implicit techniques such as FTCS, FTBSCS, BTCS, BTBSCS and Crank-Nicholson schemes. Numerical solution is given for two special cases which have been dealt with in the literature and for which an analytical solution has been provided. Comparison of the results has confirmed that the Crank-Nicholson numerical approach matches successfully with the analytical solution while the other techniques result in some levels of discrepancy.

Ndivhuwo Mphephu

Cidália Neves

Tese de doutoramento em Matematica, na especialidade de Metematica Aplicada, apresentada ao Departamento de Matematica da Faculdade de Ciencias e Tecnologia da Universidade de Coimbra

International Journal of Applied Mathematics and Theoretical Physics

KEDIR A L I Y I KOROCHE

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

RELATED PAPERS

American Scientific Research Journal for Engineering, Technology, and Sciences

Tsegaye Simon

Archives of Computational Methods in Engineering

Raghurama Rao, V. Suswaram , mayank bajpayi

MATHEMATICS OF COMPUTATION

Hillel Tal-Ezer

Abdollah Afjeh

Khalid Moinuddin

Journal of Computational and Applied Mathematics

Francisco Ureña Prieto

vijitha mukundan , Ashish Awasthi

Mathematics of Computation

Advances in Engineering Software

Halil Karahan

Numerical Methods for Partial Differential Equations

Mahaveer Jain

Raghurama Rao, V. Suswaram

Communications in Applied Mathematics and Computational Science

Ann Almgren

Journal of Computational Physics

Yong-Tao Zhang

Adrian Sescu

Ignasi Colominas

SIAM Journal on Scientific Computing

Robert C Sharpley

Siam Journal on Scientific Computing

Ángel García Gómez

Alexander Churbanov

AAAA4 TORRES

RELATED TOPICS

•   We're Hiring!
•   Help Center
• Find new research papers in:
• Health Sciences
• Earth Sciences
• Cognitive Science
• Mathematics
• Computer Science
• Academia ©2024

Department of Mathematics and Computer Science

Service navigation.

• Privacy Policy
• Accessibility Statement
• DE: Deutsch
• EN: English
• Numerical methods for PDEs and numerical software

Path Navigation

• Mathematics

Zhang-Schneider, Xingjian: Fast solvers for heterogeneous saddle point problems arising from phasefield-models

Hinrichsen, lasse: adaptive discontinuous galerkin methods for nonsmooth problems.

Numerical Methods in Scientific Computing

This is a book about numerically solving partial differential equations occurring in technical and physical contexts and the authors have set themselves a more ambitious target than to just talk about the numerics. Their aim is to show the place of numerical solutions in the general modeling process and this must inevitably lead to considerations about modeling itself. Partial differential equations usually are a consequence of applying first principles to a technical or physical problem at hand. That means, that most of the time the physics also have to be taken into account especially for validation of the numerical solution obtained. This book aims especially at engineers and scientists who have ’real world’ problems. It will concern itself less with pesky mathematical detail. For the interested reader though, we have included sections on mathematical theory to provide the necessary mathematical background. Since this treatment had to be on the superficial side we have provided further reference to the literature where necessary.

Author Biographies

Jos van Kan (1944) graduated in 1968 from Delft University of Technology, Delft, Netherlands, in Numerical Analysis and was assistant professor at the Department of Mathematics of that institute until 2009. He wrote several articles on Numerical Fluid Mechanics (pressure correction methods) and has written a multigrid pressure solver for the Delft software package to solve the Navier-Stokes equations. He was teaching classes in Numerical Analysis from 1971 until 2009, and wrote several books on the subject. Currently he is a retired professor.

Guus Segal (1948) graduated in 1971 from Delft University of Technology, Delft, Netherlands, in Numerical Analysis and was part time assistant professor at the Department of Mathematics of that institute until 2013. He also worked in the consultancy and numerical software company SEPRA in The Hague, Netherlands. He wrote a number of articles on Finite Element Methods and several articles on curvilinear Finite Volume Methods and Numerical Fluid Mechanics. He has written a book on Finite Element methods and Navier-Stokes equations. He is the main developer of the finite element package SEPRAN. He was teaching classes in Numerical Analysis from 1973 until 2013.

Fred Vermolen (1969) graduated in 1993 from Delft University of Technology, Delft, Netherlands and defended his PhD thesis on numerical methods for moving boundary problems in 1998. He has written several contributions on Stefan problems, computational mechanics, mathematical analysis and uncertainty quantification with most of the applications in medicine. He has held an assistant and associate professorship in Numerical Analysis at the Delft University from 2000 until 2020. In 2020 he started his current position as a full professor in Computational Mathematics at the University of Hasselt in Belgium.

This work is licensed under a Creative Commons Attribution 4.0 International License .

Details about the available publication format: Download PDF

Isbn-13 (15).

We use cookies on reading.ac.uk to improve your experience, monitor site performance and tailor content to you

Read our cookie policy to find out how to manage your cookie settings

This site may not work correctly on Internet Explorer. We recommend switching to a different browser for a better experience.

Mathematics PhD theses

A selection of Mathematics PhD thesis titles is listed below, some of which are available online:

2023   2022   2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Reham Alahmadi - Asymptotic Study of Toeplitz Determinants with Fisher-Hartwig Symbols and Their Double-Scaling Limits

Anne Sophie Rojahn –  Localised adaptive Particle Filters for large scale operational NWP model

Melanie Kobras –  Low order models of storm track variability

Ed Clark –  Vectorial Variational Problems in L∞ and Applications to Data Assimilation

Katerina Christou – Modelling PDEs in Population Dynamics using Fixed and Moving Meshes  

Chiara Cecilia Maiocchi –  Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems

Samuel R Harrison – Stalactite Inspired Thin Film Flow

Elena Saggioro – Causal network approaches for the study of sub-seasonal to seasonal variability and predictability

Cathie A Wells – Reformulating aircraft routing algorithms to reduce fuel burn and thus CO 2 emissions  

Jennifer E. Israelsson –  The spatial statistical distribution for multiple rainfall intensities over Ghana

Giulia Carigi –  Ergodic properties and response theory for a stochastic two-layer model of geophysical fluid dynamics

André Macedo –  Local-global principles for norms

Tsz Yan Leung  –  Weather Predictability: Some Theoretical Considerations

Jehan Alswaihli –  Iteration of Inverse Problems and Data Assimilation Techniques for Neural Field Equations

Jemima M Tabeart –  On the treatment of correlated observation errors in data assimilation

Chris Davies –  Computer Simulation Studies of Dynamics and Self-Assembly Behaviour of Charged Polymer Systems

Birzhan Ayanbayev –  Some Problems in Vectorial Calculus of Variations in L∞

Penpark Sirimark –  Mathematical Modelling of Liquid Transport in Porous Materials at Low Levels of Saturation

Adam Barker –  Path Properties of Levy Processes

Hasen Mekki Öztürk –  Spectra of Indefinite Linear Operator Pencils

Carlo Cafaro –  Information gain that convective-scale models bring to probabilistic weather forecasts

Nicola Thorn –  The boundedness and spectral properties of multiplicative Toeplitz operators

James Jackaman  – Finite element methods as geometric structure preserving algorithms

Changqiong Wang - Applications of Monte Carlo Methods in Studying Polymer Dynamics

Jack Kirk - The molecular dynamics and rheology of polymer melts near the flat surface

Hussien Ali Hussien Abugirda - Linear and Nonlinear Non-Divergence Elliptic Systems of Partial Differential Equations

Andrew Gibbs - Numerical methods for high frequency scattering by multiple obstacles (PDF-2.63MB)

Mohammad Al Azah - Fast Evaluation of Special Functions by the Modified Trapezium Rule (PDF-913KB)

Katarzyna (Kasia) Kozlowska - Riemann-Hilbert Problems and their applications in mathematical physics (PDF-1.16MB)

Anna Watkins - A Moving Mesh Finite Element Method and its Application to Population Dynamics (PDF-2.46MB)

Niall Arthurs - An Investigation of Conservative Moving-Mesh Methods for Conservation Laws (PDF-1.1MB)

Samuel Groth - Numerical and asymptotic methods for scattering by penetrable obstacles (PDF-6.29MB)

Katherine E. Howes - Accounting for Model Error in Four-Dimensional Variational Data Assimilation (PDF-2.69MB)

Jian Zhu - Multiscale Computer Simulation Studies of Entangled Branched Polymers (PDF-1.69MB)

Tommy Liu - Stochastic Resonance for a Model with Two Pathways (PDF-11.4MB)

Matthew Paul Edgington - Mathematical modelling of bacterial chemotaxis signalling pathways (PDF-9.04MB)

Anne Reinarz - Sparse space-time boundary element methods for the heat equation (PDF-1.39MB)

Adam El-Said - Conditioning of the Weak-Constraint Variational Data Assimilation Problem for Numerical Weather Prediction (PDF-2.64MB)

Nicholas Bird - A Moving-Mesh Method for High Order Nonlinear Diffusion (PDF-1.30MB)

Charlotta Jasmine Howarth - New generation finite element methods for forward seismic modelling (PDF-5,52MB)

Aldo Rota - From the classical moment problem to the realizability problem on basic semi-algebraic sets of generalized functions (PDF-1.0MB)

Sarah Lianne Cole - Truncation Error Estimates for Mesh Refinement in Lagrangian Hydrocodes (PDF-2.84MB)

Alexander J. F. Moodey - Instability and Regularization for Data Assimilation (PDF-1.32MB)

Dale Partridge - Numerical Modelling of Glaciers: Moving Meshes and Data Assimilation (PDF-3.19MB)

Joanne A. Waller - Using Observations at Different Spatial Scales in Data Assimilation for Environmental Prediction (PDF-6.75MB)

Faez Ali AL-Maamori - Theory and Examples of Generalised Prime Systems (PDF-503KB)

Mark Parsons - Mathematical Modelling of Evolving Networks

Natalie L.H. Lowery - Classification methods for an ill-posed reconstruction with an application to fuel cell monitoring

David Gilbert - Analysis of large-scale atmospheric flows

Peter Spence - Free and Moving Boundary Problems in Ion Beam Dynamics (PDF-5MB)

Timothy S. Palmer - Modelling a single polymer entanglement (PDF-5.02MB)

Mohamad Shukor Talib - Dynamics of Entangled Polymer Chain in a Grid of Obstacles (PDF-2.49MB)

Cassandra A.J. Moran - Wave scattering by harbours and offshore structures

Ashley Twigger - Boundary element methods for high frequency scattering

David A. Smith - Spectral theory of ordinary and partial linear differential operators on finite intervals (PDF-1.05MB)

Stephen A. Haben - Conditioning and Preconditioning of the Minimisation Problem in Variational Data Assimilation (PDF-3.51MB)

Jing Cao - Molecular dynamics study of polymer melts (PDF-3.98MB)

Bonhi Bhattacharya - Mathematical Modelling of Low Density Lipoprotein Metabolism. Intracellular Cholesterol Regulation (PDF-4.06MB)

Tamsin E. Lee - Modelling time-dependent partial differential equations using a moving mesh approach based on conservation (PDF-2.17MB)

Polly J. Smith - Joint state and parameter estimation using data assimilation with application to morphodynamic modelling (PDF-3Mb)

Corinna Burkard - Three-dimensional Scattering Problems with applications to Optical Security Devices (PDF-1.85Mb)

Laura M. Stewart - Correlated observation errors in data assimilation (PDF-4.07MB)

R.D. Giddings - Mesh Movement via Optimal Transportation (PDF-29.1MbB)

G.M. Baxter - 4D-Var for high resolution, nested models with a range of scales (PDF-1.06MB)

C. Spencer - A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table.

P. Jelfs - A C-property satisfying RKDG Scheme with Application to the Morphodynamic Equations (PDF-11.7MB)

L. Bennetts - Wave scattering by ice sheets of varying thickness

M. Preston - Boundary Integral Equations method for 3-D water waves

J. Percival - Displacement Assimilation for Ocean Models (PDF - 7.70MB)

D. Katz - The Application of PV-based Control Variable Transformations in Variational Data Assimilation (PDF- 1.75MB)

S. Pimentel - Estimation of the Diurnal Variability of sea surface temperatures using numerical modelling and the assimilation of satellite observations (PDF-5.9MB)

J.M. Morrell - A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian method for the numerical solution of the Euler equations (PDF-7.7MB)

L. Watkinson - Four dimensional variational data assimilation for Hamiltonian problems

M. Hunt - Unique extension of atomic functionals of JB*-Triples

D. Chilton - An alternative approach to the analysis of two-point boundary value problems for linear evolutionary PDEs and applications

T.H.A. Frame - Methods of targeting observations for the improvement of weather forecast skill

C. Hughes - On the topographical scattering and near-trapping of water waves

B.V. Wells - A moving mesh finite element method for the numerical solution of partial differential equations and systems

D.A. Bailey - A ghost fluid, finite volume continuous rezone/remap Eulerian method for time-dependent compressible Euler flows

M. Henderson - Extending the edge-colouring of graphs

K. Allen - The propagation of large scale sediment structures in closed channels

D. Cariolaro - The 1-Factorization problem and same related conjectures

A.C.P. Steptoe - Extreme functionals and Stone-Weierstrass theory of inner ideals in JB*-Triples

D.E. Brown - Preconditioners for inhomogeneous anisotropic problems with spherical geometry in ocean modelling

S.J. Fletcher - High Order Balance Conditions using Hamiltonian Dynamics for Numerical Weather Prediction

C. Johnson - Information Content of Observations in Variational Data Assimilation

M.A. Wakefield - Bounds on Quantities of Physical Interest

M. Johnson - Some problems on graphs and designs

A.C. Lemos - Numerical Methods for Singular Differential Equations Arising from Steady Flows in Channels and Ducts

R.K. Lashley - Automatic Generation of Accurate Advection Schemes on Structured Grids and their Application to Meteorological Problems

J.V. Morgan - Numerical Methods for Macroscopic Traffic Models

M.A. Wlasak - The Examination of Balanced and Unbalanced Flow using Potential Vorticity in Atmospheric Modelling

M. Martin - Data Assimilation in Ocean circulation models with systematic errors

K.W. Blake - Moving Mesh Methods for Non-Linear Parabolic Partial Differential Equations

J. Hudson - Numerical Techniques for Morphodynamic Modelling

A.S. Lawless - Development of linear models for data assimilation in numerical weather prediction .

C.J.Smith - The semi lagrangian method in atmospheric modelling

T.C. Johnson - Implicit Numerical Schemes for Transcritical Shallow Water Flow

M.J. Hoyle - Some Approximations to Water Wave Motion over Topography.

P. Samuels - An Account of Research into an Area of Analytical Fluid Mechnaics. Volume II. Some mathematical Proofs of Property u of the Weak End of Shocks.

M.J. Martin - Data Assimulation in Ocean Circulation with Systematic Errors

P. Sims - Interface Tracking using Lagrangian Eulerian Methods.

P. Macabe - The Mathematical Analysis of a Class of Singular Reaction-Diffusion Systems.

B. Sheppard - On Generalisations of the Stone-Weisstrass Theorem to Jordan Structures.

S. Leary - Least Squares Methods with Adjustable Nodes for Steady Hyperbolic PDEs.

I. Sciriha - On Some Aspects of Graph Spectra.

P.A. Burton - Convergence of flux limiter schemes for hyperbolic conservation laws with source terms.

J.F. Goodwin - Developing a practical approach to water wave scattering problems.

N.R.T. Biggs - Integral equation embedding methods in wave-diffraction methods.

L.P. Gibson - Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model.

A.K. Griffith - Data assimilation for numerical weather prediction using control theory. .

J. Bryans - Denotational semantic models for real-time LOTOS.

I. MacDonald - Analysis and computation of steady open channel flow .

A. Morton - Higher order Godunov IMPES compositional modelling of oil reservoirs.

S.M. Allen - Extended edge-colourings of graphs.

M.E. Hubbard - Multidimensional upwinding and grid adaptation for conservation laws.

C.J. Chikunji - On the classification of finite rings.

S.J.G. Bell - Numerical techniques for smooth transformation and regularisation of time-varying linear descriptor systems.

D.J. Staziker - Water wave scattering by undulating bed topography .

K.J. Neylon - Non-symmetric methods in the modelling of contaminant transport in porous media. .

D.M. Littleboy - Numerical techniques for eigenstructure assignment by output feedback in aircraft applications .

M.P. Dainton - Numerical methods for the solution of systems of uncertain differential equations with application in numerical modelling of oil recovery from underground reservoirs .

M.H. Mawson - The shallow-water semi-geostrophic equations on the sphere. .

S.M. Stringer - The use of robust observers in the simulation of gas supply networks .

S.L. Wakelin - Variational principles and the finite element method for channel flows. .

E.M. Dicks - Higher order Godunov black-oil simulations for compressible flow in porous media .

C.P. Reeves - Moving finite elements and overturning solutions .

A.J. Malcolm - Data dependent triangular grid generation. .

Alternatively, use our A–Z index

Attend an open day

Discover more about postgraduate research

PhD Numerical Analysis / Overview

Year of entry: 2024

• View full page

The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree in a discipline directly relevant to the PhD (or international equivalent) OR any upper-second class (2:1) honours degree and a Master’s degree at merit in a discipline directly relevant to the PhD (or international equivalent).

Other combinations of qualifications and research or work experience may also be considered. Please contact the admissions team to check.

Full entry requirements

Apply online

In your application you’ll need to include:

• The name of this programme
• Your research project title (i.e. the advertised project name or proposed project name) or area of research
• Your proposed supervisor’s name
• If you already have funding or you wish to be considered for any of the available funding
• A supporting statement (see 'Advice to Applicants for what to include)
• Details of your previous university level study
• Names and contact details of your two referees.

Programme options

Programme description

The Department of Mathematics has an outstanding research reputation. The research facilities include one of the finest libraries in the country, the John Rylands University Library. This library has recently made a very large commitment of resources to providing comprehensive online facilities for the free use of the University's research community. Postgraduate students in the Department benefit from direct access to all the Library electronic resources from their offices.

Many research seminars are held in the Department on a weekly basis and allow staff and research students to stay in touch with the latest developments in their fields. The Department is one of the lead partners in the MAGIC project and research students can attend any of the postgraduate courses offered by the MAGIC consortium.

For entry in the academic year beginning September 2024, the tuition fees are as follows:

• PhD (full-time) UK students (per annum): Band A £4,786; Band B £7,000; Band C £10,000; Band D £14,500; Band E £24,500 International, including EU, students (per annum): Band A £28,000; Band B £30,000; Band C £35,500; Band D £43,000; Band E £57,000
• PhD (part-time) UK students (per annum): Band A £2393; Band B £3,500; Band C £5,000; Band D £7,250; Band E 12,250 International, including EU, students (per annum): Band A £14,000; Band B £15,000; Band C £17,750; Band D £21,500; Band E £28,500

Further information for EU students can be found on our dedicated EU page.

The programme fee will vary depending on the cost of running the project. Fees quoted are fully inclusive and, therefore, you will not be required to pay any additional bench fees or administration costs.

All fees for entry will be subject to yearly review and incremental rises per annum are also likely over the duration of the course for Home students (fees are typically fixed for International students, for the course duration at the year of entry). For general fees information please visit the postgraduate fees page .

Always contact the Admissions team if you are unsure which fees apply to your project.

Scholarships/sponsorships

There are a range of scholarships, studentships and awards at university, faculty and department level to support both UK and overseas postgraduate researchers.

To be considered for many of our scholarships, you’ll need to be nominated by your proposed supervisor. Therefore, we’d highly recommend you discuss potential sources of funding with your supervisor first, so they can advise on your suitability and make sure you meet nomination deadlines.

For more information about our scholarships, visit our funding page or use our funding database to search for scholarships, studentships and awards you may be eligible for.

Contact details

Our internationally-renowned expertise across the School of Natural Sciences informs research led teaching with strong collaboration across disciplines, unlocking new and exciting fields and translating science into reality.  Our multidisciplinary learning and research activities advance the boundaries of science for the wider benefit of society, inspiring students to promote positive change through educating future leaders in the true fundamentals of science. Find out more about Science and Engineering at Manchester .

Programmes in related subject areas

Use the links below to view lists of programmes in related subject areas.

• Mathematics

Regulated by the Office for Students

The University of Manchester is regulated by the Office for Students (OfS). The OfS aims to help students succeed in Higher Education by ensuring they receive excellent information and guidance, get high quality education that prepares them for the future and by protecting their interests. More information can be found at the OfS website .

You can find regulations and policies relating to student life at The University of Manchester, including our Degree Regulations and Complaints Procedure, on our regulations website .

Doctoral Theses on SPH

Here you will find a list and links to doctoral theses on SPH that are available online and organized by year. Some of these links will take you the website of the university/institute of the student and are not hosted by SPHERIC. If you want your SPH thesis to appear online, please email the Dr. Angelo Tafuni ( [email protected] ).

Note: If you are looking for a PhD position, please see the  SPH Jobs  page.

Iván Martínez-Estévez. Coupling between the DualSPHysics solver and multiphysics libraries: implementation, validation and real engineering applications , Universidade de Vigo

Yong Yang. Modelling Extreme Wave-Current Conditions and their Interaction with Offshore Renewable Energy Systems , The University of Manchester

Francesco Ricci. Variable resolution Smoothed Particle Hydrodynamics schemes for 2-D and 3-D viscous flows , New Jersey Institute of Technology

Pawan Singh Negi. Development of Second-Order Convergent Weakly-Compressible Smoothed Particle Hydrodynamics Schemes , Indian Institute of Technology Bombay

Pablo Eleazar Merino-Alonso. Numerical Modeling of impulsive loads using the Smoothed Particle Hydrodynamics method , Universidad Politécnica de Madrid

Nicolas Quartier. Numerical Modelling of Moored Floating Structures and Energy Devices Using a Smoothed Particle Hydrodyanmics-Based Solver , Ghent University

Paulo Refachinho De Campos. A New Updated Reference Lagrangian Smooth Particle Hydrodynamics Framework for Large Strain Solid Dynamics and its Extension to Dynamic Fracture , Swansea University

Anton Esmail Yakas. Smoothed particle hydrodynamics for fluid-solid coupling: modelling fixed and mobile boundaries , Imperial College London

Muhammad Aslami. SPH Modelling of Debris in Shallow Water Flows , The University of Manchester

Rene Winchenbach.  Spatially adaptive SPH . University of Siegen

Javier Cal derón Sánchez. Numerical studies of the sloshing phenomenon using the Smoothed Particle Hydrodynamics (SPH) method , Universidad Politécnica de Madrid

Aaron English. Design of Low Residue Packs by Smoothed Particle Hydrodynamics , The University of Manchester

Thomas Fonty. Modeling air entrainment in water with the SPH method , Paris-Est University

Md Lokman Hosain: Fluid Flow and Heat Transfer Simulations for Complex Industrial Applications: From Reynolds Averages Navier-Stokes towards Smoothed Particle Hydrodynamics , Mälardalen University

Moisés Gonçalves de Brito: Numerical modeling and experimental testing of an oscillating wave surge converter , Instituto Superior Técnico de Lisboa

Tim Verbrugghe: Coupling Methodologies for Numerical Modelling of Floating Wave Energy Converters , Universiteit Gent.

Alex Chow: Incompressible SPH (ISPH) on the GPU , The University of Manchester

Mashy David Green: Sloshing simulations with the smoothed particle hydrodynamics (SPH) method,  Imperial College London

Laurent Chiron: Couplage et améliorations de la méthode SPH pour traiter des écoulements à multi- échelles temporelles et spatiales , Ecole Centrale de Nantes

Iason Zisis: From Continuum Mechanics to Smoothed Particle Hydrodynamics for Shocks through Inhomogeneous Media , Technische Universiteit Eindhoven

Alex Ghaitanellis: Modelling bed-load sediment transport through a granular approach in SPH , Universite Paris-Est

Edgar Andres Patiño Nariño: Modelagem Numérica de microfluídica através do Método Lagrangeano sem malha Smoothed Particle Hydrodynamics , Universidade Estadual de Campinas

José Luis Cercós Pita: A novel generalized diffusive SPH model: Theoretical analysis and 3D HPC implementation , Universidad Politécnica de Madrid

Angelantonio Tafuni: Smoothed Particle Hydrodynamics: development and application to problems of hydrodynamics , New York University

Abouzied Nasar: Eulerian and Lagrangian Smoothed Particle Hydrodynamics as Models for the Interaction of Fluids and Flexible Structures in Biomedical Flows , The University of Manchester

Benjamin Bouscasse: Dynamic interactions between solids and viscous liquids with free surface , Universidad Politécnica de Madrid

Ricardo Canelas: Numerical modeling of fully coupled solid-fluid flows , Universidade de Lisboa, Instituto Superior Técnico

Patrick Jonsson: Smoothed Particle Hydrodynamics of Hydraulic Jumps in Spillways , Lulea University of Technology

Domenico Davide Meringolo: Weakly-compressible SPH modeling of fluid-structure interaction problems , Università della Calabria

Georgios Fourtakas: Modelling multi-phase flows in Nuclear Decommissioning using SPH , University of Manchester

Arno Mayrhofer: An Investigation into Wall Boundary Conditions and Three-Dimensional Turbulent Flows using Smoothed Particle Hydrodynamics , University of Manchester.  Mirror link

José Domínguez: DualSPHysics: Towards High Performance Computing using SPH technique , Universidade de Vigo.  Mirror link

Swapnadip de Chowdhury: SPH Simulation of Nonlinear Water Waves , Indian Institute of Technology Madras.  Mirror link

Agnes Leroy: A New Incompressible SPH Model: Towards Industrial Applications , Université Paris-Est

Jannes Kordilla: Flow and transport in saturated and unsaturated fractured porous media: Development of particle-based modeling approaches , Georg-August-Universität Göttingen.  Mirror link

Carlos Alberto Dutra Fraga Filho: (in Portugese) Study of Gravity-Inertial Phase of Spreading of Oil on a Calm Sea employing the Lagrangian Particle Method Smoothed Particle Hydrodynamics , ( Abstract in English ), Federal University of Espírito Santo, Federal Institute of Espírito Santo

Christos Makris: (In Greek) Numerical Simulation of Coastal Wave Processes with the Use of Smoothed Particle Hydrodynamics (SPH) Method , Aristotle University of Thessaloniki

Naoki Tsuruta: Improved Particle Method with High-Resolution and Computational Stability for Solid-Liquid Two-Phase Flows , Kyoto University.  Mirror link

Athanasios Mokos: Multi-phase Modelling of Violent Hydrodynamics Using Smoothed Particle Hydrodynamics (SPH) on Graphics Processing Units (GPUs) , University of Manchester.

Daniel Barcarolo: Improvement of the precision and the efficiency of the SPH method: theoretical and numerical study , Ecole Centrale de Nantes.

Christian Ulrich: Smoothed-Particle-Hydrodynamics Simulation of Port Hydrodynamic Problems , Technische Universitat Hamburg Harburg (TUHH).  Mirror link

Abdelraheem Mahmoud Aly: A n Improved Incompressible Smoothed Particle Hydrodynamics to Simulate Fluid-Soil-Structure Interactions , Kyushu University.

Salvatore Marrone: Enhanced SPH modeling of free-surface flows with large deformations , University of Rome, La Sapienza.

Jules B. Kajtar: Smooth lattice general relativity, and SPH simulations of swimming linked bodies , Monash University.

Renato Vacondio: Shallow Water and Navier-Stokes SPH-like numerical modelling of rapidly varying free-surface flows , Università degli Studi di Parma Facoltà di Ingegneria.

Rui Xu: An Improved Incompressible Smoothed Particle Hydrodynamics Method and Its Application in Free-Surface Simulations , University of Manchester.

Ivan Federico: Simulating Open-channel Flows and Advective Diffusion Phenomena through SPH Model , Università della Calabria.

Pourya Omidvar: Wave Loading on Bodies in the Free Surface Using Smoothed Particle Hydrodynamics (SPH) , University of Manchester.

Tatiana Capone: SPH numerical modelling of impulse water waves generated by landslides , University of Rome, La Sapienza.

Martin Robinson: Turbulence and Viscous Mixing using Smoothed Particle Hydrodynamics , Monash University. Mirror link.

Nicolas Grenier: Numerical modelisation by SPH method of water-oil separation in gravity separators , Ecole Centrale de Nantes.

Muthu Narayanaswamy: A Hybrid Boussinesq-SPH Wave Propagation Model with Applications to Forced Waves in Rectangular Tanks , The Johns Hopkins University.

Abbas Khayyer: Improved Particle Methods by Refined Differential Operator Methods for Free-Surface Fluid Flows , Kyoto University.

Louis Delorme: Sloshing Flows. Experimental Investigation and numerical investigations with Smoothed Particle Hydrodynamics , Universidad Politenica de Madrid.

Alejandro J.C. Crespo: Application of the Smoothed Particle Hydrodynamics model SPHysics to free-surface hydrodynamics , Universidade de Vigo.

Shan Zou: Coastal Sediment Transport Simulation by Smoothed Particle Hydrodynamics , The Johns Hopkins University.

Hans Schwaiger: An Implementation of Smoothed Particle Hydrodynamics For Large Deformation, History Dependent Geomaterials With Applications to Tectonic Deformation , University of Washington.  Mirror link

Eun-Sug Lee: Truly incompressible approach for computing incompressible flow in SPH and comparisons with the traditional weakly compressible approach , University of Manchester (196MB).

J.A. Mansour: SPH and alpha-SPH: Applications and Analysis , Monash University.  Mirror link .

R. Ata: Development of particle methods for the resolution of free surface flows  (in French), Ecole de Technologie de Supérieure, Université de Quebec, (43MB).

Jonathan Feldman: Dynamic refinement and boundary contact forces in Smoothed Particle Hydrodynamics with applications in fluid flow problems. , University of Wales Swansea.  Mirror link

Guillaume Oger: Theoretical aspects of the SPH method and application to free surface problems in hydrodynamics. (In French) , Ecole Centrale de Nantes, France.

Andrea Colagrossi: A Meshless Lagrangian Method for free-surface and Interface Flows with Fragmentation , University of Rome, La Sapienza (45MB).

Gareth Llewellyn Vaughan: Simulating Breaking Waves Using Smoothed Particle Hydrodynamics , University of Waikato, New Zealand.

Reza Issa: Numerical assessment of the Smoothed Particle Hydrodynamics gridless method for incompressible flows and its extension to turbulent flows , UMIST.

Andrea Panizzo: Physical and Numerical Modelling of Subaerial Landslide Generated Waves , University of Rome.

On error controlled numerical model reduction for finite element squared (FE2) procedures

Fredrik Larsson is a professor in Structural Mechanics at Chalmers University of Technology in Sweden, where he also earned his PhD in 2003. He has worked on various problems in the field of Computational Mechanics and the development of finite element procedures in solid mechanics. Particular research interests are multiscale modeling, numerical model reduction and a posteriori error estimation.

The “Finite Element squared” (FE2) technique is a multiscale method for analyzing problems on two distinct length scales using the finite element method, typically a macroscale where the overall response is sought and a microscale where fine-scale features are resolved. Each macroscale quadrature point is connected to a (discretized) boundary value problem on a Representative Volume Element (RVE). For nonlinear and/or time-dependent problems, the nested problems must be solved concurrently, making the FE2 procedure is still computationally extremely demanding. However, the fact that many similar problems on RVEs are solved for with small number of input/output data makes the procedure well suited for reduction techniques applied to the discrete equations, here denoted Numerical Model Reduction. In order to obtain reliable approximations, it is of outmost importance to quantify, and control, the error associated with the reduction procedure.

In this contribution, we address a few different model problems, pertinent to heat flow, consolidation of porous media and non-linear elasticity. In particular, suitable error estimators are developed that (in the linear case) are guaranteed w.r.t. the full-fledged finite element solution. The key ingredients in the error estimator is the weak formulation of the RVE problem in space-time, construction of a suitable norm, and the definition of associated problem. The fact that the approximation is compared to the discrete finite element solution allows for explicit evaluation at low cost. Finally, the estimator is also extended to compute bounds on user-defined quantities of interest within the realm of goal-oriented error estimation.

Category:  

One-photon 3D Nanolithography using Controlled Initiator Depletion

3D printing techniques have been applied in many fields to provide a potential for complex fabrication, and photopolymerization methods are the current possible path to fabricate nanoscale 3D structures. Multi-photon lithography is the most common tool to reach below 100-nm resolution. These methods require femtosecond lasers to reliably create sophisticated 3D polymeric nanostructures using nonlinear photopolymerization of a light-sensitive resin. Though these methods provide high accuracy and flexibility in advanced fabrication, they are essentially limited by their cost and throughput. Therefore, in this work, multiple approaches were examined to develop new methods for one-photon nonlinear 3D printing. 

By controlling multiple competing processes in the radical polymerization scheme, a nonlinear photopolymerization effect is achieved using a one-photon absorption process with the assistance of inhibition radicals and controlled diffusion. This work makes use of this nonlinear response to fabricate 2D/3D structures using a continuous-wave diode laser, demonstrating a significantly more cost-efficient source for 3D nanolithography. In addition, a numerical model was constructed with the highly nonlinear response by actively controlling the consumption of the initiators with the assistance of these inhibitors, and it shows the same trend of nonlinearity from experiments. We use this model to study this dosage-based nonlinear response driven by the laser intensity in several 1D and 2D scenarios with different inputs and predicted the polymerization results in a confined voxel in the resin to support the observations from the experiments. Besides the demonstration of current one-photon nonlinear 3D printing, this work also involves some results of nonlinear response by operating local oxygen concentration and a two-step absorption nonlinear photoinitiator. These results help us to further study the potential of increasing the throughput of the one-photon nonlinear 3D printing process. 

In conclusion, a new one-photon-based dose nonlinear process is introduced in this dissertation to achieve nanoscale 3D printing with a low-cost-405-nm diode laser operating at milliwatt level. By controlling the activation and transport of initiating and inhibiting radicals, we achieve patterning of the nanoscale features at a high scanning speed.

NSF Scalable Nanomanufacturing (CMMI-1634832)

Degree type.

• Doctor of Philosophy
• Mechanical Engineering

Campus location

• West Lafayette

Advisor/Supervisor/Committee Chair

Advisor/supervisor/committee co-chair, additional committee member 2, additional committee member 3, usage metrics.

• Mechanical engineering not elsewhere classified
• Microelectromechanical systems (MEMS)