phd thesis numerical methods

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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.

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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

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

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Mathematical Modeling Doctor of Philosophy (Ph.D.) Degree

Request Info about graduate study Visit Apply

The mathematical modeling Ph.D. enables you to develop mathematical models to investigate, analyze, predict, and solve the behaviors of a range of fields from medicine, engineering, and business to physics and science.

STEM-OPT Visa Eligible

Overview for Mathematical Modeling Ph.D.

Mathematical modeling is the process of developing mathematical descriptions, or models, of real-world systems. These models can be linear or nonlinear, discrete or continuous, deterministic or stochastic, and static or dynamic, and they enable investigating, analyzing, and predicting the behavior of systems in a wide variety of fields. Through extensive study and research, graduates of the mathematical modeling Ph.D. will have the expertise not only to use the tools of mathematical modeling in various application settings, but also to contribute in creative and innovative ways to the solution of complex interdisciplinary problems and to communicate effectively with domain experts in various fields.

Plan of Study

The degree requires at least 60 credit hours of course work and research. The curriculum consists of three required core courses, three required concentration foundation courses, a course in scientific computing and high-performance computing (HPC), three elective courses focused on the student’s chosen research concentration, and a doctoral dissertation. Elective courses are available from within the School of Mathematics and Statistics as well as from other graduate programs at RIT, which can provide application-specific courses of interest for particular research projects. A minimum of 30 credits hours of course work is required. In addition to courses, at least 30 credit hours of research, including the Graduate Research Seminar, and an interdisciplinary internship outside of RIT are required.

Students develop a plan of study in consultation with an application domain advisory committee. This committee consists of the program director, one of the concentration leads, and an expert from an application domain related to the student’s research interest. The committee ensures that all students have a roadmap for completing their degree based on their background and research interests. The plan of study may be revised as needed. Learn more about our mathematical modeling doctoral students and view a selection of mathematical modeling seminars hosted by the department.

Qualifying Examinations

All students must pass two qualifying examinations to determine whether they have sufficient knowledge of modeling principles, mathematics, and computational methods to conduct doctoral research. Students must pass the examinations in order to continue in the Ph.D. program.

The first exam is based on the Numerical Analysis I (MATH-602) and Mathematical Modeling I, II (MATH-622, 722). The second exam is based on the student's concentration foundation courses and additional material deemed appropriate by the committee and consists of a short research project.

Dissertation Research Advisor and Committee

A dissertation research advisor is selected from the program faculty based on the student's research interests, faculty research interest, and discussions with the program director. Once a student has chosen a dissertation advisor, the student, in consultation with the advisor, forms a dissertation committee consisting of at least four members, including the dissertation advisor. The committee includes the dissertation advisor, one other member of the mathematical modeling program faculty, and an external chair appointed by the dean of graduate education. The external chair must be a tenured member of the RIT faculty who is not a current member of the mathematical modeling program faculty. The fourth committee member must not be a member of the RIT faculty and may be a professional affiliated with industry or with another institution; the program director must approve this committee member.

The main duties of the dissertation committee are administering both the candidacy exam and final dissertation defense. In addition, the dissertation committee assists students in planning and conducting their dissertation research and provides guidance during the writing of the dissertation.

Admission to Candidacy

When a student has developed an in-depth understanding of their dissertation research topic, the dissertation committee administers an examination to determine if the student will be admitted to candidacy for the doctoral degree. The purpose of the examination is to ensure that the student has the necessary background knowledge, command of the problem, and intellectual maturity to carry out the specific doctoral-level research project. The examination may include a review of the literature, preliminary research results, and proposed research directions for the completed dissertation. Requirements for the candidacy exam include both a written dissertation proposal and the presentation of an oral defense of the proposal. This examination must be completed at least one year before the student can graduate.

Dissertation Defense and Final Examination

The dissertation defense and final examination may be scheduled after the dissertation has been written and distributed to the dissertation committee and the committee has consented to administer the final examination. Copies of the dissertation must be distributed to all members of the dissertation committee at least four weeks prior to the final examination. The dissertation defense consists of an oral presentation of the dissertation research, which is open to the public. This public presentation must be scheduled and publicly advertised at least four weeks prior to the examination. After the presentation, questions will be fielded from the attending audience and the final examination, which consists of a private questioning of the candidate by the dissertation committee, will ensue. After the questioning, the dissertation committee immediately deliberates and thereafter notifies the candidate and the mathematical modeling graduate director of the result of the examination.

All students in the program must spend at least two consecutive semesters (summer excluded) as resident full-time students to be eligible to receive the doctoral degree.

Maximum Time Limitations

University policy requires that doctoral programs be completed within seven years of the date of the student passing the qualifying exam. All candidates must maintain continuous enrollment during the research phase of the program. Such enrollment is not limited by the maximum number of research credits that apply to the degree.

National Labs Career Fair

Hosted by RIT’s Office of Career Services and Cooperative Education, the National Labs Career Fair is an annual event that brings representatives to campus from the United States’ federally funded research and development labs. These national labs focus on scientific discovery, clean energy development, national security, technology advancements, and more. Students are invited to attend the career fair to network with lab professionals, learn about opportunities, and interview for co-ops, internships, research positions, and full-time employment.

Students are also interested in: Applied and Computational Mathematics MS

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The College of Science consistently receives research grant awards from organizations that include the National Science Foundation , National Institutes of Health , and NASA , which provide you with unique opportunities to conduct cutting-edge research with our faculty members.

Faculty in the School of Mathematics and Statistics conducts research on a broad variety of topics including:

• applied inverse problems and optimization
• applied statistics and data analytics
• biomedical mathematics
• discrete mathematics
• dynamical systems and fluid dynamics
• geometry, relativity, and gravitation
• mathematics of earth and environment systems
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Learn more by exploring the school’s mathematics research areas .

Michael Cromer

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RIT to award record number of Ph.D. degrees

RIT will confer a record 69 Ph.D. degrees during commencement May 12, marking a 53 percent increase from last year.

Curriculum for 2023-2024 for Mathematical Modeling Ph.D.

Current Students: See Curriculum Requirements

Mathematical Modeling, Ph.D. degree, typical course sequence

Concentrations, applied inverse problems, biomedical mathematics, discrete mathematics, dynamical systems and fluid dynamics, geometry, relativity and gravitation, admissions and financial aid.

This program is available on-campus only.

Full-time study is 9+ semester credit hours. International students requiring a visa to study at the RIT Rochester campus must study full‑time.

Application Details

To be considered for admission to the Mathematical Modeling Ph.D. program, candidates must fulfill the following requirements:

• Complete an online graduate application .
• Submit copies of official transcript(s) (in English) of all previously completed undergraduate and graduate course work, including any transfer credit earned.
• Hold a baccalaureate degree (or US equivalent) from an accredited university or college.
• A recommended minimum cumulative GPA of 3.0 (or equivalent).
• Submit a current resume or curriculum vitae.
• Submit a statement of purpose for research which will allow the Admissions Committee to learn the most about you as a prospective researcher.
• Submit two letters of recommendation .
• Entrance exam requirements: None
• Writing samples are optional.
• Submit English language test scores (TOEFL, IELTS, PTE Academic), if required. Details are below.

English Language Test Scores

International applicants whose native language is not English must submit one of the following official English language test scores. Some international applicants may be considered for an English test requirement waiver .

International students below the minimum requirement may be considered for conditional admission. Each program requires balanced sub-scores when determining an applicant’s need for additional English language courses.

How to Apply   Start or Manage Your Application

Cost and Financial Aid

An RIT graduate degree is an investment with lifelong returns. Ph.D. students typically receive full tuition and an RIT Graduate Assistantship that will consist of a research assistantship (stipend) or a teaching assistantship (salary).

Additional Information

Foundation courses.

Mathematical modeling encompasses a wide variety of scientific disciplines, and candidates from diverse backgrounds are encouraged to apply. If applicants have not taken the expected foundational course work, the program director may require the student to successfully complete foundational courses prior to matriculating into the Ph.D. program. Typical foundation course work includes calculus through multivariable and vector calculus, differential equations, linear algebra, probability and statistics, one course in computer programming, and at least one course in real analysis, numerical analysis, or upper-level discrete mathematics.

Classical Numerical Methods in Scientific Computing

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Jos van Kan, Delft University of Technology

Guus Segal, Delft University of Technology

Fred Vermolen, University of Hasselt

Copyright Year: 2023

ISBN 13: 9789463667326

Publisher: TU Delft Open

Language: English

Formats Available

Conditions of use.

Table of Contents

• Review of some basic mathematical concepts
• A crash course in PDEs
• Finite difference methods
• Finite volume methods
• Non-linear equations
• The heat- or diffusion equation
• The wave equation

Ancillary Material

About the book.

Partial differential equations are paramount in mathematical modelling with applications in engineering and science. The book starts with a crash course on partial differential equations in order to familiarize the reader with fundamental properties such as existence, uniqueness and possibly existing maximum principles. The main topic of the book entails the description of classical numerical methods that are used to approximate the solution of partial differential equations. The focus is on discretization methods such as the finite difference, finite volume and finite element method. The manuscript also makes a short excursion to the solution of large sets of (non)linear algebraic equations that result after application of discretization method to partial differential equations. The book treats the construction of such discretization methods, as well as some error analysis, where it is noted that the error analysis for the finite element method is merely descriptive, rather than rigorous from a mathematical point of view. The last chapters focus on time integration issues for classical time-dependent partial differential equations. After reading the book, the reader should be able to derive finite element methods, to implement the methods and to judge whether the obtained approximations are consistent with the solution to the partial differential equations. The reader will also obtain these skills for the other classical discretization methods. Acquiring such fundamental knowledge will allow the reader to continue studying more advanced methods like meshfree methods, discontinuous Galerkin methods and spectral methods for the approximation of solutions to partial differential equations.

About the Contributors

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.

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Mathematics PhD theses

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

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

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. .

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Applied mathematics has long had a central role at the Courant Institute, and roughly half of all our PhD's in Mathematics are in some applied field. There are a large number of applied fields that are the subject of research. These include:

• Atmosphere and Ocean Science
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PhD training in applied mathematics at Courant focuses on a broad and deep mathematical background, techniques of applied mathematics, computational methods, and specific application areas. Descriptions of several applied-math graduate courses are given below.

Numerical analysis is the foundation of applied mathematics, and all PhD students in the field should take the Numerical Methods I and II classes in their first year, unless they have taken an equivalent two-semester PhD-level graduate course in numerical computing/analysis at another institution. Afterwards, students can take a number of more advanced and specialized courses, some of which are detailed below. Important theoretical foundations for applied math are covered in the following courses: (1) Linear Algebra I and II, (2) Intro to PDEs, (3) Methods of Applied Math, and (4) Applied Stochastic Analysis. It is advised that students take these courses in their first year or two.

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Courses in Applied Mathematics

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(MATH-GA.2701) Methods Of Applied Math

Fall 2023, Oliver Buhler

Description:  This is a first-year course for all incoming PhD and Masters students interested in pursuing research in applied mathematics. It provides a concise and self-contained introduction to advanced mathematical methods, especially in the asymptotic analysis of differential equations. Topics include scaling, perturbation methods, multi-scale asymptotics, transform methods, geometric wave theory, and calculus of variations.

Prerequisites : Elementary linear algebra, ordinary differential equations; at least an undergraduate course on partial differential equations is strongly recommended.

(MATH-GA.2704) Applied Stochastic Analysis

Spring 2024, Jonathan Weare

This is a graduate class that will introduce the major topics in stochastic analysis from an applied mathematics perspective.  Topics to be covered include Markov chains, stochastic processes, stochastic differential equations, numerical algorithms, and asymptotics. It will pay particular attention to the connection between stochastic processes and PDEs, as well as to physical principles and applications. The class will attempt to strike a balance between rigour and heuristic arguments: it will assume that students have some familiarity with measure theory and analysis and will make occasional reference to these, but many results will be derived through other arguments. The target audience is PhD students in applied mathematics, who need to become familiar with the tools or use them in their research.

Prerequisites: Basic Probability (or equivalent masters-level probability course), Linear Algebra (graduate course), and (beginning graduate-level) knowledge of ODEs, PDEs, and analysis.

(MATH-GA.2010/ CSCI-GA.2420) Numerical Methods I

• Fall 2023, Benjamin Peherstorfer

Description:   This course is part of a two-course series meant to introduce graduate students in mathematics to the fundamentals of numerical mathematics (but any Ph.D. student seriously interested in applied mathematics should take it). It will be a demanding course covering a broad range of topics. There will be extensive homework assignments involving a mix of theory and computational experiments, and an in-class final. Topics covered in the class include floating-point arithmetic, solving large linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, linear and nonlinear least squares, and nonlinear optimization, and iterative methods. This course will not cover differential equations, which form the core of the second part of this series, Numerical Methods II.

Prerequisites:   A good background in linear algebra, and some experience with writing computer programs (in MATLAB, Python or another language).

(MATH-GA.2020 / CSCI-GA.2421) Numerical Methods II

Spring 2024, Aleksandar Donev

This course (3pts) will cover fundamental methods that are essential for the numerical solution of differential equations. It is intended for students familiar with ODE and PDE and interested in numerical computing; computer programming assignments in MATLAB/Python will form an essential part of the course. The course will introduce students to numerical methods for (approximately in this order):

• The Fast Fourier Transform and pseudo-spectral methods for PDEs in periodic domains
• Ordinary differential equations, explicit and implicit Runge-Kutta and multistep methods, IMEX methods, exponential integrators, convergence and stability
• Finite difference/element, spectral, and integral equation methods for elliptic BVPs (Poisson)
• Finite difference/element methods for parabolic (diffusion/heat eq.) PDEs (diffusion/heat)
• Finite difference/volume methods for hyperbolic (advection and wave eqs.) PDEs (advection, wave if time permits).

Prerequisites

This course requires Numerical Methods I or equivalent graduate course in numerical analysis (as approved by instructor), preferably with a grade of B+ or higher.

( MATH-GA.2011 / CSCI-GA 2945) Computational Methods For PDE

Fall 2023, Aleksandar Donev & Georg Stadler

This course follows on Numerical Methods II and covers theoretical and practical aspects of advanced computational methods for the numerical solution of partial differential equations. The first part will focus on finite element methods (FEMs), and the second part on finite volume methods (FVMs) including discontinuous Galerkin (FE+FV) methods. In addition to setting up the numerical and functional analysis theory behind these methods, the course will also illustrate how these methods can be implemented and used in practice for solving partial differential equations in two and three dimensions. Example PDEs will include the Poisson equation, linear elasticity, advection-diffusion(-reaction) equations, the shallow-water equations, the incompressible Navier-Stokes equation, and others if time permits. Students will complete a final project that includes using, developing, and/or implementing state-of-the-art solvers.

In the Fall of 2023, Georg Stadler will teach the first half of this course and cover FEMs, and Aleks Donev will teach in the second half of the course and cover FVMs.

A graduate-level PDE course, Numerical Methods II (or equivalent, with approval of syllabus by instructor(s)), and programming experience.

• Elman, Silvester, and Wathen: Finite Elements and Fast Iterative Solvers , Oxford University Press, 2014.
• Farrell: Finite Element Methods for PDEs , lecture notes, 2021.
• Hundsdorfer & Verwer: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations , Springer-Verlag, 2003.
• Leveque: Finite Volume Methods for Hyperbolic Problems , Cambridge Press, 2002.

-------------------------------------

( MATH-GA.2012 ) Immersed Boundary Method For Fluid-Structure Interaction

Not offered AY 23/24.

The immersed boundary (IB) method is a general framework for the computer simulation of flows with immersed elastic boundaries and/or complicated geometry.  It was originally developed to study the fluid dynamics of heart valves, and it has since been applied to a wide variety of problems in biofluid dynamics, such as wave propagation in the inner ear, blood clotting, swimming of creatures large and small, and the flight of insects.  Non-biological applications include sails, parachutes, flows of suspensions, and two-fluid or multifluid problems. Topics to be covered include: mathematical formulation of fluid-structure interaction in Eulerian and Lagrangian variables, with interaction equations involving the Dirac delta function; discretization of the structure, fluid, and interaction equations, including energy-based discretization of the structure equations, finite-difference discretization of the fluid equations, and IB delta functions with specified mathematical properties; a simple but effective method for adding mass to an immersed boundary; numerical simulation of rigid immersed structures or immersed structures with rigid parts; IB methods for immersed filaments with bend and twist; and a stochastic IB method for thermally fluctuating hydrodynamics within biological cells.  Some recent developments to be discussed include stability analysis of the IB method and a Fourier-Spectral IB method with improved boundary resolution.

Course requirements include homework assignments and a computing project, but no exam.  Students may collaborate on the homework and on the computing project, and are encouraged to present the results of their computing projects to the class.

Prerequisite:   Familiarity with numerical methods and fluid dynamics.

(MATH-GA.2012 / CSCI-GA.2945) :  High Performance Computing

Not offered AY 23/24

This class will be an introduction to the fundamentals of parallel scientific computing. We will establish a basic understanding of modern computer architectures (CPUs and accelerators, memory hierarchies, interconnects) and of parallel approaches to programming these machines (distributed vs. shared memory parallelism: MPI, OpenMP, OpenCL/CUDA). Issues such as load balancing, communication, and synchronization will be covered and illustrated in the context of parallel numerical algorithms. Since a prerequisite for good parallel performance is good serial performance, this aspect will also be addressed. Along the way you will be exposed to important tools for high performance computing such as debuggers, schedulers, visualization, and version control systems. This will be a hands-on class, with several parallel (and serial) computing assignments, in which you will explore material by yourself and try things out. There will be a larger final project at the end. You will learn some Unix in this course, if you don't know it already.

Prerequisites for the course are (serial) programming experience with C/C++ (I will use C in class) or Fortran, and some familiarity with numerical methods.

(MATH-GA.2011) Monte Carlo Methods

Fall 2023, Jonathan Weare and Jonathan Goodman

Topics : The theory and practice of Monte Carlo methods. Random number generators and direct sampling methods, visualization and error bars. Variance reduction methods, including multi-level methods and importance sampling. Markov chain Monte Carlo (MCMC), detailed balance, non-degeneracy and convergence theorems. Advanced MCMC, including Langevin and MALA, Hamiltonian, and affine invariant ensemble samplers. Theory and estimation of auto-correlation functions for MCMC error bars. Rare event methods including nested sampling, milestoning, and transition path sampling. Multi-step methods for integration including Wang Landau and related thermodynamic integration methods. Application to sampling problems in physical chemistry and statistical physics and to Bayesian statistics.

Required prerequisites:

• A good probability course at the level of Theory of Probability (undergrad) or Fundamentals of Probability (masters)
• Linear algebra: Factorizations (especially Cholesky), subspaces, solvability conditions, symmetric and non-symmetric eigenvalue problem and applications
• Working knowledge of a programming language such as Python, Matlab, C++, Fortran, etc.
• Familiarity with numerical computing at the level of Scientific Computing (masters)

Desirable/suggested prerequisites:

• Numerical methods for ODE
• Applied Stochastic Analysis
• Familiarity with an application area, either basic statistical mechanics (Gibbs Boltzmann distribution), or Bayesian statistics

(MATH-GA.2012 / CSCI-GA.2945) Convex & Non Smooth Optimization

Spring 2024, Michael Overton

Convex optimization problems have many important properties, including a powerful duality theory and the property that any local minimum is also a global minimum. Nonsmooth optimization refers to minimization of functions that are not necessarily convex, usually locally Lipschitz, and typically not differentiable at their minimizers. Topics in convex optimization that will be covered include duality, CVX ("disciplined convex programming"), gradient and Newton methods, Nesterov's optimal gradient method, the alternating direction method of multipliers, the primal barrier method, primal-dual interior-point methods for linear and semidefinite programs. Topics in nonsmooth optimization that will be covered include subgradients and subdifferentials, Clarke regularity, and algorithms, including gradient sampling and BFGS, for nonsmooth, nonconvex optimization. Homework will be assigned, both mathematical and computational. Students may submit a final project on a pre-approved topic or take a written final exam.

Prerequisites: Undergraduate linear algebra and multivariable calculus

Q1: What is the difference between the Scientific Computing class and the Numerical Methods two-semester sequence?

The Scientific Computing class (MATH-GA.2043, fall) is a one-semester masters-level graduate class meant for graduate or advanced undergraduate students that wish to learn the basics of computational mathematics. This class requires a working knowledge of (abstract) linear algebra (at least at the masters level), some prior programming experience in Matlab, python+numpy, Julia, or a compiled programming language such as C++ or Fortran, and working knowledge of ODEs (e.g., an undergrad class in ODEs). It only briefly mentions numerical methods for PDEs at the very end, if time allows.

The Numerical Methods I (fall) and Numerical Methods II (spring) two-semester sequence is a Ph.D.-level advanced class on numerical methods, meant for PhD students in the field of applied math, masters students in the SciComp program , or other masters or advanced undergraduate students that have already taken at least one class in numerical analysis/methods. It is intended that these two courses be taken one after the other, not in isolation . While it is possible to take just Numerical Methods I, it is instead strongly recommended to take the Scientific Computing class (fall) instead. Numerical Methods II requires part I, and at least an undergraduate class in ODEs, and also in PDEs. Students without a background in PDEs should not take Numerical Methods II; for exceptions contact Aleks Donev with a detailed justification.

The advanced topics class on Computational Methods for PDEs follows on and requires having taken NumMeth II or an equivalent graduate-level course at another institution (contact Aleks Donev with a syllabus from that course for an evaluation), and can be thought of as Numerical Methods III.

Q2: How should I choose a first graduate course in numerical analysis/methods?

• If you are an undergraduate student interested in applied math graduate classes, you should take the undergraduate Numerical Analysis course (MATH-UA.0252) first, or email the syllabus for the equivalent of a full-semester equivalent class taken elsewhere to Aleks Donev for an evaluation.
• Take the Scientific Computing class (fall), or
• Take both Numerical Methods I (fall) and II (spring), see Q1 for details. This is required of masters students in the SciComp program .

A Finite Element Method for the Dynamical Ginzburg–Landau Equations under Coulomb Gauge

• Published: 10 September 2023
• Volume 97 , article number  19 , ( 2023 )

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• Huadong Gao 1 , 2 &
• Wen Xie 1  

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This paper is concerned with numerical analysis of a finite element method for the time-dependent Ginzburg–Landau equations under the Coulomb gauge. The main challenge is that the magnetic potential \({{\varvec{A}}}\) is divergence-free and satisfies a Stokes-like structure under the Coulomb gauge. The proposed method uses linear Lagrange element \({\mathcal {P}}_1\) to solve for the order parameter \(\psi \) , the lowest order Nédélec edge element \(\mathcal {N}\! \mathcal {D}_{\! 1}\) and linear Lagrange element \({P}_1\) to approximate the magnetic potential \({{\varvec{A}}}\) and the electric potential \(\phi \) , respectively. In particular, the proposed method preserves a weakly divergence-free property for \({{\varvec{A}}}\) in the discrete level. The main aim of this work is to establish the second order spatial convergence of the most important variable \(\psi \) , though the numerical solutions of \({{\varvec{A}}}\) are only O ( h ) in space. Our analysis is based on a nonstandard quasi-projection for \(\psi \) and the corresponding \(H^{-1}\) -norm estimates for Maxwell projection. With the quasi-projection, we prove that the lower-order approximation to \({{\varvec{A}}}\) does not pollute the accuracy of \(\psi _h\) . An effective one step recovery is also proposed to obtain second order numerical solution for \({{\varvec{A}}}\) . Our numerical experiments confirm the optimal second order convergence of \(\psi _h\) and the efficiency of the recovery step.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the authors on reasonable request.

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Acknowledgements

The authors wish to thank the anonymous referees for many constructive comments that improved the paper.

This work is partially supported by National Natural Science Foundation of China under Grant Number 12231003.

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Gao, H., Xie, W. A Finite Element Method for the Dynamical Ginzburg–Landau Equations under Coulomb Gauge. J Sci Comput 97 , 19 (2023). https://doi.org/10.1007/s10915-023-02327-x

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Received : 04 February 2023

Revised : 26 June 2023

Accepted : 16 August 2023

Published : 10 September 2023

DOI : https://doi.org/10.1007/s10915-023-02327-x

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7.2.2 Numerical References

Numerical references such as those required for a Vancouver style of referencing and recommended by guidelines that call for variations on the Vancouver system are commonly used in the medical and biological sciences, so if you are working in these areas, numerical references might be the right system to use. While such references can be extremely easy to produce in running text, it is important to remember that they can also quickly become problematic if even one source is missed when the references are numbered and added to the reference list. This is because numerical references are arranged in numerical order using Arabic numerals (very rarely Roman numerals, which are overly cumbersome if many references are provided) according to when they are cited in a thesis, so the first source cited becomes reference 1, the second, reference 2, the third, reference 3 and so on. Each number is assigned to one source only and each source has only one number, which it retains throughout the thesis. Therefore, if the third source cited in your thesis is missed, for instance, while you number your references and add them to the list, not only will that source need to be numbered 3, but all sources cited after it will also need to be renumbered both in the main text and in the list, because what was reference 3 will need to be reference 4, what was reference 4 will need to be reference 5 and so on. The method for numbering any references that appear in tables, figures and other supplementary parts of a thesis differs among guidelines: some would have such references numbered after all those in the main text, while others recommend that references in a table or figure be numbered according to where the table or figure is mentioned and/or placed in the main text.

The numerals used for references are often enclosed in parentheses or provided in a superscript font: ‘A recent study(1) confirmed this result’ or ‘A recent study 1 confirmed this result.’ Occasionally, square brackets are used instead: ‘A recent study[1] confirmed this result,’ but, generally speaking, they should only be used if guidelines call for them. Groups of citations can be gathered together in numerical referencing much as they are in an author–date system simply by listing the numbers of all relevant sources separated by commas: ‘(2,3,5,8,12).’ If three or more consecutive numbers need to be listed, an en rule or hyphen (with the first preferable in theory, but the second used far more frequently in Vancouver-style referencing) should be used between the first and last numbers: ‘(2–5,8–11)’ or ‘(2-5,8-11).’ In Vancouver referencing, no spaces appear between the numbers, but spaces are occasionally used in some of the variations on Vancouver referencing, especially if the numbers appear within parentheses (or square brackets); if the numbers are superscript, it is best to avoid spaces so that the reference numbers do not become separated at line breaks. As with author–date citations, numerical references should be positioned to indicate with accuracy the use of sources in the text, but standard placement in relation to punctuation calls for the reference numbers to follow full stops and commas (According to a recent study, 1 three groups are required) and precede colons and semicolons (The following settings were considered in a recent study 1 : internal with artificial white light, internal with blue light, internal with red light and external with natural daylight). Slightly different placement is usually acceptable, however (placing the reference numbers before instead of after commas and full stops, for example), as long as the same pattern is maintained throughout a thesis.

When you directly quote sources, page numbers are required in the in-text numerical citations just as they are when quoting while using author–date referencing. If the reference numbers appear in parentheses (or square brackets), the format is ‘In this study the results were “disappointing”(1 p98)’ and for multiple pages, ‘In this study the results were “disappointing and inconsistent”[1 pp96-98]’; if the reference numbers are superscript, ‘In this study the results were “disappointing” 1(p98) ’ and, for multiple pages, ‘In this study the results were “disappointing and inconsistent” 1(pp96-98) ’ would be appropriate. Using a numerical system of referencing does not mean that the author name(s) and date of publication associated with a source cannot be mentioned in the text if you wish to include this information, but it does mean that they do not need to be included and that the number assigned to the source must always appear whether additional information is provided or not. ‘As recently as 2014, Brockle revisited this problem’ is therefore not acceptable because it provides only an author–date reference, but ‘As recently as 2014, Brockle 7 revisited this problem’ is appropriate because it includes the number of the reference as well. Any additional information provided along with a reference number must match exactly the same information for the source in the reference list, just as the reference number provided for the source must match the reference number for it in the list.

In a numerical system of in-text referencing, the full bibliographical references for the sources used in a thesis should (like those in a system based on author surnames) appear in a list usually called ‘References’ that is located at the end of the thesis. In this case, however, the references must appear in correct numerical order beginning with reference 1 (the first source cited in a thesis) and continuing through to the final source cited in the thesis. Alphabetical order should not be used, which makes the arrangement a little more straightforward, but special care must be taken to ensure that references appear in exactly the same order as they are introduced in the thesis itself, particularly whenever changes are made to the references within the thesis chapters. There is also no need in a numerical list to ensure that author surnames and publication dates appear at the beginning of each reference: author surnames still do appear first in such a system, but dates rarely follow them as they do in author–date references. The following examples present the full bibliographical reference for a single source (a chapter in an edited book) according to three different styles of numerical referencing, with the third borrowed from the guidelines of a well-known medical journal.

• Vancouver (numerical):
• Hardman P. Presenting the text: pictorial tradition in fifteenth-century manuscripts of the Canterbury Tales . In: Finley WK, Rosenblum J, editors. Chaucer illustrated: Five hundred years of the Canterbury Tales in pictures. New Castle (DE): Oak Knoll Press; 2003. Pp. 37-72.
• AMA (numerical):
• Hardman P. Presenting the text: pictorial tradition in fifteenth-century manuscripts of the Canterbury Tales . In: Finley WK, Rosenblum J, eds. Chaucer Illustrated: Five Hundred Years of the Canterbury Tales in Pictures . New Castle, De: Oak Knoll Press; 2003:37-72.
• BMC Public Health (numerical):
• Hardman P: Presenting the text: pictorial tradition in fifteenth-century manuscripts of the Canterbury Tales . In Chaucer Illustrated: Five Hundred Years of the Canterbury Tales in Pictures . Edited by Finley WK, Rosenblum J. New Castle, DE: Oak Knoll Press; 2003:37-72.

7.2.3 Footnote and Endnote References

Providing references via footnotes (at the bottom of the pages of a thesis) or endnotes (at the end of each chapter or in the final matter of the thesis) is a very different style of referencing than the author–date and numerical systems. It is now almost exclusively restricted to the humanities, where its capacity for accommodating a wide variety of sources is particularly appropriate. Footnote or endnote numbers appear in the main text in numerical order, usually in the form of superscript Arabic numerals; although the more cumbersome Roman numerals are occasionally used, they are not effective if a thesis contains many references. The note numbers can run in a single series through the whole of a thesis, in which case endnotes will appear in a single group at the end of the thesis. However, note numbers often begin again at the start of each chapter (drafting each chapter as a separate file will facilitate this), in which case the endnotes for each chapter can appear at the end of the chapter; if they instead appear at the end of the thesis, headings indicating to which chapters they apply should be included. Sometimes note numbers are enclosed in square brackets or parentheses – ‘[1]’ or ‘(1)’ – instead of being set in superscript font, which is also acceptable, but one format should be maintained throughout a thesis. The note numbers should be placed where appropriate in the text to reflect your use of sources, but, like numerical references, they usually follow full stops and commas (According to a recent study, 1 three groups are required) and precede colons and semicolons (The following settings were considered in a recent study 1 : internal with artificial white light, internal with blue light, internal with red light and external with natural daylight). However, some guidelines will insist that all footnotes appear either at the end of sentences or at the end of paragraphs, and in such cases several references will often need to be gathered into a single note. As a general rule, notes should not be attached to titles, headings or either the preliminary or final matter of a thesis. Although numerical references and note references would not both be used in a single thesis, if notes are used for supplementary information in a thesis that uses numerical references, it is essential that the numerals used for notes are clearly distinguished from those used for references: one type might appear in superscript font and the other in parentheses, or if there are very few notes, a system of symbols could be used for note indicators.

When references are provided in footnotes or endnotes, the notes generally include complete bibliographical information when the sources are first cited, and shorter versions of the references (usually author surnames and shortened titles) for all subsequent citations of the same sources. Using the Chicago style of referencing within notes as an example, the citation for a first and therefore full reference to a chapter within a book, including a page number for direct quotation, would appear as it does in this footnote. [1] Subsequent footnote/endnote references for the same source and page number would be recorded as the reference in this footnote is. [2] If more than one reference appears in a single footnote, the individual references are usually arranged in alphabetical order according to author surnames (although different arrangements are possible in order to place a quoted source first, for instance, or prioritise a certain source), and they are normally separated by semicolons, as the short references are in this footnote; [3] full references can be arranged and separated in the same way. With this style of referencing, a list of references is not strictly necessary because virtually all of the bibliographical information required to find sources has already been provided in the notes, with the single exception of full page ranges for chapters or articles. This missing element is therefore sometimes added at the end of the complete first reference, followed by the particular page on which the quotation is found, as is the case in this footnote. [4]  

[1] Linda Olson, “Romancing the Book: Manuscripts for ‘Euerich Inglische’,” in Opening Up Middle English Manuscripts: Literary and Visual Approaches , Kathryn Kerby-Fulton, Maidie Hilmo and Linda Olson (Ithaca, NY: Cornell University Press, 2012), 102.

[2] Olson, “Romancing the Book,” 102.

[3] Hilmo, “Power of Images,” 158; Kerby-Fulton, “Professional Readers,” 223; Olson, “Romancing the Book,” 102.

[4] Linda Olson, “Romancing the Book: Manuscripts for ‘Euerich Inglische’,” in Opening Up Middle English Manuscripts: Literary and Visual Approaches , Kathryn Kerby-Fulton, Maidie Hilmo and Linda Olson (Ithaca, NY: Cornell University Press, 2012), 95–151, at 102.

However, a list of complete bibliographical references is usually required in any thesis, regardless of the referencing system used, so one should be included (and usually called ‘Bibliography’) when providing references in notes, in which case the full page ranges for chapters and articles will appear in the bibliography, so there is no need to include them in the notes. Complete references in the bibliography differ slightly from those in the notes in other ways as well, so the reference to the same chapter in the bibliography for a Chicago in-note method would take this form:

Olson, Linda. “Romancing the Book: Manuscripts for ‘Euerich Inglische’.” In Opening

Up Middle English Manuscripts: Literary and Visual Approaches , Kathryn Kerby-

Fulton, Maidie Hilmo and Linda Olson, 95–151. Ithaca, NY: Cornell University Press,

As with the list of references that accompanies author–date in-text citations, the bibliography provided with an in-note referencing system should be arranged alphabetically by the surnames of authors, and it is more common to subdivide (into ‘Primary Sources’ and ‘Secondary Sources,’ for example) a bibliography accompanying footnote or endnote references than it is to subdivide a list of references for author–date citations (see Section 7.2.1).

When you are using an in-note style of referencing, the primary function of footnotes or endnotes is to provide citations and bibliographical information on sources, but additional material of all kinds can also be included in the notes, making them a useful site for introducing, comparing and contrasting a wide variety of supplementary information (see Section 3.4 above). On the other hand, when you are using an author–date or numerical referencing system, any footnotes or endnotes you use should be restricted to additional information and not be used for referencing alone. This does not mean, however, that citations cannot be provided in such notes. On the contrary, notes should be treated like any other part of the thesis and provided with the same kind of in-text references, so a note in an author–date system might read as this footnote does. [1] In a numerical system of referencing the same note would read as this footnote does. [2] When notes are included in a thesis using either an author–date or numerical system of referencing, however, they have to do more than simply provide references: they have to add information such as details, explanatory notes, alternative approaches, additional evidence and the like to the main discussion. 

[1] The problem has been explored many times with strikingly different results, but particularly relevant to my work here is the fact that the same trend was revealed in the most recent study of the problem (Brockle, 2014).

[2] The problem has been explored many times with strikingly different results, but particularly relevant to my work here is the fact that the same trend was revealed in the most recent study of the problem. 7

Whether footnotes or endnotes are used for references, supplementary information or a combination of the two, the font style used in the notes should be the same as the font style used in the main body of the thesis, but the text in notes can be a little smaller than the text in the chapters of the thesis (a 10-point font, for instance, if a 12-point font is used in the main text). Remember, however, that the text must remain clear and legible in relation to the size of the main text, which is a particular concern when complex equations, special characters and superscript elements are included in the notes. With the exception of in-note references proper, which should observe the required format of the relevant referencing style whether they are full bibliographical references or shortened versions, both footnotes and endnotes should be written in complete, grammatically correct and properly punctuated English sentences. Although notes are sometimes treated by authors as though they are appropriate places for point-form information or English that is informal or shorthand, this is not a scholarly approach and should be avoided. For further information on these and other aspects of constructing footnotes and endnotes, see Section 3.4.

PRS Tip: Even when all relevant guidelines are followed and the utmost care is taken to cite sources accurately and construct a thorough reference list or bibliography, errors and inconsistencies will inevitably creep in, and correcting them is especially time-consuming because each and every detail needs to be checked against the sources and usually against the style guide or guidelines you are using. A professional proofreader familiar with academic and scientific referencing systems and styles can be of enormous assistance in checking the accuracy and consistency of all the niggling details associated with both in-text references and reference lists, but a proofreader cannot provide citations or construct a reference list for you. Remember that you are the only one who can truly judge when a reference to a source is appropriate, and in most cases you are also the only one among your first readers (including proofreaders) who has access to the complete bibliographical information for the sources you have used. So the key to making a proofreader’s work serve you and your references well is to do your very best to provide appropriate citations in your thesis chapters and complete bibliographical information in your list in a consistent manner that conforms as closely as you can manage to the style with which you are working. Then send your thesis and your reference list, along with information about any style or guidelines you are following, to PRS for a final check and polishing. If you are using your own format for your references, be especially careful to be as consistent as possible throughout your text and list, because only if you set the pattern clearly can a proofreader determine what that pattern is and help you conform to it as he or she ensures that all your references are complete and adhere to good academic or scientific practice.

Why PhD Success?

To Graduate Successfully

This article is part of a book called "PhD Success" which focuses on the writing process of a phd thesis, with its aim being to provide sound practices and principles for reporting and formatting in text the methods, results and discussion of even the most innovative and unique research in ways that are clear, correct, professional and persuasive.

The assumption of the book is that the doctoral candidate reading it is both eager to write and more than capable of doing so, but nonetheless requires information and guidance on exactly what he or she should be writing and how best to approach the task. The basic components of a doctoral thesis are outlined and described, as are the elements of complete and accurate scholarly references, and detailed descriptions of writing practices are clarified through the use of numerous examples.

The basic components of a doctoral thesis are outlined and described, as are the elements of complete and accurate scholarly references, and detailed descriptions of writing practices are clarified through the use of numerous examples. PhD Success provides guidance for students familiar with English and the procedures of English universities, but it also acknowledges that many theses in the English language are now written by candidates whose first language is not English, so it carefully explains the scholarly styles, conventions and standards expected of a successful doctoral thesis in the English language.

Individual chapters of this book address reflective and critical writing early in the thesis process; working successfully with thesis supervisors and benefiting from commentary and criticism; drafting and revising effective thesis chapters and developing an academic or scientific argument; writing and formatting a thesis in clear and correct scholarly English; citing, quoting and documenting sources thoroughly and accurately; and preparing for and excelling in thesis meetings and examinations. 

Completing a doctoral thesis successfully requires long and penetrating thought, intellectual rigour and creativity, original research and sound methods (whether established or innovative), precision in recording detail and a wide-ranging thoroughness, as much perseverance and mental toughness as insight and brilliance, and, no matter how many helpful writing guides are consulted, a great deal of hard work over a significant period of time. Writing a thesis can be an enjoyable as well as a challenging experience, however, and even if it is not always so, the personal and professional rewards of achieving such an enormous goal are considerable, as all doctoral candidates no doubt realise, and will last a great deal longer than any problems that may be encountered during the process.

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Rene Tetzner

Rene Tetzner's blog posts dedicated to academic writing. Although the focus is on How To Write a Doctoral Thesis, many other important aspects of research-based writing, editing and publishing are addressed in helpful detail.

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