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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

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Podlesny, Joscha: Multiscale modelling and simulation of deformation accumulation in fault networks

Kahnt, max: numerical approximation of non-isothermal multi-component, multi-phase field systems.

The subject of this thesis is the derivation and analysis of numerical approximations of multi-component, multi-phase field systems. Recent approximations of solutions to such models are mostly based on explicit time stepping schemes and require the computation of many time steps. Implicit methods exhibit inherent numerical challenges, in particular due to the non-smoothness of the underlying energy functionals.

Our focus lies on the derivation of numerical approximations within the thermodynamically consistent context with high efficiency and robustness. We aim to exploit the special mathematical structure of the model and the underlying thermodynamics without introducing additional regularizations.

We introduce the thermodynamic and multi-phase setting in chapter 2 and continue by motivating and presenting a thermodynamically consistent multi-component, multi-phase field model in chapter 3. Based on Rothe’s method, we obtain a semi-discretization allowing for adaptive meshes in chapter 4 and the implicit problems are analyzed. In chapter 5, a full discretization with adaptive finite elements based on hierarchical a posteriori error estimation is set up. We transition to a purely algebraic formulation and present the iterative approximation of solutions with a nonsmooth Schur–Newton multigrid approach in chapter 6. Finally, in chapter 7, we perform numerical experiments to underline the thermodynamical consistency and numerical efficiency of our method.

Kies, Tobias: Gradient methods for membrane-mediated particle interactions

Discrete-continuous hybrid models are a popular means for describing elastic membrane-mediated particle interactions in and on lipid bilayers. Here, the continuous part is usually given by an approximation of the lipid membrane by an infinitely thin and sufficiently smooth hypersurface, whose elastic energy is determined by a Canham-Helfrich type functional. The discrete component results from modeling non-membrane particles as rigid discrete entities, which, depending on their configuration, induce local constraints on the membrane along the membrane-particle interfaces. In this context, the interaction potential describes the optimal elastic energy of such hybrid systems with a fixed particle configuration. Correspondingly, the energy minimization principle yields that stationary particle configurations are given by the local minima of the interaction potential. The main goal of this work is the proof of differentiability of the interaction potential for a selected class of models. This is accomplished using a variational approach that is already established in the literature in order to develop and apply robust numerical optimization methods for computing stationary particle configurations. Correspondingly, an additional focus is the derivation of a numerically accessible representation of the gradient, including its discretization and relevant numerical analysis. The proof of differentiability is brought forward by an application of the implicit function theorem. The basis for this is so-called boundary preserving domain transformations, which are induced by suitable families of vector fields and which locally admit the reformulation of the minimization problem that is implicitly defined by the interaction potential with respect to a fixed particle configuration. This subsequently enables the representation of the gradient as a volume integral using matrix analysis methods. The discretization of the partial differential equations for describing optimal membrane shapes is done via finite element methods. For particle methods with so-called curve restrictions a fictitious domain stabilized Nitsche method is developed, and for models with point value restrictions a conforming Galerking discretization is made possible by local QR transformations of the nodal finite element basis. For both cases suitable a priori error estimates are proven, and in addition also error estimates for the volume representation of the gradient are shown within that context. These developed methods open up the domain of efficient simulation of macro structures by isotropic and anisotropic particles, which is illustrated with the aid of various example applications and by means of perturbed gradient methods.

Djurdjevac, Ana: Random partial differential equations on evolving hypersurfaces

Partial differential equations with random coefficients (random PDEs) is a very developed and popular field. The variety of applications, especially in biology, motivate us to consider the random PDEs on curved moving domains. We introduce and analyse the advection-diffusion equations with random coefficients on moving hypersurfaces. We consider both cases, uniform and log-normal distributions of coefficients. Furthermore, we will introduce and analyse a surface finite element discretisation of the equation. We show unique solvability of the resulting semi-discrete problem and prove optimal error bounds for the semi-discrete solution and Monte Carlo samplings of its expectation. Our theoretical findings are illustrated by numerical experiments. In the end we present an outlook for the case when the velocity of a hypersurface is an uniformly bounded random field and the domain is flat.

Youett, Evgenia: Adaptive multilevel Monte Carlo methods for random elliptic problems

In this thesis we introduce a novel framework for uncertainty quantification in problems with random coefficients. The developed framework utilizes the ideas of multilevel Monte Carlo (MLMC) methods and allows for exploiting the advantages of adaptive finite element techniques. In contrast to the standard MLMC method, where levels are characterized by a hierarchy of uniform meshes, we associate the MLMC levels with a chosen sequence of tolerances. Each deterministic problem corresponding to a MC sample on a given level is then approximated up to the corresponding accuracy. This can be done, for example, using pathwise a posteriori error estimation and adaptive mesh refinement techniques. We further introduce an adaptive MLMC finite element method for random linear elliptic problems based on a residual-based a posteriori error estimation technique. We provide a careful analysis of the novel method based on a generalization of existing results, for deterministic residual-based error estimation, to the random setting. We complement our theoretical results by numerical simulations illustrating the advantages of our approach compared to the standard MLMC finite element method when applied to problems with random singularities.

Youett, Jonathan William: Dynamic large deformation contact problems and applications in virtual medicine

Dynamic large deformation contact problems arise in many industrial applications like auto mobile engineering or biomechanics but only few methods exists for their numerical solution, all having their advantages and disadvantages. In this thesis the numerical solution of large deformation contact problems is tackled from an optimisation point of view and an application of this approach within a femoroacetabular impingement analysis is described. In this thesis we use a non-smooth Hamilton principle and Fréchet subdifferential calculus to derive a weak formulation of the problem. The resulting subdifferential inclusion is discretised in time by constructing a contact-stabilised midpoint rule. For the spatial discretisation the state-of- the-art dual mortar method is applied which results in non-convex constrained minimisation problems that have to be solved solved during each time step. For the solution of these problems an inexact filter trust-region method is derived which allows to use inexact linearisations of the non-penetration constraints. This method in combination with fast monotone multigrid method is then shown to be globally convergent.

Hardering, Hanne: Intrinsic Discretization Error Bounds for Geodesic Finite Elements

This work is concerned with the proof of optimal error bounds for the discretization of $H^1$-elliptic minimization problem with solutions taking values in a Riemannian manifold. The discretization is done using Geodesic Finite Elements, a method of arbitrary order that is invariant under isometries. The discretization error is considered both intrinsically in a specially introduced Sobolev-distance as well as extrinsically. Optimal estimates of $H^1$- and $L^2$-type are shown, that have been observed experimentally in previous works of other authors. Using the Rothe method consisting of an implicit Euler method for the time discretization and Geodesic Finite Elements for the spatial discretization, error estimates for $L^2$-gradient flows of $H^1$-elliptic energies are derived as well. The core of the work is formed by the discretization error estimates for minimization problems in instrinsic $H^1$- and $L^2$-distances. To derive these, inverse estimates and interpolation errors for Geodesic Finite Elements and their discrete variations are shown. Using a nonlinear Cea's Lemma, this leads to the $H^1$-error estimate for minimizers of $H^1$-elliptic energies. A generalization of the Aubin-Nitsche-Lemma shows optimal $L^2$-error estimates for (essentially) semilinear energies, as long as the dimension of the domain of the minimizer is limited to $d<4$ for technical reasons. All results are illustrated using harmonic maps into a smooth Riemannian manifold satisfying certain curvature bounds as an example.

Pipping, Elias: Dynamic problems of rate-and-state friction in viscoelasticity

In this work, the model of rate-and-state friction, which can be viewed as central to the numerical simulation of earthquakes, is considered from a mathematical point of view. First, a framework is presented through which a general class of such friction laws can be understood and analysed. A prototypical viscoelastic problem of earthquake rupture is then formulated, both in strong and in variational form. Analysis of this problem is difficult, since the incorporation of rate-and-state friction leads to a coupling of variables. In a time-discrete setting, nonetheless, results on existence, uniqueness, and continuous parameter dependence of solutions can be obtained. The principal idea is to reformulate the variable interdependence as a fixed point problem and to prove convergence for a corresponding iteration. With that in mind, next, a numerical algorithm is proposed that resolves the coupling through a fixed point iteration. Since it puts a state-of-the-art solver and adaptive time stepping to use, it is not only stable but also fast. Its applicability to problems of interest is demonstrated in the penultimate chapter, which focuses on simulations of megathrust earthquakes that form at the base of a subduction zone. The main assumptions made throughout this work are summarised and discussed in the last chapter.

Sack, Uli: Numerical Simulation of Phase Separation in Binary and Multicomponent Systems

The ban of lead in electronics solder by EU directives results in the technological challenge to develop lead-free alternatives with comparable life span and processing properties. Numerical simulations of the microstructure evolution may contribute to identify promising candidates and thus focus the immense experimental effort. Aim of this work is on the one hand to develop a numerical framework for the efficient and robust simulation of the microstructure evolution in binary alloys combining adaptive finite element methods with fast solvers for the Cahn-Hilliard model. On the other hand we will extend the existing fast solvers for the discrete scalar Cahn-Hilliard equation to the vector-valued case. After some preliminary remarks on phase diagrams, phase separation, and phasefield models in Chapter 1 we will firstly discuss anisotropic Allen-Cahn equations in Chapter 2. Alle-Cahn-like problems arise as subproblems in the Nonsmooth Schur-Newton (NSNMG) method for Cahn- Hilliard equations in Chapters 3 and 4. Here we prove existence and uniqueness of solutions to the anisotropic Allen-Cahn equation with logarithmic potential using the theory of maximal monotone operators. For the numerical solution we introduce an adaptive spatial mesh refinement cycle for evolution problems and several variants of implicit Euler time discretization. We prove stability for the latter and numerical experiments conclude the chapter. Chapter 3 combines existing and newly developed numerical tools to a simulation software for microstructure evolution in binary alloys. Key ingredients are the adaptive mesh refinement cycle of Chapter 2, the NSNMG solver, a quantification algorithm for measuring "coarseness" of microstructures and a quotient space multigrid method for indefinite problems. An application of this software to simulate the microstructure evolution in a eutectic AgCu alloy shows only marginal impact of elastic stresses on coarsening in the setting considered; while the use of a smooth interpolant of the logarithmic potential affects the coarsening dynamics considerably. In the final chapter we consider the multicomponent Cahn-Hilliard equation and derive a unified formulation for the discrete problems which allows a direct application of the NSNMG method. Existence and uniqueness of discrete solution are proved and numerical examples illustrate the robustness of the scheme with respect to temperature, mesh size, and number of components.

Gräser, Carsten: Convex minimization and phase field models

Phase field models are a widely used approach to describe physical processes that are characterized by thin interfacial regions between large almost homogeneous domains. Important application areas of phase field models are transition processes of the state of matter and the separation of alloys. A fundamental property of these models is, that the transition and separation of phases is driven by a double-well potential with distinct minima for the different phases. Already the pioneering work of Cahn and Hilliard used a temperature dependent logarithmic potential that is differentiable with singular derivatives. If the temperature tends to zero it degenerates to the non-differentiable obstacle potential. The goal of this thesis is to develop methods for the efficient numerical solution of such equations that are also robust for nonsmooth potentials and anisotropic surface energies. These methods are derived for the Cahn-Hilliard equation that are prototypic for a multitude of such models. The main result of the thesis is the development of a fast iterative solver for nonlinear saddle point problems like the ones that arise from finite element discretization of Cahn-Hilliard equations. The solver relies on a reformulation of the problem as dual minimization problem whose energy functional is differentiable. The gradient of this functional turns out to be the nonlinear Schur complement of the saddle point problem. Generalized linearizations for the Schur complement are derived and used for a nonsmooth Newton method. Global convergence for this 'Schur Nonsmooth Newton' method and inexact versions is proved using the fact the equivalence to a descent method for the dual minimization problem. Each step of this method requires the solution of a nonlinear convex minimization problem. To tackle this problem the 'Truncated Nonsmooth Newton Multigrid' (TNNMG) method is developed. In contrast to other nonlinear multigrid methods the TNNMG method is significantly easier to implement and can also be applied to anisotropic problems while its convergence speed is in general comparable or sometimes even faster. Numerical examples show that the derived methods exhibit mesh independent convergence. Furthermore they turn out to be robust with respect to the temperature including the limiting case zero. The reason for this robustness is, that all methods do not rely on smoothness but on the inherent convex structure of the problems.

Forster, Ralf: On the stochastic Richards equation

A frequent problem during numerical computations consists in the uncertainty of certain model parameters due to measuring errors or their high variability. In the last years, one could observe an increasing interest in the quantification of these uncertainties and their effects to the solution of numerical simulations; a powerful tool which has been proven to be an efficient approach in this context is the so-called polynomial chaos method which is based on a spectral decomposition of the covariance function of the uncertain parameters and a representation of the solution in a polynomial basis. The aim of this thesis is the application of this method to the Richards equation modeling groundwater flow in saturated and unsaturated porous media. The main difficulty consists in the saturation and the hydraulic conductivity appearing in the time derivative and in the spatial derivatives, since both depend nonlinearly on the pressure. Considering uncertain parameters like random initial and boundary conditions and, in particular, a random permeability leads to a stochastic variational inequality of second kind with obstacle conditions and a nonlinear convex functional as superposition operator. Considering variational inequalities in the context of uncertain parameters and the polynomial chaos method is new, and we start by deriving a weak formulation of the problem and approximating the parameters by a Karhunen-Loève expansion. The existence of a unique solution u in a tensor space can be proven for the time-discrete problem by reformulation as a convex minimization problem. We proceed by discretizing with finite elements and polynomial ansatz functions and by approximating the convex functional with Gaussian quadrature. The convergence of the solution of the discretized problem to the solution u is proved in a special case for a stochastic obstacle problem. Moreover, we perform numerical experiments to determine the discretization error. In the second part of this thesis, we develop an efficient numerical method to solve the discretized minimization problems. It is based on a global converging Block Gauß Seidel method and exploits a transformation which decouples the stochastic coefficients and connects the stochastic Galerkin with the stochastic collocation approach. This also allows us to establish a multigrid solver to accelerate the convergence. We conclude this thesis by demonstrating the power of our approach on a realistic example with lognormal permeability and exponential covariance.

Sander, Oliver: Multidimensional coupling in a human knee model

The thesis presents a new model for the numerical simulation of the mechanics of the human knee. In this model bones are described using linear elasticity. Ligaments instead are modelled as one-dimensional Cosserat rods. The simulations give insight into the mechanical behavior of human joints. This can be helpful for a number of applications. For example, it is possible to estimate the long-term effect of certain surgical interventions. Also, the design of prosthetic devices can be improved. The main mathematical focus is on the correct formulation of the coupling conditions between one- and three- dimensional objects. Starting from the case of two three-dimensional objects, for which coupling conditions can be derived rigorously, conditions for the multidimensional case are formulated. A solution algorithm for this coupled problem is presented, and the existence of solutions is shown under certain symmetry assumptions. For the subproblems, large contact problems and minimization problems on Riemannian manifolds have to be solved. For both problems, robust and efficient numerical methods are introduced. Numerical experiments show the applicability for real-world problems.

Berninger, Heiko: Domain decomposition methods for elliptic problems with jumping nonlinearities and application to the Richards equation

The thesis presents a new method for the solution of saturated-unsaturated groundwater flow problems in heterogeneous porous media. Concretely, highly nonlinear degenerate elliptic problems arising from a certain time discretization of the Richards equation are the basis of this work. The problems are considered as homogeneous in subdomains where a single soil prevails and, therefore, the parameter functions do not depend on space. These nonlinearities, however, may jump across the interfaces between the subdomains and, thus, account for the heterogeneous setting of different soil types in different subdomains. As a consequence, non-overlapping domain decomposition problems in which subproblems are coupled via nonlinear transmission conditions are obtained. In this work these problems are solved without any linearization. By Kirchhoff transformation the homogeneous subproblems are transformed into convex minimization problems. Here, additional constraints like Signorini-type boundary conditions, which occur on seapage faces around lakes, can be taken into account. Finite elements are chosen for the space discretization, and convex analysis is applied as the solution theory. Finally, monotone multigrid methods provide efficient solvers which are robust with respect to degenerating soil parameters. In order to deal with the coupling of the homogeneous subproblems, nonlinear Dirichlet-Neumann and Robin methods are used. Here, the thesis provides new convergence results for these iterations applied to nonlinear elliptic problems in 1D as well as well- posedness results, which generalize existing linear theory. On the other hand, detailed numerical experiments demonstrate that the methods can also be applied successfully to problems in 2D. Finally, based on the artificial viscosity method, an upwind discretization with finite elements is developed in order to account for gravity. Hence, stability of the numerical solutions is obtained. In a closing numerical example the Richards equation is solved in 2D with four different soils and coupled to a surface water reservoir. The result demonstrates the applicability of the developed solution technique to a heterogeneous problem with realistic hydrological data.

Gebauer, Susanna: Hierarchical Domain Decomposition Methods for saturated Groundwaterflow in fractured porous Media

The focus of this thesis ist the numerical computation of flow in special geometries dominated by jumps in the flow coefficients and large differences in the scales of the main flow pathes and the surrounding materials. These characteristics result in difficulties in the numerical computation of the modelling equations. By means of groundwaterflow in fractured porous media we present a hierarchical domain decomposition method for the numerical computation of flow. Under certain assumptions this new method converges independently of the fracture width, the refinement depth and the jump in the flow coefficient. The theoretical results are confirmed by practical computations of a model problem and a fracture network. Thus for a new class of completely overlapping domain decomposition methods multigrid efficiency is shown for a class of problems, for which so far no comparably theoretically validated method existed.

Krause, Rolf H.: Monotone Multigrid Methods for Signorini's Problem with Friction

In this work, we consider the numerical simulation of contact problems. Since the numerical realization of contact problems is of high importance in many application areas, there is a strong demand for fast and reliable simulation method. We introduce and analyze a new nonlinear multigrid method for solving contact problems with and without friction. As it turns out, by means of our new method nonlinear contact problems can be solved with a computational amount comparable that of linear problems. In particular, in our numerical experiments we observe our method to be of optimal complexity. Moreover, since we do not use any regularization techniques, the computed discrete boundary stresses as well as the computed displacements turn out to be highly accurate. The new method is based on the succesive minimization of the associated energy functional in direction of properly choosen functions. We show the global convergence of our method and give several numerical examples in two and three space dimensions, illustrating the robustness and the performance of the method. In addition to the theoretical analysis, the method has been implemented in an object oriented way. We explain the concepts of our implementation and show the flexibility of our approach by deriving a nonlinear algebraic multigrid method. To include frictional effects, we use a discrete fixed point iteration. As a faster alternative, also a Gauss-Seidel like iteration scheme is proposed. Both methods are compared in numerical examples. The resulting nonlinear algorithm turns out to be fast and reliable. Finally, we consider the case of contact between elastic bodies. Here, the information transfer at the interface is realized by means of non conforming domain decomposition methods (mortar methods). This gives rise to a non-linear Dirichlet Neumann Algorithm.

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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.J. 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. 

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.

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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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Journal of Applied Mathematics, 2013

Three numerical methods have been used to solve the one-dimensional advection-diffusion equation with constant coefficients. This partial differential equation is dissipative but not dispersive. We consider the Lax-Wendroff scheme which is explicit, the Crank-Nicolson scheme which is implicit, and a nonstandard finite difference scheme (Mickens 1991). We solve a 1D numerical experiment with specified initial and boundary conditions, for which the exact solution is known using all these three schemes using some different values for the space and time step sizes denoted byhandk, respectively, for which the Reynolds number is 2 or 4. Some errors are computed, namely, the error rate with respect to theL1norm, dispersion, and dissipation errors. We have both dissipative and dispersive errors, and this indicates that the methods generate artificial dispersion, though the partial differential considered is not dispersive. It is seen that the Lax-Wendroff and NSFD are quite good methods to ...

Applied Mathematics and Computation, 2016

Three numerical methods have been used to solve two problems described by advection-diffusion equations with specified initial and boundary conditions. The methods used are the third order upwind scheme [4], fourth order upwind scheme [4] and Non-Standard Finite Difference scheme (NSFD) [9]. We considered two test problems. The first test problem has steep boundary layers near x = 1 and this is challenging problem as many schemes are plagued by non-physical oscillation near steep boundaries [15]. Many methods suffer from computational noise when modelling the second test problem especially when the coefficient of diffusivity is very small for instance 0.01. We compute some errors, namely L 2 and L ∞ errors, dissipation and dispersion errors, total variation and the total mean square error for both problems and compare the computational time when the codes are run on a matlab platform. We then use the optimization technique devised by Appadu [1] to find the optimal value of the time step at a given value of the spatial step which minimizes the dispersion error and this is validated by some numerical experiments.

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...

Numerical Heat Transfer, Part B: Fundamentals, 2019

The aim of the present work is to propose and modify several numerical methods including the classes of temporal discretization methods for hyperbolic conservation laws. The first order in space standard Lax approximation is updated to modified first-order and newly proposed third-order accurate approximation. Presently proposed methods can be coupled with the modified and newly proposed Lax approximations and this coupling make the methods conservative. Some additional new classes of explicit and implicit methods for PDEs in time are proposed. Additionally, some new methods are given to reduce oscillations in the solutions. These new methods of reducing oscillations provide the conditions for coupling of first and higher-order methods.

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, 2009

This article deals with the study of the development and application of the high-order upwind ADBQUICKEST scheme, an adaptative bounded version of the QUICKEST for unsteady problems (Commun. Numer. Meth. Engng 2007; 23:419-445), employing both linear and nonlinear convection term discretization. This scheme is applicable to a wide range of computational fluid dynamics problems, where transport phenomena are of special importance. In particular, the performance of the scheme is assessed through an extensive numerical simulation study of advection-diffusion problems. The scheme, implemented in the context of finite difference methodology, combines a good approximation of shocks (or discontinuities) with a good approximation of the smooth parts of the solutions. In order to assess the performance of the scheme, seven problems are solved, namely (a) advection of scalars; (b) non-linear viscous Burgers equation; (c) Euler equations of gas dynamics; (d) Newtonian flow in a channel; (e) axisymmetric Newtonian jet flow; (f) axisymmetric non-Newtonian (generalized Newtonian) flow in a pipe; and (g) collapse of a fluid column. The numerical experiments clearly show that the scheme provides more consistent solutions than those found in the literature. From the study, the flexibility and robustness of the ADBQUICKEST scheme is confirmed by demonstrating its capability to solve a variety of linear and nonlinear problems with and without discontinuous solutions.

AL-Rafidain Journal of Computer Sciences and Mathematics

The objective of this paper is to construct numerical schemes using finite difference methods for the one-dimensional general hyperbolicparabolic-reaction problem. The finite difference method with the exponential transformation form is used to solve the problem, and employs difference approximation technique to obtain the numerical solutions. Computational examples are presented and compared with the exact solutions. We obtained that the Crank-Nicholson scheme is more accurate than Forward scheme. Therefore the form of exponential transformation for the problem yields a stable solution compared with exact solution.

This paper presents the comparison of three different and unique finite difference schemes used for finding the solutions of parabolic partial differential equations (PPDE). Knowing fully that the efficiency of a numerical schemes depends solely on their stability therefore, the schemes were compared based on their stability using von Newmann method. The implicit scheme and Dufort-Frankel schemes using von Newmann stability method are unconditionally stable, while the explicit scheme is conditionally stable. The schemes were also applied to solve a one dimensional parabolic partial differential equations (heat equation) numerically and their results compared for best in efficiency. The numerical experiments as seen in the tables presented and also the percentage errors, which proves that the implicit scheme is good compare to the other two schemes. Also, the implementation of the implicit scheme is faster than that of the explicit and Dufort-Frankel schemes. The results obtained in work also compliment and agrees with the results in literature.

In cognitive science, creative ideas are defined as original and feasible solutions in response to problems. A common proposal is that creative ideas are generated across dedicated cognitive pathways. Only after creative ideas have emerged, they can be enacted to solve the problem. We present an alternative viewpoint, based upon the dynamic systems approach to perception and action, that creative solutions emerge in the act rather than before. Creative actions, thus, are as much a product of individual constraints as they are of the task and environment constraints. Accordingly, we understand creative motor actions as functional movement patterns that are new to the individual and/or group and adapted to satisfy the constraints on the motor problem at hand. We argue that creative motor actions are promoted by practice interventions that promote exploration by manipulating constraints. Exploration enhances variability of functional movement patterns in terms of either coordination or control solutions. At both levels, creative motor actions can emerge from finding new and degenerate adaptive motor solutions. Generally speaking, we anticipate that in most cases, when exposed to variation in constraints, people are not looking for creative motor actions, but discover them while doing an effort to satisfy constraints. For future research, this paper achieves two important aspects: it delineates how adaptive (movement) variability is at the heart of (motor) creativity, and it sets out methodologies toward stimulating adaptive variability.

Annals of Public and Cooperative Economics, 2015

recognized as the valuable tool for economic development in low-income countries. However, recently researchers reported that one of their primary challenges to their expansion is the high level of inefficient. In this study, we analyzed the relationship between growth and efficiency of SACCOS using economies of scale concept. Then we address the role of management of the capital structure and allocation of resources in the expansion of SACCOS. The study used financial statement data from 60 SACCOS in Tanzania for the period of 2004-2011. The findings supports that most of SACCOS are small and cost inefficient because the industry is young, but, the efficiency increases as SACCOS expand. Second the allocation of resources in liquid, financial and nonfinancial investment leads to no expansion in SACCOS. Thus, the growth of SACCOS via increasing loan to members, members' savings, shares, and institutional capital should be encouraged as it increases the efficiency of SACCOS. Also, SACCOS should minimize the allocation of assets in other investments which are different from credit to members.

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