On the application of partial differential equations and fractional partial differential equations to images and their methods of solution

This body of work examines the plausibility of applying partial di erential equations and time-fractional partial di erential equations to images. The standard di usion equation is coupled with a nonlinear cubic source term of the Fitzhugh-Nagumo type to obtain a model with di usive properties and a binarizing e ect due to the source term. We examine the e ects of applying this model to a class of images known as document images; images that largely comprise text. The e ects of this model result in a binarization process that is competitive with the state-of-the-art techniques. Further to this application, we provide a stability analysis of the method as well as high-performance implementation on general purpose graphical processing units. The model is extended to include time derivatives to a fractional order which a ords us another degree of control over this process and the nature of the fractionality is discussed indicating the change in dynamics brought about by this generalization. We apply a semi-discrete method derived by hybridizing the Laplace transform and two discretization methods: nite-di erences and Chebyshev collocation. These hybrid techniques are coupled with a quasi-linearization process to allow for the application of the Laplace transform, a linear operator, to a nonlinear equation of fractional order in the temporal domain. A thorough analysis of these methods is provided giving rise to conditions for solvability. The merits and demerits of the methods are discussed indicating the appropriateness of each method

Lectures on Computational Numerical Analysis of Partial Differential Equations

From Chapter 1: The purpose of these lectures is to present a set of straightforward numerical methods with applicability to essentially any problem associated with a partial differential equation (PDE) or system of PDEs independent of type, spatial dimension or form of nonlinearity.https://uknowledge.uky.edu/me_textbooks/1002/thumbnail.jp

Numerical methods for solving hyperbolic and parabolic partial differential equations

The main object of this thesis is a study of the numerical 'solution of hyperbolic and parabolic partial differential equations. The introductory chapter deals with a general description and classification of partial differential equations. Some useful mathematical preliminaries and properties of matrices are outlined. Chapters Two and Three are concerned with a general survey of current numerical methods to solve these equations. By employing finite differences, the differential system is replaced by a large matrix system. Important concepts such as convergence, consistency, stability and accuracy are discussed with some detail. The group explicit (GE) methods as developed by Evans and Abdullah on parabolic equations are now applied to first and second order (wave equation) hyperbolic equations in Chapter 4. By coupling existing difference equations to approximate the given hyperbolic equations, new GE schemes are introduced. Their accuracies and truncation errors are studied and their stabilities established. Chapter 5 deals with the application of the GE techniques on some commonly occurring examples possessing variable coefficients such as the parabolic diffusion equations with cylindrical and spherical symmetry. A complicated stability analysis is also carried out to verify the stability, consistency and convergence of the proposed scheme. In Chapter 6 a new iterative alternating group explicit (AGE) method with the fractional splitting strategy is proposed to solve various linear and non-linear hyperbolic and parabolic problems in one dimension. The AGE algorithm with its PR (Peaceman Rachford) and DR (Douglas Rachford) variants is implemented on tridiagonal systems of difference schemes and proved to be stable. Its rate of convergence is governed by the acceleration parameter and with an optimum choice of this parameter, it is found that the accuracy of this method, in general, is better if not comparable to that of the GE class of problems as well as other existing schemes. The work on the AGE algorithm is extended to parabolic problems of two and three space dimensions in Chapter 7. A number of examples are treated and the DR variant is used because of consideration of stability requirement. The thesis ends with a summary and recommendations for future work

Error analysis of truncated expansion solutions to high-dimensional parabolic PDEs

We study an expansion method for high-dimensional parabolic PDEs which constructs accurate approximate solutions by decomposition into solutions to lower-dimensional PDEs, and which is particularly effective if there are a low number of dominant principal components. The focus of the present article is the derivation of sharp error bounds for the constant coefficient case and a first and second order approximation. We give a precise characterisation when these bounds hold for (non-smooth) option pricing applications and provide numerical results demonstrating that the practically observed convergence speed is in agreement with the theoretical predictions

Multiscale modelling of evolving foams

We present a set of multi-scale interlinked algorithms to model the dynamics of evolving foams. These algorithms couple the key effects of macroscopic bubble rearrangement, thin film drainage, and membrane rupture. For each of the mechanisms, we construct consistent and accurate algorithms, and couple them together to work across the wide range of space and time scales that occur in foam dynamics. These algorithms include second order finite difference projection methods for computing incompressible fluid flow on the macroscale, second order finite element methods to solve thin film drainage equations in the lamellae and Plateau borders, multiphase Voronoi Implicit Interface Methods to track interconnected membrane boundaries and capture topological changes, and Lagrangian particle methods for conservative liquid redistribution during rearrangement and rupture. We derive a full set of numerical approximations that are coupled via interface jump conditions and flux boundary conditions, and show convergence for the individual mechanisms. We demonstrate our approach by computing a variety of foam dynamics, including coupled evolution of three-dimensional bubble clusters attached to an anchored membrane and collapse of a foam cluster

High Performance Computing Based Methods for Simulation and Optimisation of Flow Problems

The thesis is concerned with the study of methods in high-performance computing for simulation and optimisation of flow problems that occur in the framework of microflows. We consider the adequate use of techniques in parallel computing by means of finite element based solvers for partial differential equations and by means of sensitivity- and adjoint-based optimisation methods. The main focus is on three-dimensional, low Reynolds number flows described by the instationary Navier-Stokes equations

Locally Implicit Time Integration for Linear Maxwell\u27s Equations

This thesis is concerned with the full discretization of Maxwell\u27s equations in cases where the spatial discretization has to be carried out with a locally refined grid. In such situations locally implicit time integrators are an appealing choice for the time discretization since they overcome the grid-induced stiffness of these problems. We analyze such a locally implicit time integrator in the case where the space discretization stems from a central fluxes discontinuous Galerkin method. In fact, we prove its stability under a CFL condition which solely depends on the coarse part of the spatial grid and give a rigorous error analysis showing that the integrator is second order convergent. Moreover, we extend this time integrator so that it can be applied to an upwind fluxes discontinuous Galerkin space discretization. We show that this novel integrator preserves the second order temporal convergence and that it inherits the improved properties of an upwind fluxes discretization (better stability, higher spatial convergence rate) compared to the central fluxes case

Glosarium Matematika

273 p.; 24 cm

Numerical simulation of shock wave/turbulent boundary layer interactions in over-expanded nozzles

The performances of first-stage liquid rocket engines are highly dependent on the fluid dynamic behaviour of the expansion nozzle and for launch-trajectory optimisation purposes, large values of the ratio between the exit and throat areas are desirable. The maximum limit to this ratio is imposed by the need to avoid internal flow separation, since at sea level the flow is highly overexpanded. However, during the start-up phase the chamber pressure is below the design pressure and the flow separates from the nozzle wall. This condition is characterised by complex physical features, including the formation of a shock-wave system that adapts the exhaust flow to the higher ambient pressure, shock-wave/boundary-layer interactions (SWBLI), and a turbulent recirculation zone. As a global effect, the nozzle experiences non-axial forces, known as side-loads, which can be of sufficient strength to cause structural damage to the engine. Despite several studies in the last decades, a clear physical understanding of the driving factors of the unsteadiness is still lacking. The experiments on axi-symmetric nozzles suffer from the lack of flow measurements inside the nozzle itself, due to the challenging flow conditions and absence of optical access. Therefore, numerical simulations represent an important complementary tool to gain a more complete insight into the physics of separated rocket nozzle flows, giving the opportunity to address important open questions. The present thesis investigates shock wave induced flow separation in over-expanded rocket nozzles by means of large-scale high-fidelity numerical computations based on the delayed detached eddy simulation (DDES) methodology, a hybrid RANS/LES method that allows the simulation of high-Reynolds number flows involving massive flow separation. In this approach, attached boundary layers are treated in RANS mode, lowering the computational requirements, while the most energetic turbulent scales of separated shear layers and turbulent recirculating zones are directly treated by the LES mode of the method. The potential of DDES has been first tested on a simple planar nozzle configuration for which experimental and numerical studies are available, with the main aim of highlighting the strengths and weaknesses of the approach. The results indicate that the DDES is able to capture the shock oscillations and that the computed characteristic frequency is close to that reported in literature for the same test case. The study then focuses on the investigation of the unsteadiness in a truncated ideal contoured (TIC) nozzle, a configuration for which experimental data are available. The numerical data agree well with the experimental results in terms of mean and fluctuating wall pressure statistics. The frequency spectra are characterised by the presence of a large bump in the low-frequency range associated to an axi-symmetric (piston-like) motion of the shock system and a broad and high amplitude peak at high frequencies generated by the turbulent activity of the detached shear layer. Moreover, a distinct peak at an intermediate frequency (f « 1 kHz) is observed in the wall-pressure spectra downstream of the separation shock. A Fourier-based spectral analysis performed in both time and azimuthal wave number space, reveals that this peak is associated with the first (non- symmetrical) pressure mode and is thus related to the generation of the aerodynamic side loads. Furthermore, it is found that the unsteady Mach disk is characterised by an intense vortex shedding activity that, together with the vortical structures of the annular shear layer, contributes to the sustainment of an aeroacoustic feedback loop occurring within the nozzle