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

Coupled/combined compact IRBF schemes for fluid flow and FSI problems

The thesis is concerned with the development of compact approximation methods based on Integrated Radial Basis Functions (IRBFs) and their applications in fluid flows and FSI problems. The contributions include (i) new compact IRBF stencils where first- and second-order derivatives are included; (ii) a preconditioning technique where a preconditioner to enhance the stability of the flat IRBF solutions; and, (iii) the incorporation of the proposed stencils into the immersed boundary methods. Numerical experiments show the present schemes generally produce more accurate solutions and better convergence rates than existing methods (e.g. FDM, high-order compact FDM and compact IRBF methods)

Competition between transport phenomena in a Reaction-Diffusion-Convection system

This doctoral dissertation consists of three main parts. In part one, a general overview of the basic concepts of nonlinear science, nonlinear analysis and non-equilibrium thermodynamics is presented. Kinetics of chemical oscillations and the well known Belousov-Zhabotinsky reaction are also illustrated. In part two, a Reaction-Diffusion-Convection (RDC) model is introduced as a convenient framework for studying instability scenarios by which chemical oscillators are driven to chaos, along with its translation to an opportune code for numerical simulations. In part three, we report the methods and the data obtained. We observe that distinct bifurcation points are found in the oscillating patterns as Diu-sion coecients (di) or Grashof numbers (Gri) vary. Singularly there emerge peculiar bifurcation paths, inscribed in a general scenario of the RTN type, in which quasi{periodicity transmutes into a period-doubling sequence to chemical chaos. The opposite influence exhibited by the two parameters in these transitions clearly indicate that diusion of active species and natural convection are in `competition` for the stability of ordered dynamics. Moreover, a mirrored behavior between chemical oscillations and spatio-temporal dynamics is observed, suggesting that the emergence of the two observables are a manifestation of the same phenomenon. The interplay between chemical and transport phenomena instabilities is at the general origin of chaos for these systems. Further, a molecular dynamics study has been carried out for the calculation of diusion coecients of active species in the Belousov-Zhabotinsky reaction, namely HBrO2 and Ce(III), by means of mean square displacement and velocity autocorrelation function. These data have been used for a deeper comprehension of the hydrodynamic competition observed between diusion and convective motions for the stability of the system.</br

The Du Fort and Frankel finite difference scheme applied to and adapted for a class of finance problems

We consider the finite difference method applied to a class of financial problems. Specifically, we investigate the properties of the Du Fort and Frankel finite difference scheme and experiment with adaptations of the scheme to improve on its consistency properties. The Du Fort and Frankel finite difference scheme is applied to a number of problems that frequently occur in finance. We specifically investigate problems associated with jumps, discontinuous behavior, free boundary problems and multi dimensionality. In each case we consider adaptations to the Du Fort and Frankel scheme in order to produce reliable results. CopyrightDissertation (MSc)--University of Pretoria, 2009.Mathematics and Applied Mathematicsunrestricte

Analysis of Heat Partitioning During Sliding Contact At High Speed and Pressure

This research develops a mathematical formulation and an analytical solution to frictional heat partitioning in a high speed sliding system. Frictional heating at the interface of sliding materials impacts temperature and the wear mechanisms. The heat partition fraction for a sliding system is an important parameter in calculating the distribution of frictional heat flux between the contacting surfaces. The solution presented in this dissertation considers the characteristics of the slipper\u27s frictional heat partition values along with the experimental loading data. With a physics based, rather than a phenomenological approach, this solution improves the estimate for the slipper\u27s heat partition function. Moreover, this analytical solution is practical in calculating the average surface temperature and estimating the total melt wear volume. The heat partition function compares favorably with existing experimental and analytical data. Using the Strang\u27s Splitting and ADI methods, a numerical method for surface temperature and corresponding wear percentage under dynamic bounce conditions was extensively developed

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

A Simulation of the Mississippi River Salt Wedge Estuary Using a Three-Dimensional Cartesian Z Coordinate Model

The stratified flow of the lower Mississippi River due to density gradients is a well documented phenomenon. This stratification of fresh and saline water manifests itself as a heavier wedge of saline water that extends upriver and a buoyant fresh water plume extending into the Gulf of Mexico past the Southwest Pass jetties. The maximum absolute distance of saltwater intrusion observed anywhere in the world occurred on the Mississippi River in 1939 and 1940 when saltwater was observed approximately 225 km upstream from the mouth of Southwest Pass. The U. S. Army Corps of Engineers now prevents the wedge from migrating upstream by constructing a subaqueous barrier in the river channel. A curvilinear grid was constructed representative of the modern Mississippi River delta. Boundary conditions were developed for the drought year of 2012 and the grid was tested in order to evaluate the salinity intrusion and sediment transport abilities of the Cartesian Z-coordinate Delft3D code. The Z-model proved to have the ability to propagate the saline density current as observed in the prototype. The effect of salinity on fine sediment transport is evaluated by manipulation of the settling velocity through a cosine function provided in the model code. Manipulation of the fine sediment fall velocity through the cosine function was an effective means to simulate the re-circulation of flocculated sediments in the saline wedge turbidity maxima. In addition, the Z-model capably reproduced the fine sediment concentration profiles in a fully turbulent shear flow environment. With the ability to reproduce the seasonal saline density current and its effect on sedimentation within the turbidity maxima as well as sedimentation characteristics in a fully turbulent shear flow, a model capable of analyzing all of the major processes affecting fine sediment transport within the Mississippi River salt wedge estuary has been developed

Finite Difference Computing with PDEs: A Modern Software Approach

finite difference methods; programming; python; verification; numerical methods; differential equation