Dissertations / Theses: 'Numerical solution of partial differential equations'

1

Williamson, Rosemary Anne. "Numerical solution of hyperbolic partial differential equations." Thesis, University of Cambridge, 1985. https://www.repository.cam.ac.uk/handle/1810/278503.

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Tråsdahl, Øystein. "Numerical solution of partial differential equations in time-dependent domains." Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2008. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-9752.

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Numerical solution of heat transfer and fluid flow problems in two spatial dimensions is studied. An arbitrary Lagrangian-Eulerian (ALE) formulation of the governing equations is applied to handle time-dependent geometries. A Legendre spectral method is used for the spatial discretization, and the temporal discretization is done with a semi-implicit multi-step method. The Stefan problem, a convection-diffusion boundary value problem modeling phase transition, makes for some interesting model problems. One problem is solved numerically to obtain first, second and third order convergence in time, and another numerical example is used to illustrate the difficulties that may arise with distribution of computational grid points in moving boundary problems. Strategies to maintain a favorable grid configuration for some particular geometries are presented. The Navier-Stokes equations are more complex and introduce new challenges not encountered in the convection-diffusion problems. They are studied in detail by considering different simplifications. Some numerical examples in static domains are presented to verify exponential convergence in space and second order convergence in time. A preconditioning technique for the unsteady Stokes problem with Dirichlet boundary conditions is presented and tested numerically. Free surface conditions are then introduced and studied numerically in a model of a droplet. The fluid is modeled first as Stokes flow, then Navier-Stokes flow, and the difference in the models is clearly visible in the numerical results. Finally, an interesting problem with non-constant surface tension is studied numerically.

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Ibrahem, Abdul Nabi Ismail. "The numerical solution of partial differential equations on unbounded domains." Thesis, Keele University, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.279648.

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Bratsos, A. G. "Numerical solutions of nonlinear partial differential equations." Thesis, Brunel University, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.332806.

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Sundqvist, Per. "Numerical Computations with Fundamental Solutions." Doctoral thesis, Uppsala : Acta Universitatis Upsaliensis : Univ.-bibl. [distributör], 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-5757.

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Pun, K. S. "The numerical solution of partial differential equations with the Tau method." Thesis, Imperial College London, 1985. http://hdl.handle.net/10044/1/37823.

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7

Pratt, P. "Problem solving environments for the numerical solution of partial differential equations." Thesis, University of Leeds, 1996. http://etheses.whiterose.ac.uk/1267/.

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The complexity and sophistication of numerical codes for the simulation of complex problems modelled by partial differential equations (PDEs) has increased greatly over the last decade. This makes it difficult for those without direct knowledge of the PDE software to employ it efficiently. Problem Solving Environments (PSEs) are seen as a way of making it possible to provide an easy-to-use layer surrounding the numerical software. The users can then concentrate on gaining an understanding of the physical problem through the results the code is providing. PSEs aim to aid novice and expert users in the problem specification process and to provide a natural way to solve the problem. They also decrease the time spent on the problem solving process. This study is concerned with the construction of a PSE for the numerical solution of PDEs. This is one area where PSEs can be used to particularly good effect because the solution process is complicated and error prone. The driving of numerical software and the construction of mathematical models used by the software pose problems for users of the software. The interpretation of results provided by the numerical code may also be difficult. It will be shown how PSEs can remedy these issues by allowing the user to easily specify and solve the problem. The construction of a prototype PSE is achieved through the utilisation and integration of existing scientific software tools and systems. An examination of the solution process of PDEs is used to identify the various components required in a PSE for such problems. The PSE makes use of an open design environment and incorporates the knowledge of the users and developers of the numerical code together with a set of generic software tools based on emerging standards. This combination of tools allows the PSE to automate the solution procedure for a number of PDE problems. Finally, the success of this approach to building PSEs is examined by reference to an engineering PDE problem.

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Barreira, Maria Raquel. "Numerical solution of non-linear partial differential equations on triangulated surfaces." Thesis, University of Sussex, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.496863.

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This work aims to solve numerically non-linear partial differential equations on surfaces, that may evolve in time, for a set of different applications. The core of all the numerical schemes is a finite element method recently introduced for triangulated surfaces. The main classes of applications under appreciation are the motion of curves on surfaces, segmentation of images painted on surfaces and the formation of Turing patterns on surfaces. For the first one, three different approaches are considered and compared: the level set method, the phase field framework and the diffusion generated motion method. The formation of patterns leads to an interesting application to the growth of tumours which is also investigated. Implementation of all numerical schemes proposed is carried out and some analysis on the convergence and stability of the approximations is presented. The finite element method has shown efficiency and great flexibility when it comes to the equations it can approximate and the surfaces it can handle.

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Qiao, Zhonghua. "Numerical solution for nonlinear Poisson-Boltzmann equations and numerical simulations for spike dynamics." HKBU Institutional Repository, 2006. http://repository.hkbu.edu.hk/etd_ra/727.

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Kwok, Ting On. "Adaptive meshless methods for solving partial differential equations." HKBU Institutional Repository, 2009. http://repository.hkbu.edu.hk/etd_ra/1076.

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11

Malek, Alaeddin. "Numerical spectral solution of elliptic partial differential equations using domain decomposition techniques." Thesis, Cardiff University, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.241798.

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Palitta, Davide. "Preconditioning strategies for the numerical solution of convection-diffusion partial differential equations." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2014. http://amslaurea.unibo.it/7464/.

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Il trattamento numerico dell'equazione di convezione-diffusione con le relative condizioni al bordo, comporta la risoluzione di sistemi lineari algebrici di grandi dimensioni in cui la matrice dei coefficienti è non simmetrica. Risolutori iterativi basati sul sottospazio di Krylov sono ampiamente utilizzati per questi sistemi lineari la cui risoluzione risulta particolarmente impegnativa nel caso di convezione dominante. In questa tesi vengono analizzate alcune strategie di precondizionamento, atte ad accelerare la convergenza di questi metodi iterativi. Vengono confrontati sperimentalmente precondizionatori molto noti come ILU e iterazioni di tipo inner-outer flessibile. Nel caso in cui i coefficienti del termine di convezione siano a variabili separabili, proponiamo una nuova strategia di precondizionamento basata sull'approssimazione, mediante equazione matriciale, dell'operatore differenziale di convezione-diffusione. L'azione di questo nuovo precondizionatore sfrutta in modo opportuno recenti risolutori efficienti per equazioni matriciali lineari. Vengono riportati numerosi esperimenti numerici per studiare la dipendenza della performance dei diversi risolutori dalla scelta del termine di convezione, e dai parametri di discretizzazione.

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Jayes, Mohd Idris. "Numerical solution of ordinary and partial differential equations occurring in scientific applications." Thesis, Loughborough University, 1992. https://dspace.lboro.ac.uk/2134/32103.

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Kadhum, Nashat Ibrahim. "The spline approach to the numerical solution of parabolic partial differential equations." Thesis, Loughborough University, 1988. https://dspace.lboro.ac.uk/2134/6725.

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This thesis is concerned with the Numerical Solution of Partial Differential Equations. Initially some definitions and mathematical background are given, accompanied by the basic theories of solving linear systems and other related topics. Also, an introduction to splines, particularly cubic splines and their identities are presented. The methods used to solve parabolic partial differential equations are surveyed and classified into explicit or implicit (direct and iterative) methods. We concentrate on the Alternating Direction Implicit (ADI), the Group Explicit (GE) and the Crank-Nicolson (C-N) methods. A new method, the Splines Group Explicit Iterative Method is derived, and a theoretical analysis is given. An optimum single parameter is found for a special case. Two criteria for the acceleration parameters are considered; they are the Peaceman-Rachford and the Wachspress criteria. The method is tested for different numbers of both parameters. The method is also tested using single parameters, i. e. when used as a direct method. The numerical results and the computational complexity analysis are compared with other methods, and are shown to be competitive. The method is shown to have good stability property and achieves high accuracy in the numerical results. Another direct explicit method is developed from cubic splines; the splines Group Explicit Method which includes a parameter that can be chosen to give optimum results. Some analysis and the computational complexity of the method is given, with some numerical results shown to confirm the efficiency and compatibility of the method. Extensions to two dimensional parabolic problems are given in a further chapter. In this thesis the Dirichlet, the Neumann and the periodic boundary conditions for linear parabolic partial differential equations are considered. The thesis concludes with some conclusions and suggestions for further work.

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Zeng, Suxing. "Numerical solutions of boundary inverse problems for some elliptic partial differential equations." Morgantown, W. Va. : [West Virginia University Libraries], 2009. http://hdl.handle.net/10450/10345.

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Thesis (Ph. D.)--West Virginia University, 2009.
Title from document title page. Document formatted into pages; contains v, 58 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 56-58).

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Cheung, Ka Chun. "Meshless algorithm for partial differential equations on open and singular surfaces." HKBU Institutional Repository, 2016. https://repository.hkbu.edu.hk/etd_oa/278.

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Radial Basis function (RBF) method for solving partial differential equation (PDE) has a lot of applications in many areas. One of the advantages of RBF method is meshless. The cost of mesh generation can be reduced by playing with scattered data. It can also allow adaptivity to solve some problems with special feature. In this thesis, RBF method will be considered to solve several problems. Firstly, we solve the PDEs on surface with singularity (folded surface) by a localized method. The localized method is a generalization of finite difference method. A priori error estimate for the discreitzation of Laplace operator is given for points selection. A stable solver (RBF-QR) is used to avoid ill-conditioning for the numerical simulation. Secondly, a {dollar}H^2{dollar} convergence study for the least-squares kernel collocation method, a.k.a. least-square Kansa's method will be discussed. This chapter can be separated into two main parts: constraint least-square method and weighted least-square method. For both methods, stability and consistency analysis are considered. Error estimate for both methods are also provided. For the case of weighted least-square Kansa's method, we figured out a suitable weighting for optimal error estimation. In Chapter two, we solve partial differential equation on smooth surface by an embedding method in the embedding space {dollar}\R^d{dollar}. Therefore, one can apply any numerical method in {dollar}\R^d{dollar} to solve the embedding problem. Thus, as an application of previous result, we solve embedding problem by least-squares kernel collocation. Moreover, we propose a new embedding condition in this chapter which has high order of convergence. As a result, we solve partial differential equation on smooth surface with a high order kernel collocation method. Similar to chapter two, we also provide error estimate for the numerical solution. Some applications such as pattern formation in the Brusselator system and excitable media in FitzHughNagumo model are also studied.

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17

Postell, Floyd Vince. "High order finite difference methods." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/28876.

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Yang, Xue-Feng. "Extensions of sturm-liouville theory : nodal sets in both ordinary and partial differential equations." Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/28021.

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19

Perella, Andrew James. "A class of Petrov-Galerkin finite element methods for the numerical solution of the stationary convection-diffusion equation." Thesis, Durham University, 1996. http://etheses.dur.ac.uk/5381/.

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A class of Petrov-Galerkin finite element methods is proposed for the numerical solution of the n dimensional stationary convection-diffusion equation. After an initial review of the literature we describe this class of methods and present both asymptotic and nonasymptotic error analyses. Links are made with the classical Galerkin finite element method and the cell vertex finite volume method. We then present numerical results obtained for a selection of these methods applied to some standard test problems. We also describe extensions of these methods which enable us to solve accurately for derivative values of the solution.

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Platte, Rodrigo B. "Accuracy and stability of global radial basis function methods for the numerical solution of partial differential equations." Access to citation, abstract and download form provided by ProQuest Information and Learning Company; downloadable PDF file 8.72Mb, 143 p, 2005. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:3181853.

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Stern, Louis G. "An explicitly conservative method for time-accurate solution of hyperbolic partial differential equations on embedded Chimera grids /." Thesis, Connect to this title online; UW restricted, 1996. http://hdl.handle.net/1773/6758.

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Luo, Wuan Hou Thomas Y. "Wiener chaos expansion and numerical solutions of stochastic partial differential equations /." Diss., Pasadena, Calif. : Caltech, 2006. http://resolver.caltech.edu/CaltechETD:etd-05182006-173710.

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23

Al-Muslimawi, Alaa Hasan A. "Numerical analysis of partial differential equations for viscoelastic and free surface flows." Thesis, Swansea University, 2013. https://cronfa.swan.ac.uk/Record/cronfa42876.

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He, Chuan. "Numerical solutions of differential equations on FPGA-enhanced computers." [College Station, Tex. : Texas A&M University, 2007. http://hdl.handle.net/1969.1/ETD-TAMU-1248.

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Gyurko, Lajos Gergely. "Numerical methods for approximating solutions to rough differential equations." Thesis, University of Oxford, 2008. http://ora.ox.ac.uk/objects/uuid:d977be17-76c6-46d6-8691-6d3b7bd51f7a.

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The main motivation behind writing this thesis was to construct numerical methods to approximate solutions to differential equations driven by rough paths, where the solution is considered in the rough path-sense. Rough paths of inhomogeneous degree of smoothness as driving noise are considered. We also aimed to find applications of these numerical methods to stochastic differential equations. After sketching the core ideas of the Rough Paths Theory in Chapter 1, the versions of the core theorems corresponding to the inhomogeneous degree of smoothness case are stated and proved in Chapter 2 along with some auxiliary claims on the continuity of the solution in a certain sense, including an RDE-version of Gronwall's lemma. In Chapter 3, numerical schemes for approximating solutions to differential equations driven by rough paths of inhomogeneous degree of smoothness are constructed. We start with setting up some principles of approximations. Then a general class of local approximations is introduced. This class is used to construct global approximations by pasting together the local ones. A general sufficient condition on the local approximations implying global convergence is given and proved. The next step is to construct particular local approximations in finite dimensions based on solutions to ordinary differential equations derived locally and satisfying the sufficient condition for global convergence. These local approximations require strong conditions on the one-form defining the rough differential equation. Finally, we show that when the local ODE-based schemes are applied in combination with rough polynomial approximations, the conditions on the one-form can be weakened. In Chapter 4, the results of Gyurko & Lyons (2010) on path-wise approximation of solutions to stochastic differential equations are recalled and extended to the truncated signature level of the solution. Furthermore, some practical considerations related to the implementation of high order schemes are described. The effectiveness of the derived schemes is demonstrated on numerical examples. In Chapter 5, the background theory of the Kusuoka-Lyons-Victoir (KLV) family of weak approximations is recalled and linked to the results of Chapter 4. We highlight how the different versions of the KLV family are related. Finally, a numerical evaluation of the autonomous ODE-based versions of the family is carried out, focusing on SDEs in dimensions up to 4, using cubature formulas of different degrees and several high order numerical ODE solvers. We demonstrate the effectiveness and the occasional non-effectiveness of the numerical approximations in cases when the KLV family is used in its original version and also when used in combination with partial sampling methods (Monte-Carlo, TBBA) and Romberg extrapolation.

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Zhang, Jiwei. "Local absorbing boundary conditions for some nonlinear PDEs on unbounded domains." HKBU Institutional Repository, 2009. http://repository.hkbu.edu.hk/etd_ra/1074.

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27

Watson, Aaron Michael. "The WN adaptive method for numerical solution of particle transport problems." Texas A&M University, 2005. http://hdl.handle.net/1969.1/3133.

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The source and nature, as well as the history of ray-effects, is described. A benchmark code, using piecewise constant functions in angle and diamond differencing in space, is derived in order to analyze four sample problems. The results of this analysis are presented showing the ray effects and how increasing the resolution (number of angles) eliminates them. The theory of wavelets is introduced and the use of wavelets in multiresolution analysis is discussed. This multiresolution analysis is applied to the transport equation, and equations that can be solved to calculate the coefficients in the wavelet expansion for the angular flux are derived. The use of thresholding to eliminate wavelet coefficients that are not required to adequately solve a problem is then discussed. An iterative sweeping algorithm, called the SN-WN method, is derived to solve the wavelet-based equations. The convergence of the SN-WN method is discussed. An algorithm for solving the equations is derived, by solving a matrix within each cell directly for the expansion coefficients. This algorithm is called the CWWN method. The results of applying the CW-WN method to the benchmark problems are presented. These results show that more research is needed to improve the convergence of the SN-WN method, and that the CW-WN method is computationally too costly to be seriously considered.

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28

ROEHL, NITZI MESQUITA. "NUMERICAL SOLUTIONS FOR SHAPE OPTIMIZATION PROBLEMS ASSOCIATED WITH ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 1991. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=9277@1.

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COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
Essa dissertação visa à obtenção de soluções numéricas para problemas de otimização de formas geométricas associados a equações diferenciais parciais elípticas. A principal motivação é um problema termal, onde deseja-se determinar a fronteira ótima, para um volume de material isolante fixo, tal que a perda de calor de um corpo seja minimizada. Realiza-se a análise e implementação numérica de uma abordagem via método das penalidades dos problemas de minimização. O método de elementos finitos é utilizado para discretizar o domínio em questão. A formulação empregada possui a característica atrativa da minimização ser conduzida sobre um espaço de funções lineares. Uma série de resultados numéricos são obtidos. Propõe-se, ainda, um algoritmo para a solução de problemas termais que envolvem material isolante composto.
This work is directed at the problem of determining numerical solutions for shape optimization problems associated with elliptic partial differential equations. Our primarily motivation is the problem of determining optimal shapes in order to minimize the heat lost of a body, given a fixed volume of insulation and a fixed internal (or external) geometry. The analysis and implementation of a penaly approach of the heat loss minimization problem are achieved. The formulation employed has the attractive feature that minimization is conducted over a linear function space. The algrithm adopted is based on the finite element method. Many numerical results are presented. We also propose an algorithm for the numerical solution of termal problems wich are concerned with multiple insulation layers.

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Wells, B. V. "A moving mesh finite element method for the numerical solution of partial differential equations and systems." Thesis, University of Reading, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.414567.

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30

Pitts, George Gustav. "Domain decomposition and high order discretization of elliptic partial differential equations." Diss., Virginia Tech, 1994. http://hdl.handle.net/10919/39143.

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Pitts, George G. "Domain decomposition and high order discretization of elliptic partial differential equations." Diss., Virginia Tech, 1994. http://hdl.handle.net/10919/39143.

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Numerical solutions of partial differential equations (PDEs) resulting from problems in both the engineering and natural sciences result in solving large sparse linear systems Au = b. The construction of such linear systems and their solutions using either direct or iterative methods are topics of continuing research. The recent advent of parallel computer architectures has resulted in a search for good parallel algorithms to solve such systems, which in turn has led to a recent burgeoning of research into domain decomposition algorithms. Domain decomposition is a procedure which employs subdivision of the solution domain into smaller regions of convenient size or shape and, although such partitionings have proven to be quite effective on serial computers, they have proven to be even more effective on parallel computers. Recent work in domain decomposition algorithms has largely been based on second order accurate discretization techniques. This dissertation describes an algorithm for the numerical solution of general two-dimensional linear elliptic partial differential equations with variable coefficients which employs both a high order accurate discretization and a Krylov subspace iterative solver in which a preconditioner is developed using domain decomposition. Most current research into such algorithms has been based on symmetric systems; however, variable PDE coefficients generally result in a nonsymmetric A, and less is known about the use of preconditioned Krylov subspace iterative methods for the solution of nonsymmetric systems. The use of the high order accurate discretization together with a domain decomposition based preconditioner results in an iterative technique with both high accuracy and rapid convergence. Supporting theory for both the discretization and the preconditioned iterative solver is presented. Numerical results are given on a set of test problems of varying complexity demonstrating the robustness of the algorithm. It is shown that, if only second order accuracy is required, the algorithm becomes an extremely fast direct solver. Parallel performance of the algorithm is illustrated with results from a shared memory multiproces-SOr.
Ph. D.

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32

Sweet, Erik. "ANALYTICAL AND NUMERICAL SOLUTIONS OF DIFFERENTIALEQUATIONS ARISING IN FLUID FLOW AND HEAT TRANSFER PROBLEMS." Doctoral diss., University of Central Florida, 2009. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/2585.

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The solutions of nonlinear ordinary or partial differential equations are important in the study of fluid flow and heat transfer. In this thesis we apply the Homotopy Analysis Method (HAM) and obtain solutions for several fluid flow and heat transfer problems. In chapter 1, a brief introduction to the history of homotopies and embeddings, along with some examples, are given. The application of homotopies and an introduction to the solutions procedure of differential equations (used in the thesis) are provided. In the chapters that follow, we apply HAM to a variety of problems to highlight its use and versatility in solving a range of nonlinear problems arising in fluid flow. In chapter 2, a viscous fluid flow problem is considered to illustrate the application of HAM. In chapter 3, we explore the solution of a non-Newtonian fluid flow and provide a proof for the existence of solutions. In addition, chapter 3 sheds light on the versatility and the ease of the application of the Homotopy Analysis Method, and its capability in handling non-linearity (of rational powers). In chapter 4, we apply HAM to the case in which the fluid is flowing along stretching surfaces by taking into the effects of "slip" and suction or injection at the surface. In chapter 5 we apply HAM to a Magneto-hydrodynamic fluid (MHD) flow in two dimensions. Here we allow for the fluid to flow between two plates which are allowed to move together or apart. Also, by considering the effects of suction or injection at the surface, we investigate the effects of changes in the fluid density on the velocity field. Furthermore, the effect of the magnetic field is considered. Chapter 6 deals with MHD fluid flow over a sphere. This problem gave us the first opportunity to apply HAM to a coupled system of nonlinear differential equations. In chapter 7, we study the fluid flow between two infinite stretching disks. Here we solve a fourth order nonlinear ordinary differential equation. In chapter 8, we apply HAM to a nonlinear system of coupled partial differential equations known as the Drinfeld Sokolov equations and bring out the effects of the physical parameters on the traveling wave solutions. Finally, in chapter 9, we present prospects for future work.
Ph.D.
Department of Mathematics
Sciences
Mathematics PhD

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Bujok, Karolina Edyta. "Numerical solutions to a class of stochastic partial differential equations arising in finance." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:d2e76713-607b-4f26-977a-ac4df56d54f2.

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We propose two alternative approaches to evaluate numerically credit basket derivatives in a N-name structural model where the number of entities, N, is large, and where the names are independent and identically distributed random variables conditional on common random factors. In the ﬁrst framework, we treat a N-name model as a set of N Bernoulli random variables indicating a default or a survival. We show that certain expected functionals of the proportion L_N of variables in a given state converge at rate 1/N as N [right arrow - infinity]. Based on these results, we propose a multi-level simulation algorithm using a family of sequences with increasing length, to obtain estimators for these expected functionals with a mean-square error of epsilon ² and computational complexity of order epsilon⁻², independent of N. In particular, this optimal complexity order also holds for the inﬁnite-dimensional limit. Numerical examples are presented for tranche spreads of basket credit derivatives. In the second framework, we extend the approximation of Bush et al. [13] to a structural jump-diﬀusion model with discretely monitored defaults. Under this approach, a N-name model is represented as a system of particles with an absorbing boundary that is active in a discrete time set, and the loss of a portfolio is given as the function of empirical measure of the system. We show that, for the inﬁnite system, the empirical measure has a density with respect to the Lebesgue measure that satisﬁes a stochastic partial differential equation. Then, we develop an algorithm to efficiently estimate CDO index and tranche spreads consistent with underlying credit default swaps, using a ﬁnite diﬀerence simulation for the resulting SPDE. We verify the validity of this approximation numerically by comparison with results obtained by direct Monte Carlo simulation of the basket constituents. A calibration exercise assesses the ﬂexibility of the model and its extensions to match CDO spreads from precrisis and crisis periods.

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Trojan, Alice von. "Finite difference methods for advection and diffusion." Title page, abstract and contents only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phv948.pdf.

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Includes bibliographical references (leaves 158-163). Concerns the development of high-order finite-difference methods on a uniform rectangular grid for advection and diffuse problems with smooth variable coefficients. This technique has been successfully applied to variable-coefficient advection and diffusion problems. Demonstrates that the new schemes may readily be incorporated into multi-dimensional problems by using locally one-dimensional techniques, or that they may be used in process splitting algorithms to solve complicatef time-dependent partial differential equations.

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Asner, Liya. "Efficient numerical methods for the solution of coupled multiphysics problems." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:a38b0c37-f18e-4c05-adbb-b57c911feeb9.

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Multiphysics systems with interface coupling are used to model a variety of physical phenomena, such as arterial blood flow, air flow around aeroplane wings, or interactions between surface and ground water flows. Numerical methods enable the practical application of these models through computer simulations. Specifically a high level of detail and accuracy is achieved in finite element methods by discretisations which use extremely large numbers of degrees of freedom, rendering the solution process challenging from the computational perspective. In this thesis we address this challenge by developing a twofold strategy for improving the efficiency of standard finite element coupled solvers. First, we propose to solve a monolithic coupled problem using block-preconditioned GMRES with a new Schur complement approximation. This results in a modular and robust method which significantly reduces the computational cost of solving the system. In particular, numerical tests show mesh-independent convergence of the solver for all the considered problems, suggesting that the method is well-suited to solving large-scale coupled systems. Second, we derive an adjoint-based formula for goal-oriented a posteriori error estimation, which leads to a time-space mesh refinement strategy. The strategy produces a mesh tailored to a given problem and quantity of interest. The monolithic formulation of the coupled problem allows us to obtain expressions for the error in the Lagrange multiplier, which often represents a physically relevant quantity, such as the normal stress on the interface between the problem components. This adaptive refinement technique provides an effective tool for controlling the error in the quantity of interest and/or the size of the discrete system, which may be limited by the available computational resources. The solver and the mesh refinement strategy are both successfully employed to solve a coupled Stokes-Darcy-Stokes problem modelling flow through a cartridge filter.

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Thorne, Jr Daniel Thomas. "Multigrid with Cache Optimizations on Adaptive Mesh Refinement Hierarchies." UKnowledge, 2003. http://uknowledge.uky.edu/gradschool_diss/325.

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This dissertation presents a multilevel algorithm to solve constant and variable coeffcient elliptic boundary value problems on adaptively refined structured meshes in 2D and 3D. Cacheaware algorithms for optimizing the operations to exploit the cache memory subsystem areshown. Keywords: Multigrid, Cache Aware, Adaptive Mesh Refinement, Partial Differential Equations, Numerical Solution.

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何正華 and Ching-wah Ho. "Iterative methods for the Robbins problem." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2000. http://hub.hku.hk/bib/B31222572.

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Macias, Diaz Jorge. "A Numerical Method for Computing Radially Symmetric Solutions of a Dissipative Nonlinear Modified Klein-Gordon Equation." ScholarWorks@UNO, 2004. http://scholarworks.uno.edu/td/167.

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In this paper we develop a finite-difference scheme to approximate radially symmetric solutions of a dissipative nonlinear modified Klein-Gordon equation in an open sphere around the origin, with constant internal and external damping coefficients and nonlinear term of the form G' (w) = w ^p, with p an odd number greater than 1. We prove that our scheme is consistent of quadratic order, and provide a necessary condition for it to be stable order n. Part of our study will be devoted to study the effects of internal and external damping.

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Li, Hongwei. "Local absorbing boundary conditions for wave propagations." HKBU Institutional Repository, 2012. https://repository.hkbu.edu.hk/etd_ra/1434.

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Røvik, Camilla. "Fast Tensor-Product Solvers for the Numerical Solution of Partial Differential Equations : Application to Deformed Geometries and to Space-Time Domains." Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2010. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-10814.

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Spectral discretization in space and time of the weak formulation of a partial differential equations (PDE) is studied. The exact solution to the PDE, with either Dirichlet or Neumann boundary conditions imposed, is approximated using high order polynomials. This is known as a spectral Galerkin method. The main focus of this work is the solution algorithm for the arising algebraic system of equations. A direct fast tensor-product solver is presented for the Poisson problem in a rectangular domain. We also explore the possibility of using a similar method in deformed domains, where the geometry of the domain is approximated using high order polynomials. Furthermore, time-dependent PDE's are studied. For the linear convection-diffusion equation in $mathbb{R}$ we present a tensor-product solver allowing for parallel implementation, solving $mathcal{O}(N)$ independent systems of equations. Lastly, an iterative tensor-product solver is considered for a nonlinear time-dependent PDE. For most algorithms implemented, the computational cost is $mathcal O (N^{p+1})$ floating point operations and a memory required of $mathcal O (N^{p})$ floating point numbers for $mathcal O (N^{p})$ unknowns. In this work we only consider $p=2$, but the theory is easily extended to apply in higher dimensions. Numerical results verify the expected convergence for both the iterative method and the spectral discretization. Exponential convergence is obtained when the solution and domain geometry are infinitely smooth.

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Ozdemir, Nilufer A. "The method of moments solution of a nonconformal volume integral equation via the IE-FFT algorithm for electromagnetic scattering from penetrable objects." Columbus, Ohio : Ohio State University, 2007. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1182258230.

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Li, Siqing. "Kernel-based least-squares approximations: theories and applications." HKBU Institutional Repository, 2018. https://repository.hkbu.edu.hk/etd_oa/539.

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Kernel-based meshless methods for approximating functions and solutions of partial differential equations have many applications in engineering fields. As only scattered data are used, meshless methods using radial basis functions can be extended to complicated geometry and high-dimensional problems. In this thesis, kernel-based least-squares methods will be used to solve several direct and inverse problems. In chapter 2, we consider discrete least-squares methods using radial basis functions. A general l^2-Tikhonov regularization with W_2^m-penalty is considered. We provide error estimates that are comparable to kernel-based interpolation in cases in which the function being approximated is within and is outside of the native space of the kernel. These results are extended to the case of noisy data. Numerical demonstrations are provided to verify the theoretical results. In chapter 3, we apply kernel-based collocation methods to elliptic problems with mixed boundary conditions. We propose some weighted least-squares formulations with different weights for the Dirichlet and Neumann boundary collocation terms. Besides fill distance of discrete sets, our weights also depend on three other factors: proportion of the measures of the Dirichlet and Neumann boundaries, dimensionless volume ratios of the boundary and domain, and kernel smoothness. We determine the dependencies of these terms in weights by different numerical tests. Our least-squares formulations can be proved to be convergent at the H^2 (Ω) norm. Numerical experiments in two and three dimensions show that we can obtain desired convergent results under different boundary conditions and different domain shapes. In chapter 4, we use a kernel-based least-squares method to solve ill-posed Cauchy problems for elliptic partial differential equations. We construct stable methods for these inverse problems. Numerical approximations to solutions of elliptic Cauchy problems are formulated as solutions of nonlinear least-squares problems with quadratic inequality constraints. A convergence analysis with respect to noise levels and fill distances of data points is provided, from which a Tikhonov regularization strategy is obtained. A nonlinear algorithm is proposed to obtain stable solutions of the resulting nonlinear problems. Numerical experiments are provided to verify our convergence results. In the final chapter, we apply meshless methods to the Gierer-Meinhardt activator-inhibitor model. Pattern transitions in irregular domains of the Gierer-Meinhardt model are shown. We propose various parameter settings for different patterns appearing in nature and test these settings on some irregular domains. To further simulate patterns in reality, we construct different kinds of domains and apply proposed parameter settings on different patches of domains found in nature.

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Li, Wen. "Numerical methods for the solution of the HJB equations arising in European and American option pricing with proportional transaction costs." University of Western Australia. School of Mathematics and Statistics, 2010. http://theses.library.uwa.edu.au/adt-WU2010.0098.

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This thesis is concerned with the investigation of numerical methods for the solution of the Hamilton-Jacobi-Bellman (HJB) equations arising in European and American option pricing with proportional transaction costs. We first consider the problem of computing reservation purchase and write prices of a European option in the model proposed by Davis, Panas and Zariphopoulou [19]. It has been shown [19] that computing the reservation purchase and write prices of a European option involves solving three different fully nonlinear HJB equations. In this thesis, we propose a penalty approach combined with a finite difference scheme to solve the HJB equations. We first approximate each of the HJB equations by a quasi-linear second order partial differential equation containing two linear penalty terms with penalty parameters. We then develop a numerical scheme based on the finite differencing in both space and time for solving the penalized equation. We prove that there exists a unique viscosity solution to the penalized equation and the viscosity solution to the penalized equation converges to that of the original HJB equation as the penalty parameters tend to infinity. We also prove that the solution of the finite difference scheme converges to the viscosity solution of the penalized equation. Numerical results are given to demonstrate the effectiveness of the proposed method. We extend the penalty approach combined with a finite difference scheme to the HJB equations in the American option pricing model proposed by Davis and Zarphopoulou [20]. Numerical experiments are presented to illustrate the theoretical findings.

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Heryudono, Alfa R. H. "Adaptive radial basis function methods for the numerical solution of partial differential equations, with application to the simulation of the human tear film." Access to citation, abstract and download form provided by ProQuest Information and Learning Company; downloadable PDF file, 178 p, 2008. http://proquest.umi.com/pqdweb?did=1601513551&sid=5&Fmt=2&clientId=8331&RQT=309&VName=PQD.

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Zhou, Jian Ming. "A multi-grid method for computation of film cooling." Thesis, University of British Columbia, 1990. http://hdl.handle.net/2429/29414.

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This thesis presents a multi-grid scheme applied to the solution of transport equations in turbulent flow associated with heat transfer. The multi-grid scheme is then applied to flow which occurs in the film cooling of turbine blades. The governing equations are discretized on a staggered grid with the hybrid differencing scheme. The momentum and continuity equations are solved by a nonlinear full multi-grid scheme with the SIMPLE algorithm as a relaxation smoother. The turbulence k — Є equations and the thermal energy equation are solved on each grid without multi-grid correction. Observation shows that the multi-grid scheme has a faster convergence rate in solving the Navier-Stokes equations and that the rate is not sensitive to the number of mesh points or the Reynolds number. A significant acceleration of convergence is also produced for the k — Є and the thermal energy equations, even though the multi-grid correction is not applied to these equations. The multi-grid method provides a stable and efficient means for local mesh refinement with only little additional computational and.memory costs. Driven cavity flows at high Reynolds numbers are computed on a number of fine meshes for both the multi-grid scheme and the local mesh-refinement scheme. Two-dimensional film cooling flow is studied using multi-grid processing and significant improvements in the results are obtained. The non-uniformity of the flow at the slot exit and its influence on the film cooling are investigated with the fine grid resolution. A near-wall turbulence model is used. Film cooling results are presented for slot injection with different mass flow ratios.
Science, Faculty of
Mathematics, Department of
Graduate

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Murali, Vasanth Kumar. "Code verification using the method of manufactured solutions." Master's thesis, Mississippi State : Mississippi State University, 2002. http://library.msstate.edu/etd/show.asp?etd=etd-11112002-121649.

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Chen, Meng. "Intrinsic meshless methods for PDEs on manifolds and applications." HKBU Institutional Repository, 2018. https://repository.hkbu.edu.hk/etd_oa/528.

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Radial basis function (RBF) methods for partial differential equations (PDEs), either in bulk domains, on surfaces, or in a combination of the formers, arise in a wide range of practical applications. This thesis proposes numerical approaches of RBF-based meshless techniques to solve these three kinds of PDEs on stationary and nonstationary surfaces and domains. In Chapter 1, we introduce the background of RBF methods, some basic concepts, and error estimates for RBF interpolation. We then provide some preliminaries for manifolds, restricted RBFs on manifolds, and some convergence properties of RBF interpolation. Finally, implicit-explicit time stepping schemes are briefly presented. In Chapter 2, we propose methods to implement meshless collocation approaches intrinsically to solve elliptic PDEs on smooth, closed, connected, and complete Riemannian manifolds with arbitrary codimensions. Our methods are based on strong-form collocations with oversampling and least-squares minimizations, which can be implemented either analytically or approximately. By restricting global kernels to the manifold, our methods resemble their easy-to-implement domain-type analogies, that is, Kansa methods. Our main theoretical contribution is a robust convergence analysis under some standard smoothness assumptions for high-order convergence. We simulate reaction-diffusion equations to generate Turing patterns and solve shallow water problems on manifolds. In Chapter 3, we consider convective-diffusion problems that model surfactants or heat transport along moving surfaces. We propose two time-space algorithms by combining the methods of lines and kernel-based meshless collocation techniques intrinsic to surfaces. We use a low-order time discretization for fair comparison, and higher-order schemes in time are possible. The proposed methods can achieve second-order convergence. They use either analytic or approximated spatial discretization of the surface operators, which do not require regeneration of point clouds at each temporal iteration. Thus, they are alternatively applied to handle models on two types of evolving surfaces, which are defined as prescribed motions and governed by geometric evolution laws, respectively. We present numerical examples on various evolving surfaces for the performance of our algorithms and apply the approximated one to merging surfaces. In Chapter 4, a kernel-based meshless method is developed to solve coupled second-order elliptic PDEs in bulk domains and on surfaces, subject to Robin boundary conditions. It combines a least-squares kernel-based collocation method with a surface-type intrinsic approach. We can thus use each pair for discrete point sets, RBF kernels (globally and restrictedly), trial spaces, and some essential assumptions, to search for least-squares solutions in bulks and on surfaces, respectively. We first analyze error estimates for a domain-type Robin-boundary problem. Based on this analysis and the existing results for surface PDEs, we discuss the theoretical requirements for the Sobolev kernels used. We then select the orders of smoothness for the kernels in bulks and on surfaces. Finally, several numerical experiments are demonstrated to test the robustness of the coupled method in terms of accuracy and convergence rates under different settings.

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Rebaza-Vasquez, Jorge. "Computation and continuation of equilibrium-to-periodic and periodic-to-periodic connections." Diss., Georgia Institute of Technology, 2002. http://hdl.handle.net/1853/28991.

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Shu, Yupeng. "Numerical Solutions of Generalized Burgers' Equations for Some Incompressible Non-Newtonian Fluids." ScholarWorks@UNO, 2015. http://scholarworks.uno.edu/td/2051.

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The author presents some generalized Burgers' equations for incompressible and isothermal flow of viscous non-Newtonian fluids based on the Cross model, the Carreau model, and the Power-Law model and some simple assumptions on the flows. The author numerically solves the traveling wave equations for the Cross model, the Carreau model, the Power-Law model by using industrial data. The author proves existence and uniqueness of solutions to the traveling wave equations of each of the three models. The author also provides numerical estimates of the shock thickness as well as maximum strain $\varepsilon_{11}$ for each of the fluids.

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Maroofi, Hamed. "Applications of the Monge - Kantorovich theory." Diss., Georgia Institute of Technology, 2002. http://hdl.handle.net/1853/29197.

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Dissertations / Theses on the topic 'Numerical solution of partial differential equations'

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