Teses / dissertações: "Partial Numerical solutions"

1

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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2

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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3

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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4

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

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5

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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6

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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7

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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8

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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9

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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10

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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11

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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12

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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13

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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14

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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15

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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16

何正華 e 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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17

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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18

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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19

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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20

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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21

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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22

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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23

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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24

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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25

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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Bénézet, Cyril. "Study of numerical methods for partial hedging and switching problems with costs uncertainty". Thesis, Université de Paris (2019-....), 2019. http://www.theses.fr/2019UNIP7079.

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Nous apportons dans cette thèse quelques contributions à l’étude théorique et numérique de certains problèmes de contrôle stochastique, ainsi que leurs applications aux mathématiques financières et à la gestion des risques financiers. Ces applications portent sur des problématiques de valorisation et de couverture faibles de produits financiers, ainsi que sur des problématiques réglementaires. Nous proposons des méthodes numériques afin de calculer efficacement ces quantités pour lesquelles il n’existe pas de formule explicite. Enfin, nous étudions les équations différentielles stochastiques rétrogrades liées à de nouveaux problèmes de switching, avec incertitude sur les coûts
In this thesis, we give some contributions to the theoretical and numerical study to some stochastic optimal control problems, and their applications to financial mathematics and risk management. These applications are related to weak pricing and hedging of financial products and to regulation issues. We develop numerical methods in order to compute efficiently these quantities, when no closed formulae are available. We also study backward stochastic differential equations linked to some new switching problems, with costs uncertainty

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29

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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30

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

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Moita, Daniel. "Desempenho de esquemas numéricos na modelagem da não linearidade da precipitação em um modelo atmosférico simplificado". Laboratório Nacional de Computação científica, 2011. https://tede.lncc.br/handle/tede/159.

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The interaction between large-scale ux fields and precipitation fields in the tropical atmosphere can produce sharp boundaries between dry and humid regions (limited ahead by the rainfall front). As the speed of propagation of disturbances in these regions are di_erent, they form a discontinuity (Days and Pauluis, 2009). This phenomenon can be represented by a system of equations formed by the shallow water equations and the conservation equation of water vapor coupled by a nonlinear source term (Frierson et al., 2004). The aim of this study is to compare the consistency of results and performance of numerical simulation models that use ten di_erent numerical methods for solving these equations, including the finite di_erence methods: Leapfrog, Lax-Wendro_ and Leapfrog with _lters; the upwind method developed by Walcek (2000) and used in Freitas et al. (2011); and the finite volume methods: Godunov, Lax-Wendro_, Minmod, MC, Superbee and Bean-Warming. These models are tested with nonlinear conditions and, at the end, the results show that, albeit more complex and with a higher computational cost in identical simulations, the finite volume method is more appropriate to simulate such a phenomenon because it provides a more accurate solution when dealing with these discontinuities, thereby producing more realistic results than in the other cases. These results may have impact on the design of new models for the operational centers of weather and climate.
A interação entre campos de fluxos de larga escala e campos de precipitação na atmosfera tropical pode apresentar fronteiras abruptas entre regiões secas e regiões úmidas (limitadas pela frente de precipitação). Como a velocidade de propagação de distúrbios nessas regiões é diferente, forma-se uma descontinuidade (Dias e Pauluis, 2009). Tal fenômeno pode ser representado por um sistema de equações formado pelas equações da água rasa e pela equação de conservação do vapor d'água acopladas por um termo fonte não linear (Frierson et al., 2004) . O objetivo do presente trabalho é comparar a consistência dos resultados e o desempenho de modelos de simulação numérica que utilizam dez métodos numéricos diferentes para resolver essas equações, dentre eles os métodos das diferenças finitas: Leapfrog, Lax-Wendroff e Leapfrog com filtros; Upwind desenvolvido por Walcek (2000) e utilizado em Freitas et al. (2011); e os métodos de volumes finitos: Godunov, Lax-Wendroff, Minmod, MC, Superbee e Bean-Warming. Estes modelos são testados com condições não lineares e, ao final, os resultados mostram que o método dos volumes finitos, mesmo sendo mais complexo e ter um custo computacional maior em simulações idênticas, é mais adequado para simular tal fenômeno pois fornece uma solução mais precisa ao lidar com as descontinuidades, gerando, assim, resultados mais realísticos que nos outros casos. Esses resultados podem ter impacto no desenho dos novos modelos para os centros operacionais de previsão de tempo e clima.

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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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Mohd, Damanhuri Nor Alisa. "The numerical approximation to solutions for the double-slip and double-spin model for the deformation and flow of granular materials". Thesis, University of Manchester, 2017. https://www.research.manchester.ac.uk/portal/en/theses/the-numerical-approximation-to-solutions-for-the-doubleslip-and-doublespin-model-for-the-deformation-and-flow-of-granular-materials(9986ac45-e48c-4061-a299-a80b2e665c3e).html.

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The aim of this thesis is to develop a numerical method to find approximations to solutions of the double-slip and double-spin model for the deformation and flow of granular materials. The model incorporates the physical and kinematic concepts of yield, shearing motion on slip lines, dilatation and average grain rotation. The equations governing the model comprise a set of five first order partial differential equations for the five dependent variables comprising two stress variables, two velocity components and the density. For steady state flows, the model is hyperbolic and the characteristic directions and relations along the characteristics are presented. The numerical approximation for the rate of working of the stresses are also presented. The model is then applied to a number of granular flow problems using the numerical method.

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Figueroa, Leonardo E. "Deterministic simulation of multi-beaded models of dilute polymer solutions". Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:4c3414ba-415a-4109-8e98-6c4fa24f9cdc.

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We study the convergence of a nonlinear approximation method introduced in the engineering literature for the numerical solution of a high-dimensional Fokker--Planck equation featuring in Navier--Stokes--Fokker--Planck systems that arise in kinetic models of dilute polymers. To do so, we build on the analysis carried out recently by Le~Bris, Leli\`evre and Maday (Const. Approx. 30: 621--651, 2009) in the case of Poisson's equation on a rectangular domain in $\mathbb{R}^2$, subject to a homogeneous Dirichlet boundary condition, where they exploited the connection of the approximation method with the greedy algorithms from nonlinear approximation theory explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173--187, 1996). We extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le~Bris, Leli\`evre and Maday to the technically more complicated situation of the elliptic Fokker--Planck equation, where the role of the Laplace operator is played out by a high-dimensional Ornstein--Uhlenbeck operator with unbounded drift, of the kind that appears in Fokker--Planck equations that arise in bead-spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space $\mathsf{D} = D_1 \times \dotsm \times D_N$ contained in $\mathbb{R}^{N d}$, where each set $D_i$, $i=1, \dotsc, N$, is a bounded open ball in $\mathbb{R}^d$, $d = 2, 3$. We exploit detailed information on the spectral properties and elliptic regularity of the Ornstein--Uhlenbeck operator to give conditions on the true solution of the Fokker--Planck equation which guarantee certain rates of convergence of the greedy algorithms. We extend the analysis to discretized versions of the greedy algorithms.

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Al, Zohbi Maryam. "Contributions to the existence, uniqueness, and contraction of the solutions to some evolutionary partial differential equations". Thesis, Compiègne, 2021. http://www.theses.fr/2021COMP2646.

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Dans cette thèse, nous nous sommes principalement intéressés à l’étude théorique et numérique de quelques équations qui décrivent la dynamique des densités des dislocations. Les dislocations sont des défauts microscopiques qui se déplacent dans les matériaux sous l’effet des contraintes extérieures. Dans un premier travail, nous démontrons un résultat d’existence globale en temps des solutions discontinues pour un système hyperbolique diagonal qui n’est pas nécessairement strictement hyperbolique, dans un espace unidimensionnel. Ainsi dans un deuxième travail, nous élargissons notre portée en démontrant un résultat similaire pour un système d’équations de type eikonal non-linéaire qui est en fait une généralisation du système hyperbolique déjà étudié. En effet, nous prouvons aussi l’existence et l’unicité d’une solution continue pour le système eikonal. Ensuite, nous nous sommes intéressés à l’analyse numérique de ce système en proposant un schéma aux différences finies, par lequel nous montrons la convergence vers le problème continu et nous consolidons nos résultats avec quelques simulations numériques. Dans une autre direction, nous nous sommes intéressés à la théorie de contraction différentielle pour les équations d’évolutions. Après avoir introduit une nouvelle distance, nous construisons une nouvelle famille des solutions contractantes positives pour l’équation d’évolution p-Laplace
In this thesis, we are mainly interested in the theoretical and numerical study of certain equations that describe the dynamics of dislocation densities. Dislocations are microscopic defects in materials, which move under the effect of an external stress. As a first work, we prove a global in time existence result of a discontinuous solution to a diagonal hyperbolic system, which is not necessarily strictly hyperbolic, in one space dimension. Then in another work, we broaden our scope by proving a similar result to a non-linear eikonal system, which is in fact a generalization of the hyperbolic system studied first. We also prove the existence and uniqueness of a continuous solution to the eikonal system. After that, we study this system numerically in a third work through proposing a finite difference scheme approximating it, of which we prove the convergence to the continuous problem, strengthening our outcomes with some numerical simulations. On a different direction, we were enthused by the theory of differential contraction to evolutionary equations. By introducing a new distance, we create a new family of contracting positive solutions to the evolutionary p-Laplacian equation

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Martel, Sofiane. "Theoretical and numerical analysis of invariant measures of viscous stochastic scalar conservation laws". Thesis, Paris Est, 2019. http://www.theses.fr/2019PESC1040.

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Cette thèse se consacre à une analyse théorique puis numérique d'une certaine classe d'équations aux dérivées partielles stochastiques (EDPS) : les lois de conservation scalaires avec viscosité et avec un forçage aléatoire de type additif et bruit blanc en temps. Un exemple typique est l'équation de Burgers stochastique, motivée par la théorie de la turbulence. On s'intéresse particulièrement au comportement en temps long des solutions de ces équations à travers une étude des mesures invariantes. La partie théorique de la thèse constitue le chapitre 2. Dans ce chapitre, on prouve l'existence et l'unicité d'une solution au sens fort. Pour cela, des estimations sur les normes de Sobolev jusqu'à l'ordre 2 sont établies. Dans la seconde partie du chapitre 2, on montre que la solution de l'EDPS admet une unique mesure invariante. On se propose dans le chapitre 3 d'approcher numériquement cette mesure invariante. À cette fin, on introduit un schéma numérique dont la discrétisation spatiale est de type Volumes Finis et dont la discrétisation temporelle est une méthode d'Euler semi-implicite. Il est montré que ce type de schéma respecte certaines propriétés fondamentales de l'EDPS telles que la dissipation d'énergie et la contraction L1. Ces propriétés assurent l'existence et l'unicité d'une mesure invariante pour le schéma. À l'aide d'un certain nombre d'estimations de régularité, on montre ensuite que cette mesure invariante discrète converge, lorsque le pas de temps et le pas d'espace tendent vers zéro, vers l'unique mesure invariante pour l'EDPS au sens de la distance de Wasserstein d'ordre 2. Enfin, des expériences numériques sont effectuées sur l'équation de Burgers pour illustrer cette convergence ainsi que des propriétés à petites échelles spatiale relatives à la turbulence
This devoted to the theoretical and numerical analysis of a certain class of stochastic partial differential equations (SPDEs), namely scalar conservation laws with viscosity and with a stochastic forcing which is an additive white noise in time. A particular case of interest is the stochastic Burgers equation, which is motivated by turbulence theory. We focus on the long time behaviour of the solutions of these equations through a study of the invariant measures. The theoretical part of the thesis constitutes the second chapter. In this chapter, we prove the existence and uniqueness of a solution in a strong sense. To this end, estimates on Sobolev norms up to the second order are established. In the second part of Chapter~2, we show that the solution of the SPDE admits a unique invariant measure. In the third chapter, we aim to approximate numerically this invariant measure. For this purpose, we introduce a numerical scheme whose spatial discretisation is of the finite volume type and whose temporal discretisation is a split-step backward Euler method. It is shown that this kind of scheme preserves some fundamental properties of the SPDE such as energy dissipation and L^1-contraction. Those properties ensure the existence and uniqueness of an invariant measure for the numerical scheme. Thanks to a few regularity estimates, we show that this discrete invariant measure converges, as the space and time steps tend to zero, towards the unique invariant measure for the SPDE in the sense of the second order Wasserstein distance. Finally, numerical experiments are performed on the Burgers equation in order to illustrate this convergence as well as some small-scale properties related to turbulence

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McCoy, James A. (James Alexander) 1976. "The surface area preserving mean curvature flow". Monash University, Dept. of Mathematics, 2002. http://arrow.monash.edu.au/hdl/1959.1/8291.

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Brubaker, Lauren P. "Completely Residual Based Code Verification". University of Akron / OhioLINK, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=akron1132592325.

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Bréhier, Charles-Edouard. "Numerical analysis of highly oscillatory Stochastic PDEs". Phd thesis, École normale supérieure de Cachan - ENS Cachan, 2012. http://tel.archives-ouvertes.fr/tel-00824693.

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In a first part, we are interested in the behavior of a system of Stochastic PDEs with two time-scales- more precisely, we focus on the approximation of the slow component thanks to an efficient numerical scheme. We first prove an averaging principle, which states that the slow component converges to the solution of the so-called averaged equation. We then show that a numerical scheme of Euler type provides a good approximation of an unknown coefficient appearing in the averaged equation. Finally, we build and we analyze a discretization scheme based on the previous results, according to the HMM methodology (Heterogeneous Multiscale Method). We precise the orders of convergence with respect to the time-scale parameter and to the parameters of the numerical discretization- we study the convergence in a strong sense - approximation of the trajectories - and in a weak sense - approximation of the laws. In a second part, we study a method for approximating solutions of parabolic PDEs, which combines a semi-lagrangian approach and a Monte-Carlo discretization. We first show in a simplified situation that the variance depends on the discretization steps. We then provide numerical simulations of solutions, in order to show some possible applications of such a method.

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Paillere, Henri J. "Multidimensional upwind residual distribution schemes for the Euler and Navier-Stokes equations on unstructured grids". Doctoral thesis, Universite Libre de Bruxelles, 1995. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/212553.

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Une approche multidimensionelle pour la résolution numérique des équations d'Euler et de Navier-Stokes sur maillages non-structurés est proposée. Dans une première partie, un exposé complet des schémas de distribution, dits de "fluctuation-splitting" ,est décrit, comprenant une étude comparative des schémas décentrés, positifs et de 2ème ordre, pour résoudre l'équation de convection à coefficients constants, ainsi qu'une étude théorique et numérique de la précision des schémas sur maillages réguliers et distordus. L'extension à des lois de conservation non-linéaires est aussi abordée, et une attention particulière est portée au problème de la linéarisation conservative. Dans une deuxième partie, diverses discrétisations des termes visqueux pour l'équation de convection-diffusion sont développées, avec pour but de déterminer l'approche qui offre le meilleur compromis entre précision et coût. L'extension de la méthode aux systèmes des lois de conservation, et en particulier à celui des équations d'Euler de la dynamique des gaz, représente le noyau principal de la thèse, et est abordée dans la troisième partie. Contrairement aux schémas de distribution classiques, qui reposent sur une extension formelle du cas scalaire, l'approche développée ici repose sur une décomposition du résidu par élément en équations scalaires, modélisant le transport de variables caracteristiques. La difficulté vient du fait que les équations d'Euler instationnaires ne se diagonalisent pas, et admettent une infinité de solutions élémentaires (ondes simples) se propageant dans toutes les directions d'espace. En régime stationnaire, en revanche, les équations se diagonalisent complètement dans le cas des écoulements supersoniques, et partiellement dans le cas des écoulements subsoniques. Ainsi, les équations sous forme conservative peuvent être remplacées par un système équivalent comprenant deux équations totalement découplées, exprimant l'invariance de l'entropie et de l'enthalpie totale le long des lignes de courant, et deux autres équations, modélisant les effets purement acoustiques. En régime supersonique, celles-ci se découplent aussi, et expriment la convection le long des lignes de Mach d'invariants de Riemann généralisés. La discrétisation de ces équations par des schémas scalaires décentrés permet de simuler des écoulements continus et discontinus avec une grande précision et sans oscillations. Finalement, dans une dernière partie, l'extension aux équations de Navier-Stokes est abordée, et la discrétisation des termes visqueux par une approche éléments finis est proposée. Les résultats numériques confirment la précision et la robustesse de la méthode.

Doctorat en sciences appliquées
info:eu-repo/semantics/nonPublished

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Coimbra, Tiago Antonio Alves 1981. "Solução da equação da onda imagem para continuação do afastamento mediante o metodo das caracteristicas". [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306030.

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Orientadores: Maria Amelia Novais Schleicher, Joerg Dietrich Wilhelm Schleicher
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: O deslocamento de um evento sísmico sob a chamada operação de continuação de afastamento (Offset Continuation Operation - OCO) pode ser descrita por uma equação diferencial parcial de segunda ordem que foi denominada de equação da onda imagem para OCO. Por substituição de uma solução tentativa da forma da teoria dos raios, pode se deduzir uma equação iconal OCO que descreve os aspectos cinemáticos da propagação da onda imagem OCO. Neste trabalho, resolvemos a equação da onda imagem OCO por meio do método das características. As características desta equação são as trajetórias OCO que descrevem o caminho do deslocamento de um evento sísmico sob variação do afastamento entre fonte e receptor. O conjunto de pontos finais de diversas trajetórias OCO, traçadas a partir do mesmo afastamento inicial até o mesmo afastamento final, define o raio de velocidade OCO ou, mais breve, raio OCO. Este raio OCO pode ser empregado para análise de velocidade. O algoritmo consiste do traçamento de raios OCO e então encontrar o ponto de interseção entre o raio OCO e o evento de reflexão sísmica dentro da seção final de afastamento comum. O procedimento tem a vantagem sobre a análise de velocidade convencional de que está baseado numa comparação de dados simulados com dados adquiridos ao invés de dois conjuntos de dados simulados. Exemplos numéricos demonstram que o traçamento de raios OCO pode ser executado de maneira precisa e de que a análise de velocidade resultante fornece velocidades confiáveis. Além disso, baseado nas expressões analíticas para os raios OCO que começam a partir do afastamento zero (migraton to common offset - MCO), deduzimos uma equação da onda imagem para continuação de velocidade MCO. Demonstramos que, em muitas situações práticas, esta equação pode ser empregada diretamente para OCO, assim evitando a necessidade de traçar trajetórias e raios OCO
Abstract: The dislocation of a seismic event under the so-called Offset Continuation Operation (OCO) can be described by a second-order partial differential equation, which has been called the OCO image-wave equation. By substitution of a ray-like trial solution, an OCO image-wave eikonal equation is obtained that describes the kinematic aspects of OCO imagewave propagation. In this work, we solve the OCO image-wave eikonal equation by means of the method of characteristics. The characteristics of this equation are the OCO trajectories that describe the path of dislocation of a seismic event under variation of the source-receiver offset. The set of endpoints of several OCO trajectories traced from the same initial to the same final offset under varying values for the medium velocity defines the OCO velocity ray or briefly OCO ray. This OCO ray can be employed for velocity analysis. The algorithm consists of OCO ray tracing an then finding the intersection point of the OCO ray with the seismic reflection event in the final common-offset section. The procedure has the advantage over conventional velocity analysis that it is based on a comparison of simulated and acquired data rather than two sets of simulated data. Numerical examples demonstrate that the OCO ray tracing can be accurately executed and that the resulting velocity analysis yields reliable velocities. Moreover, based on the analytic expressions for the OCO rays starting from zero-offset (migraton to common offset, MCO), we derived an image-wave equation for MCO velocity continuation. We demonstrate that in many practical situations this equation can be directly employed for OCO, thus avoiding the need to trace OCO trajectories and OCO rays
Mestrado
Geofisica Computacional
Mestre em Matemática Aplicada

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Lao, Kun Leng. "Multigrid algorithm based on cyclic reduction for convection diffusion equations". Thesis, University of Macau, 2010. http://umaclib3.umac.mo/record=b2148274.

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Landry, Richard S. Jr. "An Application of M-matrices to Preserve Bounded Positive Solutions to the Evolution Equations of Biofilm Models". ScholarWorks@UNO, 2017. https://scholarworks.uno.edu/td/2418.

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In this work, we design a linear, two step implicit finite difference method to approximate the solutions of a biological system that describes the interaction between a microbial colony and a surrounding substrate. Three separate models are analyzed, all of which can be described as systems of partial differential equations (PDE)s with nonlinear diffusion and reaction, where the biological colony grows and decays based on the substrate bioavailability. The systems under investigation are all complex models describing the dynamics of biological films. In view of the difficulties to calculate analytical solutions of the models, we design here a numerical technique to consistently approximate the system evolution dynamics, guaranteeing that nonnegative initial conditions will evolve uniquely into new, nonnegative approximations. This property of our technique is established using the theory of M-matrices, which are nonsingular matrices where all the entries of their inverses are positive numbers. We provide numerical simulations to evince the preservation of the nonnegative character of solutions under homogeneous Dirichlet and Neumann boundary conditions. The computational results suggest that the method proposed in this work is stable, and that it also preserves the bounded character of the discrete solutions.

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Pinheiro, Nathalie Carvalho. "Controle impulsional limitação da variação de nivel com minimização das atuações". [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/259269.

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Orientador: João Bosco Ribeiro do Val
Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação
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Resumo: Os elevados custos de produção na extração de petróleo marítimo são reduzidos com um sistema inovador que a Petrobras está desenvolvendo, denominado VASPS (do inglês, Vertical Annular Separation and Pumping System), capaz de separar gás e líquido ainda no assoalho oceânico. Esta dissertação trata do ajuste do nível de l'iquido no reservatório do VASPS, via controle impulsional estocástico, que se distancia bastante de um regulador convencional. A vazão de entrada flutua e o controle consiste em alterar a velocidade da bomba de saída do tanque, mantendo o nível numa faixa de operação, na qual ele varia livremente sem prejuízo. Todavia, é necessário de um lado observar o risco de operar próximo aos extremos de nível e de outro minimizar o número de intervenções na velocidade da bomba para prolongar sua vida útil. Optou-se por modelar o sistema por meio de um processo de difusão e formular por controle impulsional. Para a sua solução, o controle impulsional é convertido em uma sequência de problemas de parada ótima iterados, resolvidos utilizando-se a Discretização do Valor Médio, MVS (do inglês, Mean Value Scheme). Esta dissertação introduz o uso do controle impulsional nesta aplicação além da técnica citada para resolver problemas de parada ótima.
Abstract: The high costs in offshore oil production are reduced with the use of an innovative system which has been developed by Petrobras, the Brazilian oil company. Called VASPS (Vertical Annular Separation and Pumping System), it consists of an undersea gas/liquid separator. This work presents a strategy for the liquid level adjustment in the VASPS tank, which is subject to uncertain liquid inflow. This is far from a strict control regulator problem, since the liquid level may drifts freely inside an operation range. Although, in one hand, it is necessary to account for a risky operation near the limits, on the other hand, acting freely and continuously in the controlled pump may drastically shorten the lifetime of the equipment. To prevent premature worn with halt in the oil production, the control input variations should be meager. We propose a stochastic impulse control for varying the outflow pump speed. This formulation is transformed in a sequence of iterated optimal stopping problems, which results in a sequence of variational inequalities. We employ the numerical method called Mean Value Scheme (MVS) to solve this type of problem. This monograph introduces impulse control to the resevoir level adjustment of VASPS, together with the application of the MVS to its solution.
Mestrado
Automação
Mestre em Engenharia Elétrica

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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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Cardoso, André da Silva. "DFLD-EXP: uma solução semi-analítica para a equação de advecção-dispersão". Universidade do Estado do Rio de Janeiro, 2008. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=771.

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A equação de advecção-dispersão possui grande importância na engenharia e nas ciências aplicadas. No entanto, como é bem conhecido, a obtenção de uma solução numérica apropriada para essa equação é um problema desafiador tanto para engenheiros como para matemáticos, físicos e outros profissionais que trabalham com a modelagem de fenômenos associados a ela. Muitos métodos numéricos desenvolvidos podem apresentar uma série de inconvenientes, tais como oscilações, dispersão e/ou dissipação numérica e instabilidade, além de serem inapropriados para determinadas condições de contorno. O presente trabalho apresenta e analisa a metodologia DFLD-exp, uma nova abordagem para a obtenção de soluções semi-analíticas da equação de advecção-dispersão, a qual utiliza um tipo particular de diferenças finitas para a discretização espacial juntamente com técnicas de exponencial de matrizes para a resolução temporal. Uma cuidadosa análise numérica mostra que a metodologia resultante é não-oscilatória, essencialmente não-dispersiva e não-dissipativa, e incondicionalmente estável. Resoluções de vários exemplos numéricos, através de um código desenvolvido em linguagem MATLAB, confirmam os resultados teóricos.
The advection-dispersion equation has been very important in engineering and the applied sciences. However, the obtainment of an appropriate numerical solution to that equation has been challenging problem to engineers, mathematicians, physicians and others that work in the modeling of phenomena associate to advection-dispersion equation. Many developed numerical methods may produce a succession of mistakes, just as oscillations, numerical dispersion and/or dissipation, instability and those methods also may be inappropriate to determined boundary conditions. The present work shows and analyses the DFLD-exp methodology, a new way to obtain semi-analytic solutions to advection-dispersion equation, that make use of a particular form of finite differencing to the spatial discretization with techniques of matrix exponential to the time solving. A detailed numerical analysis shows the methodology is non-oscillatory, essentially non-dispersive and non-dissipative, and unconditionally stable. Resolutions of any numerical examples, by a computational code developed in MATLAB language, confirm the theoretical results.

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Morris, Graham Peter. "Parameter recovery in AC solution-phase voltammetry and a consideration of some issues arising when applied to surface-confined reactions". Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:1b1d40f3-ef1a-4f64-b500-17ce34630c43.

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A major problem in the quantitative analysis of AC voltammetric data has been the variance in results between laboratories, often resulting from a reliance on "heuristic" methods of parameter estimation that are strongly dependent on the choices of the operator. In this thesis, an automatic method for parameter estimation will be tested in the context of experiments involving electron-transfer processes in solution-phase. It will be shown that this automatic method produces parameter estimates consistent with those from other methods and the literature in the case of the ferri-/ferrocyanide couple, and is able to explain inconsistency in published values of the rate parameter for the ferrocene/ferrocenium couple. When a coupled homogeneous reaction is considered in a theoretical study, parameter recovery is achieved with a higher degree of accuracy when simulated data resulting from a high frequency AC voltammetry waveform are used. When surface-confined reactions are considered, heterogeneity in the rate constant and formal potential make parameter estimation more challenging. In the final study, a method for incorporating these "dispersion" effects into voltammetric simulations is presented, and for the first time, a quantitive theoretical study of the impact of dispersion on measured current is undertaken.

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48

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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49

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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50

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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