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Dissertations / Theses on the topic 'Poisson's equation Numerical solutions'

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Published: 10 December 2022
Last updated: 28 January 2023

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1

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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2

Nystrand, Thomas. "Summation By Part Methods for Poisson's Equation with Discontinuous Variable Coefficients." Thesis, Uppsala universitet, Institutionen för informationsteknologi, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-235418.

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Nowadays there is an ever increasing demand to obtain more accurate numericalsimulation results while at the same time using fewer computations. One area withsuch a demand is oil reservoir simulations, which builds upon Poisson's equation withvariable coefficients (PEWVC). This thesis focuses on applying and testing a high ordernumerical scheme to solve the PEWVC, namely Summation By Parts - SimultaneousApproximation Term (SBP-SAT). The thesis opens with proving that the method isconvergent at arbitrary high orders given sufficiently smooth coefficients. Theconvergence is furthermore verified in practice by test cases on the Poisson'sequation with smoothly variable permeability coefficients. To balance observed lowerboundary flux convergence, the SBP-SAT method was modified with additionalpenalty terms that were subsequently shown to work as expected. Finally theSBP-SAT method was tested on a semi-realistic model of an oil reservoir withdiscontinuous permeability. The correctness of the resulting pressure distributionvaried and it was shown that flux leakage was the probable cause. Hence theproposed SBP-SAT method performs, as expected, very well in continuous settingsbut typically allows undesirable leakage in discontinuous settings. There are possiblefixes, but these are outside the scope of this thesis.
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3

Simmel, Martin. "Two numerical solutions for the stochastic collection equation." Universitätsbibliothek Leipzig, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-215378.

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Two different methods are used to solve the stochastic collection equation (SCE) numerically. They are called linear discrete method (LDM) and bin shift method (BSM), respectively. Conceptually, both of them are similar to the well-known discrete method (DM) of Kovetz and Olund. For LDM and BSM, their concept is extended to two prognostic moments. Therefore, the \"splitting factors\" (which are constant in time for DM) become time-dependent for LDM and BSM. Simulations are shown for the Golovin kernel (for which an analytical solution is available) and the hydrodynamic kernel after Hall. Different bin resolutions and time steps are investigated. As expected, the results become better with increasing bin resolution. LDM and BSM do not show the anomalous dispersion which is a weakness of DM
Es werden zwei verschiedene Methoden zur numerischen Lösung der \"Gleichung für stochastisches Einsammeln\" (stochastic collection equation, SCE) vorgestellt. Sie werden als Lineare Diskrete Methode (LDM) bzw. Bin Shift Methode (BSM) bezeichnet. Konzeptuell sind beide der bekannten Diskreten Methode (DM) von Kovetz und Olund ähnlich. Für LDM und BSM wird deren Konzept auf zwei prognostische Momente erweitert. Für LDM und BSM werden die\" Aufteil-Faktoren\" (die für DM zeitlich konstant sind) dadurch zeitabhängig. Es werden Simulationsrechnungen für die Koaleszenzfunktion nach Golovin (für die eine analytische Lösung existiert) und die hydrodynamische Koaleszenzfunktion nach Hall gezeigt. Verschiedene Klassenauflösungen und Zeitschritte werden untersucht. Wie erwartet werden die Ergebnisse mit zunehmender Auflösung besser. LDM und BSM zeigen nicht die anomale Dispersion, die eine Schwäche der DM ist
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4

Simmel, Martin. "Two numerical solutions for the stochastic collection equation." Wissenschaftliche Mitteilungen des Leipziger Instituts für Meteorologie ; 17 = Meteorologische Arbeiten aus Leipzig ; 5 (2000), S. 61-73, 2000. https://ul.qucosa.de/id/qucosa%3A15149.

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Two different methods are used to solve the stochastic collection equation (SCE) numerically. They are called linear discrete method (LDM) and bin shift method (BSM), respectively. Conceptually, both of them are similar to the well-known discrete method (DM) of Kovetz and Olund. For LDM and BSM, their concept is extended to two prognostic moments. Therefore, the \"splitting factors\" (which are constant in time for DM) become time-dependent for LDM and BSM. Simulations are shown for the Golovin kernel (for which an analytical solution is available) and the hydrodynamic kernel after Hall. Different bin resolutions and time steps are investigated. As expected, the results become better with increasing bin resolution. LDM and BSM do not show the anomalous dispersion which is a weakness of DM.
Es werden zwei verschiedene Methoden zur numerischen Lösung der \"Gleichung für stochastisches Einsammeln\" (stochastic collection equation, SCE) vorgestellt. Sie werden als Lineare Diskrete Methode (LDM) bzw. Bin Shift Methode (BSM) bezeichnet. Konzeptuell sind beide der bekannten Diskreten Methode (DM) von Kovetz und Olund ähnlich. Für LDM und BSM wird deren Konzept auf zwei prognostische Momente erweitert. Für LDM und BSM werden die\" Aufteil-Faktoren\" (die für DM zeitlich konstant sind) dadurch zeitabhängig. Es werden Simulationsrechnungen für die Koaleszenzfunktion nach Golovin (für die eine analytische Lösung existiert) und die hydrodynamische Koaleszenzfunktion nach Hall gezeigt. Verschiedene Klassenauflösungen und Zeitschritte werden untersucht. Wie erwartet werden die Ergebnisse mit zunehmender Auflösung besser. LDM und BSM zeigen nicht die anomale Dispersion, die eine Schwäche der DM ist.
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5

Sjölander, Filip. "Numerical solutions to the Boussinesq equation and the Korteweg-de Vries equation." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-297544.

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The aim of the report is to numerically construct solutions to two analytically solvable non-linear differential equations: the Korteweg–De Vries equation and the Boussinesq equation. To accomplish this, a range of numerical methods where implemented, including Galerkin methods. To asses the accuracy of the solutions, analytic solutions were derived for reference. Characteristic of both equations is that they support a certain type of wave-solutions called "soliton solutions", which admit an intuitive physical interpretation as solitary traveling waves. Theses solutions are the ones simulated. The solitons are also qualitatively studied in the report.
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6

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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7

Maral, Tugrul. "Spectral (h-p) Element Methods Approach To The Solution Of Poisson And Helmholtz Equations Using Matlab." Master's thesis, METU, 2006. http://etd.lib.metu.edu.tr/upload/3/12607945/index.pdf.

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A spectral element solver program using MATLAB is written for the solution of Poisson and Helmholtz equations. The accuracy of spectral methods (p-type high order) and the geometric flexibility of the low-order h-type finite elements are combined in spectral element methods. Rectangular elements are used to solve Poisson and Helmholtz equations with Dirichlet and Neumann boundary conditions which are homogeneous or non homogeneous. Robin (mixed) boundary conditions are also implemented. Poisson equation is also solved by discretising the domain with curvilinear quadrilateral elements so that the accuracy of both isoparametric quadrilateral and rectangular element stiffness matrices and element mass matrices are tested. Quadrilateral elements are used to obtain the stream functions of the inviscid flow around a cylinder problem. Nonhomogeneous Neumann boundary conditions are imposed to the quadrilateral element stiffness matrix to solve the velocity potentials.
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8

Sundström, Carl. "Numerical solutions to high frequency approximations of the scalar wave equation." Thesis, Uppsala universitet, Tillämpad beräkningsvetenskap, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-429072.

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Throughout many fields of science and engineering, the need for describing waveequations is crucial. Solving the wave equation for high-frequency waves istime-consuming, requires a fine mesh size and memory usage. The main goal wasimplementing and comparing different solution methods for high-frequency waves.Four different methods have been implemented and compared in terms of runtimeand discretization error. From my results, the method which performs the best is thefast sweeping method. For the fast marching method, the time-complexity of thenumerical solver was higher than expected which indicates an error in myimplementation.
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9

Hayman, Kenneth John. "Finite-difference methods for the diffusion equation." Title page, table of contents and summary only, 1988. http://web4.library.adelaide.edu.au/theses/09PH/09phh422.pdf.

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10

Pusch, Gordon D. "Differential algebraic methods for obtaining approximate numerical solutions to the Hamilton-Jacobi equation." Diss., This resource online, 1990. http://scholar.lib.vt.edu/theses/available/etd-07282008-135711/.

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11

Agiza, Hamdy N. "A numerical and theoretical study of solutions to a damped nonlinear wave equation." Thesis, Heriot-Watt University, 1987. http://hdl.handle.net/10399/1058.

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12

Sagheer, Muhammad. "Mathematical analysis and numerical solutions of an integral equation arising from population dynamics." Thesis, University of Sussex, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.420495.

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13

Chiang, Shihchung. "Numerical solutions for a class of singular integrodifferential equations." Diss., This resource online, 1996. http://scholar.lib.vt.edu/theses/available/etd-06062008-151231/.

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14

Huang, Jeffrey. "Numerical solutions of continuous wave beam in nonlinear media." PDXScholar, 1987. https://pdxscholar.library.pdx.edu/open_access_etds/3742.

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Deformation of a Gaussian beam is observed when it propagates through a plasma. Self-focusing of the beam may be observed when the intensity of the laser increases the index of refraction of plasma gas. Due to the difficulties in solving the nonlinear partial differential equation in Maxwell's wave equation, a numerical technique has been developed in favor of the traditional analytical method. Result of numerical solution shows consistency with the analytical method. This further suggests the validity of the numerical technique employed. A three dimensional graphics package was used to depict the numerical data obtained from the calculation. Plots from the data further show the deformation of the Gaussian beam as it propagates through the plasma gas.
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15

Hårderup, Peder, and William Brorsson. "A Numerical and Analytical Investigation of The sine-Gordon Equation and Its Soliton Solutions." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-297564.

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This thesis investigates the nonlinear partial differential equation known as sine-Gordon and its special soliton solutions.Simpler analytical results are derived and more advanced methods and their results are discussed.Further, a finite difference scheme is derived, implemented and compared against a known energy conserving scheme of sine-Gordon in terms of stability, accuracy, convergence and computation time.The complete solvability of the equation enables comparison between numerical solutions and their analytical counterparts.No unified answer to which numerical scheme is best was determined as they both were shown to have pros and cons.
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16

Porter, Annabelle Louise. "The evolution of equation-solving: Linear, quadratic, and cubic." CSUSB ScholarWorks, 2006. https://scholarworks.lib.csusb.edu/etd-project/3069.

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This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work. It uses a historical perspective to highlight many of the concepts underlying modern equation solving.
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17

Yevik, Andrei. "Numerical approximations to the stationary solutions of stochastic differential equations." Thesis, Loughborough University, 2011. https://dspace.lboro.ac.uk/2134/7777.

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This thesis investigates the possibility of approximating stationary solutions of stochastic differential equations using numerical methods. We consider a particular class of stochastic differential equations, which are known to generate random dynamical systems. The existence of stochastic stationary solution is proved using global attractor approach. Euler's numerical method, applied to the stochastic differential equation, is proved to generate a discrete random dynamical system. The existence of stationary solution is proved again using global attractor approach. At last we prove that the approximate stationary point converges in mean-square sense to the exact one as the time step of the numerical scheme diminishes.
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18

Keeve, Michael Octavis. "Study and implementation of Gauss Runge-Kutta schemes and application to Riccati equations." Diss., Georgia Institute of Technology, 1997. http://hdl.handle.net/1853/30956.

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19

Zheng, Bing. "Incorporating equation solving into unification through stratified term rewriting." Thesis, Virginia Polytechnic Institute and State University, 1989. http://hdl.handle.net/10919/52096.

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This thesis studies equational theories incorporated into unification and describes STAR, a stratified term rewriting system that achieves a full integration. STAR is an advance over existing systems because it integrates an equational theory with unification at a lower, more fundamental level. Certain properties of STAR are proven including termination and confluence. We also discuss the algorithmic complexity of the reduction algorithm, a vital component of STAR. We compare our system with narrowing and discuss the merits and drawbacks of each technique. Since our system is an experimental integration of equation solving and unification, we are not concerned with the efficiency of the implementation. We do propose, however, some future improvements.
Master of Science
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20

Al-Hussyni, Saad Kohel Ali. "Numerical study of turbulent plane jets in still and flowing environments employing two-equation k-ε model." Thesis, University of Edinburgh, 1987. http://hdl.handle.net/1842/11065.

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21

Volkin, Robert P. "Spherical Shell Solutions to the Radially Symmetric Aggregation Equation: Analysis and a Novel Numerical Method." Case Western Reserve University School of Graduate Studies / OhioLINK, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=case1575639958498416.

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22

Koutoumbas, Anastasios M. "Bidirectional and unidirectional spectral representations for the scalar wave equation." Thesis, Virginia Tech, 1990. http://hdl.handle.net/10919/41904.

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The Cauchy problem associated with the scalar wave equation in free space is used as a vehicle for a critical examination and assessment of the bidirectional and unidirectional spectral representations. These two novel methods for synthesizing wave signals are distinct from the superposition principle underlying the conventional Fourier method and they can effectively be used to derive a large class of localized solutions to the scalar wave equation. The bidirectional spectral representation is presented as an extension of Brittingham's ansatz and Ziolkowski's Focus Wave Mode spectral representations. On the other hand, the unidirectional spectral representation is motivated through a group-theoretic similarity reduction of the scalar wave equation.


Master of Science
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23

Dubois, Olivier 1980. "Optimized Schwarz methods for the advection-diffusion equation and for problems with discontinuous coefficients." Thesis, McGill University, 2007. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=103379.

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Optimized Schwarz methods are iterative domain decomposition procedures with greatly improved convergence properties, for solving second order elliptic boundary value problems. The enhanced convergence is obtained by replacing the Dirichlet transmission conditions in the classical Schwarz iteration with more general conditions that are optimized for performance. The convergence is optimized through the solution of a min-max problem. The theoretical study of the min-max problems gives explicit formulas or characterizations for the optimized transmission conditions for practical use, and it permits the analysis of the asymptotic behavior of the convergence.
In the first part of this work, we continue the study of optimized transmission conditions for advection-diffusion problems with smooth coefficients. We derive asymptotic formulas for the optimized parameters for small mesh sizes, in the overlapping and non-overlapping cases, and show that these formulas are accurate when the component of the advection tangential to the interface is not too large.
In a second part, we consider a diffusion problem with a discontinuous coefficient and non-overlapping domain decompositions. We derive several choices of optimized transmission conditions by thoroughly solving the associated min-max problems. We show in particular that the convergence of optimized Schwarz methods improves as the jump in the coefficient increases, if an appropriate scaling of the transmission conditions is used. Moreover, we prove that optimized two-sided Robin conditions lead to mesh-independent convergence. Numerical experiments with two subdomains are presented to verify the analysis. We also report the results of experiments using the decomposition of a rectangle into many vertical strips; some additional analysis is carried out to improve the optimized transmission conditions in that case.
On a third topic, we experiment with different coarse space corrections for the Schwarz method in a simple one-dimensional setting, for both overlapping and non-overlapping subdomains. The goal is to obtain a convergence that does not deteriorate as we increase the number of subdomains. We design a coarse space correction for the Schwarz method with Robin transmission conditions by considering an augmented linear system, which avoids merging the local approximations in overlapping regions. With numerical experiments, we demonstrate that the best Robin conditions are very different for the Schwarz iteration with, and without coarse correction.
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24

Liu, Fang-Lan. "Some asymptotic stability results for the Boussinesq equation." Diss., Virginia Tech, 1993. http://hdl.handle.net/10919/40052.

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25

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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26

Kurianski, Kristin Marie-Dettmers. "Estimates for solutions to the Dysthe equation and numerical simulations of walking droplets in harmonic potentials." Thesis, Massachusetts Institute of Technology, 2019. https://hdl.handle.net/1721.1/122173.

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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 119-124).
In this thesis, we study wave-type phenomena both from a numerical point of view and a theoretical one. We first present the results of a numerical investigation of droplets walking in a harmonic potential on a vibrating fluid bath. The droplet's trajectory is described by an integro-differential equation, which is simulated numerically in various parameter regimes. We produce a regime diagram that summarizes the dependence of the walker's behavior on the system parameters for a droplet of fixed size. At relatively low vibrational forcing, a number of periodic and quasiperiodic trajectories emerge. In the limit of large vibrational forcing, the walker's trajectory becomes chaotic, but the resulting trajectories can be decomposed into portions of unstable quasiperiodic states. We then recast the integro-differential equation as a coupled system of ordinary differential equations in time. This method is used to simulate droplet lattices in various configurations and in the presence of a harmonic potential, creating structures reminiscent of Wigner molecules. The development of this approach is presented in detail along with its future applications. We then switch focus to a fluid system described by a modified nonlinear Schrödinger equation. The surface of an incompressible, inviscid, irrotational fluid of infinite depth can be described in two dimensions by the Dysthe equation. Recently, this equation has been used to model extraordinarily large waves occurring on the ocean's surface called rogue waves. In this thesis, we prove dispersive estimates and Strichartz estimates for the Dysthe equation. We then prove a Kato-type smoothing effect in which we are able to bound uniformly in space the L² norm in time of a fractional derivative of the linear solution by the L² norm in space of the initial data. This section of the thesis lays the groundwork for further developments in proving well-posedness via a contraction argument.
Financial support from National Science Foundation and the MIT School of Science
by Kristin Marie-Dettmers Kurianski.
Ph. D.
Ph.D. Massachusetts Institute of Technology, Department of Mathematics
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27

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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28

Liu, Guanhui, and 刘冠辉. "Formulation of multifield finite element models for Helmholtzproblems." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2010. http://hub.hku.hk/bib/B44204875.

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29

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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30

Fok, Chin Man. "Numerical solutions for the Navier-Stokes equations and the Fokker-Planck equations using spectral methods." HKBU Institutional Repository, 2002. http://repository.hkbu.edu.hk/etd_ra/435.

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31

Lampshire, Gregory B. "Review of random media homogenization using effective medium theories." Thesis, Virginia Tech, 1992. http://hdl.handle.net/10919/40659.

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Calculation of propagation constants in particulate matter is an important aspect of wave propagation analysis in engineering disciplines such as satellite comnlunication, geophysical exploration, radio astronomy and material science. It is important to understand why different propagation constants produced by different theories are not applicable to a particular problem. Homogenization of the random media using effective medium theories yields the effective propagation constants by effacing the particulate, microscopic nature of the medium. The Maxwell-Gamet and Bruggeman effective medium theories are widely used but their limitations are not always well understood.

In this thesis, some of the more complex homogenization theories will only be partially derived or heuristically constructed in order to avoid unnecessary mathematical complexity which does not yield additional physical insight. The intent of this thesis is to elucidate the nature of effective medium theories, discuss the theories' approximations and gain a better global understanding of wave propagation equations. The focus will be on the Maxwell-Garnet and Bruggeman theories because they yield simple relationships and therefore serve as anchors in a sea of myriad approximations.


Master of Science
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32

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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33

Pinilla, Camilo Ernesto. "Numerical simulation of shear instability in shallow shear flows." Thesis, McGill University, 2008. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=115697.

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The instabilities of shallow shear flows are analyzed to study exchanges processes across shear flows in inland and coastal waters, coastal and ocean currents, and winds across the thermal-and-moisture fronts. These shear flows observed in nature are driven by gravity and governed by the shallow water equations (SWE). A highly accurate, and robust, computational scheme has been developed to solve these SWE. Time integration of the SWE was carried out using the fourth-order Runge-Kutta scheme. A third-order upwind bias finite difference approximation known as QUICK (Quadratic Upstream Interpolation of Convective Kinematics) was employed for the spatial discretization. The numerical oscillations were controlled using flux limiters for Total Variation Diminishing (TVD). Direct numerical simulations (DNS) were conducted for the base flow with the TANH velocity profile, and the base flow in the form of a jet with the SECH velocity profile. The depth across the base flows was selected for the' balance of the driving forces. In the rotating flow simulation, the Coriolis force in the lateral direction was perfectly in balance with the pressure gradient across the shear flow during the simulation. The development of instabilities in the shear flows was considered for a range of convective Froude number, friction number, and Rossby number. The DNS of the SWE has produced linear results that are consistent with classical stability analyses based on the normal mode approach, and new results that had not been determined by the classical method. The formation of eddies, and the generation of shocklets subsequent to the linear instabilities were computed as part of the DNS. Without modelling the small scales, the simulation was able to produce the correct turbulent spreading rate in agreement with the experimental observations. The simulations have identified radiation damping, in addition to friction damping, as a primary factor of influence on the instability of the shear flows admissible to waves. A convective Froude number correlated the energy lost due to radiation damping. The friction number determined the energy lost due to friction. A significant fraction of available energy produced by the shear flow is lost due the radiation of waves at high convective Froude number. This radiation of gravity waves in shallow gravity-stratified shear flow, and its dependence on the convective Froude number, is shown to be analogous to the Mach-number effect in compressible flow. Furthermore, and most significantly, is the discovery from the simulation the crucial role of the radiation damping in the development of shear flows in the rotating earth. Rings and eddies were produced by the rotating-flow simulations in a range of Rossby numbers, as they were observed in the Gulf Stream of the Atlantic, Jet Stream in the atmosphere, and various fronts across currents in coastal waters.
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34

Pack, Jeong-Ki. "A wave-kinetic numerical method for the propagation of optical waves." Thesis, Virginia Polytechnic Institute and State University, 1985. http://hdl.handle.net/10919/104527.

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35

Ting, Lycretia Englang. "Sturm-Liouville theory." CSUSB ScholarWorks, 1996. https://scholarworks.lib.csusb.edu/etd-project/1206.

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36

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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37

Li, Yao. "Stochastic perturbation theory and its application to complex biological networks -- a quantification of systematic features of biological networks." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/49013.

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The primary objective of this thesis is to make a quantitative study of complex biological networks. Our fundamental motivation is to obtain the statistical dependency between modules by injecting external noise. To accomplish this, a deep study of stochastic dynamical systems would be essential. The first chapter is about the stochastic dynamical system theory. The classical estimation of invariant measures of Fokker-Planck equations is improved by the level set method. Further, we develop a discrete Fokker-Planck-type equation to study the discrete stochastic dynamical systems. In the second part, we quantify systematic measures including degeneracy, complexity and robustness. We also provide a series of results on their properties and the connection between them. Then we apply our theory to the JAK-STAT signaling pathway network.
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38

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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39

Fahs, Amin. "Modeling of naturel convection in porous media : development of semi-analytical and spectral numerical solutions of heat transfer problem in special domains." Thesis, Strasbourg, 2021. https://publication-theses.unistra.fr/restreint/theses_doctorat/2021/Fahs_Amin_2021_ED269.pdf.

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Le problème de la cavité poreuse carrée est largement utilisé comme cas de référence courant pour les problèmes de Convection Naturelle (CN) en milieux poreux. Il peut être utilisé pour plusieurs applications numériques, théoriques et pratiques. Par ailleurs, toutes les solutions de haute précision existantes dans la littérature scientifique sont développées dans des conditions de régime permanent. Cependant, il est bien connu que les processus de CN dans les milieux poreux se produisent naturellement dans un régime dépendant du temps, car les conditions aux limites peuvent être variables dans le temps. Pour surmonter cette difficulté, la solution en régime permanent est souvent simulée comme une solution transitoire qui évolue jusqu'à atteindre l'état d'équilibre. Ces régimes dépendant du temps sont très efficaces pour détecter les effets des variations de paramètres sur le processus physique de CN, en particulier pour les sujets d'intérêt de cette thèse: la variation du niveau d'inclinaison du domaine et la prise en compte des variations de température de la paroi chaude dans le temps. À cet effet, trois objectifs sont identifiés dans cette thèse: 1. Développer une solution de convection naturelle en fonction du temps dans des milieux poreux en utilisant le Modèle Darcy en deux modes: transitoire et instable. 2. Étudier le comportement en fonction du temps de la convection naturelle dans des milieux poreux ayant le niveau d'inclinaison du domaine comme paramètre variable dans deux modes: transitoire et instable. 3. Développer une solution de convection naturelle en fonction du temps dans des milieux poreux en utilisant le Modèle Darcy-Lapwood-Brinkman en deux modes: transitoire et instable. Pour ce faire, du fait de la grande précision dans les domaines simplement connectés, une méthode spectrale de résidus pondérés de type Galerkin est choisie pour développer une solution au problème de CN dans une cavité carrée poreuse. L’application de la procédure de Fourier-Galerkin (FG), deux configurations traitant des régimes instables sont considérées où chaque solution est dérivée pour une large gamme des nombres de Rayleigh (Ra) avec d'autres conditions spéciales. Ce travail de thèse est subdivisé en cinq chapitres. Dans le premier chapitre, nous avons présenté un aperçu physique du processus de convection naturelle en milieux poreux. Dans le deuxième chapitre, le développement mathématique des équations, la méthode de résolution et la procédure de résolution sont décrits en détails. Dans le chapitre trois, la première étude de cas de cette thèse, la solution dépendante du temps de la convection naturelle dans une cavité carrée remplie de milieux poreux saturé utilisant le modèle de Darcy est développé. Dans le chapitre quatre, le problème de variation temporelle de Darcy-Lapwood- Brinkman de CN dans une enceinte poreuse saturée carrée est étudié. Dans le chapitre cinq, les solutions dépendant du temps sont développées pour le problème de convection naturelle utilisant la loi de Darcy dans une cavité poreuse inclinée et considéré comme une étude complète sur les effets de l'inclinaison du domaine sur le processus physique du problème de convection libre. Pour tous les cas, les régimes transitoires et instables sont considérés
The problem of the porous square cavity is extensively used as a common benchmark case for Natural convection (NC) problem in porous media. It can be used for several numerical, theoretical, and practical purposes. All the existing high accurate solutions are developed under steady-state conditions. However, it is well known that the processes of NC in porous media occurs naturally in a time-dependent procedure, as boundary conditions can be variable in time. Also, the convergence of the steady-state solution is known to be difficult. To overcome this difficulty, the steady-state solution is often simulated as a transient solution that evolves until reaching the steady-state condition. These time-dependent modes are very efficient to detect the effects of the parameter variations on the physical process of NC, especially for the subject of interest in this thesis: the domain inclination level and hot wall temperature variation in time. For this purpose, three goals are identified in this Thesis: 1. Developing a time-dependent solution of natural convection in porous media using the Darcy model in two modes: Transient and unsteady. 2. Investigating the time-dependent behavior of natural convection in porous media having the domain inclination level as a variable parameter in two modes: Transient and unsteady. 3. Developing a time-dependent solution of natural convection in porous media using the Darcy-Lapwood-Brinkman model in two modes: Transient and unsteady. To do so, according to the high accuracy in the simply connected domains, one of the Galerkin spectral weighted residual method is chosen to develop a space-time dependent solution for NC problem in a square porous cavity. Applying the Fourier-Galerkin (FG) procedure, two configurations dealing with transient and unsteady regimes are considered where each solution is derived for a wide range of Rayleigh numbers with other special conditions. This work of thesis is explained in details as five chapters.The NC physical process with the time-dependent variations is described in the transient mode to reach the steady-state solution and for the unsteady mode during a one period using periodic sinusoidal boundary conditions on the cavity hot wall. Finally, the work of this thesis is described in details in five chapters; while the sixth and last chapter is devoted to the summary and conclusion.The results in this thesis work provide a set of high-accurate data that are published in three papers to be used for testing numerical codes of heat transfer in time-dependent configurations
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40

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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41

Tovar, Anthony Alan. "Off-axis multimode light beam propagation in tapered lenslike media including those with spatial gain or loss variation." PDXScholar, 1988. https://pdxscholar.library.pdx.edu/open_access_etds/3839.

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The propagation of light beams in inhomogeneous dielectric media is considered. The derivation begins with first principles and remains general enough to include off-axis asymmetric multimode input beams in tapered lenslike media with spatial variations of gain or loss. The tapering of lenslike media leads to a number of important applications. A parabolic taper is proposed as a model for a heated axially stretched fiber taper, and beams in such media are fully characterized. Other models are proposed by the concatenation of a parabola with other taper functions.
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42

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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43

Israel, Haydi. "Comportement asymptotique de modèles en séparation de phases." Thesis, Poitiers, 2013. http://www.theses.fr/2013POIT2308/document.

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Dans cette thèse, on étudie l'existence, l'unicité et la régularité des solutionsd'équation de type Cahn-Hilliard ainsi que son comportement asymptotiqueen termes d'existence de l'attracteur global et d'un attracteur exponentiel. Cetteéquation est considérée dans un domaine borné et régulier pour différents types denonlinéarités et de conditions au bord.D'abord, on étudie l'équation avec des conditions de type Dirichlet sur le bord etune nonlinéarité régulière. Après, on considère une perturbation du problème et ondémontre l'existence d'une famille robuste d'attracteurs exponentiels lorsque ε tendvers 0.Ensuite, on étudie l'équation avec des conditions dynamiques sur le bord. On considèretout d'abord une nonlinéarité régulière et on donne une étude théorique etnumérique. Après, on illustre ces résultats par des simulations numériques en dimensiondeux d'espace qui permettent d'étudier l'influence des différents paramètres.On termine par une étude du modèle considéré avec une nonlinéarité singulière quel'on approche par des fonctions régulières et on introduit une notion de solutionappropriée
This thesis is devoted to the study of the existence, uniqueness andregularity of solutions for a Cahn-Hilliard type equation, as well as the asymptoticbehavior in terms of existence of the global attractor and of an exponential attractor.This equation is considered in a bounded and smooth domain under variousassumptions on the nonlinear terms and with different boundary conditions.We start by studying the equation with Dirichlet boundary conditions and a regularnonlinearity. Then, we consider a perturbation of the problem and we prove theexistence of a robust family of exponential attractors as ε tends to 0.For the equation endowed with dynamic boundary conditions, we first consider aregular nonlinearity and we treat the theoretical and numerical analysis. Then, weillustrate the results by numerical simulations in two space dimension which allow usto study the influence of different parameters. Finally, we treat the problem consideredwith a singular nonlinearity which is approximated by regular functions andwe give a suitable notion of solutions
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44

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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45

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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46

Ferreira, Fábio Freitas. "Problemas inversos sobre a esfera." Universidade do Estado do Rio de Janeiro, 2008. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=889.

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Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro
O objetivo desta tese é o desenvolvimento de algoritmos para determinar as soluções, e para determinação de fontes, das equações de Poisson e da condução de calor definidas em uma esfera. Determinamos as formas das equações de Poisson e de calor sobre a esfera, e desenvolvemos métodos iterativos, baseados em uma malha icosaedral e sua respectiva malha dual, para obter as soluções das mesmas. Mostramos que os métodos iterativos convergem para as soluções das equações discretizadas. Empregamos o método de regularização iterada de Alifanov para resolver o problema inverso, de determinação de fonte, definido na esfera.
The objective of this thesis is the development of algorithms to determine the solutions, and for determination of sources of, the equations of Poisson and heat conduction for a sphere. We establish the form of equations of Poisson and heat on the sphere, and developed iterative methods, based on a icosaedral mesh and its dual mesh, to obtain the solutions for them. It is shown that the iterative methods converge to the solutions of the equations discretizadas. It employed the method of settlement of Alifanov iterated to solve the inverse problem, determination of source, set in the sphere.
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47

Rizik, Vivian. "Analysis of an elasto-visco-plastic model describing dislocation dynamics." Thesis, Compiègne, 2019. http://www.theses.fr/2019COMP2505.

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Dans cette thèse on s'intéresse à l'analyse théorique et numérique de la dynamique des densités des dislocations, où les dislocations sont des défauts cristallins, apparaissant à l'échelle microscopique dans les alliages métalliques. En particulier, on considère en premier temps l'étude du modèle de Groma-Czikor-Zaiser (GCZ) et en second temps l'étude du modèle de Groma-Balog (GZ). Il s'agit en réalité d'un système d'équations de type paraboliques et de type Hamilton-Jacobi non-linéaires. Au départ, nous démontrons un résultat d'existence et d'unicité d'une solution régulière en utilisant le principe de comparaison et un argument de point fixe pour concernant le modèle GCZ. Ensuite, nous démontrons un résultat d'existence global en temps pour le modèle de GB, en se basant sur les notions des solutions de viscosités discontinues et sur une nouvelle estimation sur la variation totale de la solution, ainsi que sur la propagation à vitesse finie des équations régissantes. Ce résultat est étendu aussi au cas des systèmes d'équations d'Hamilton-Jacobi général. Enfin, nous proposons un schéma numérique semi-explicite permettant la discrétisation du modèle de GB. Nous montons, en s'appuyant sur l'étude théorique, que la solution discrète convergent vers la solution continue, ainsi qu'une estimation d'erreur entre la solution continue et la solution numérique. Des simulations montrant la robustesse du schème numériques sont également présentées
In this thesis, we are interested in the theoretical and numerical analysis o the dynamics of dislocation densities, where dislocations are crystalline defects appearing at the microscopic scale in metallic alloys. Particularly, the study of the Groma-Czikor-Zaiser model (GCZ) and the study of the Groma-Balog model (GB) are considered. The first is actually a system of parabolic type equations, where as, the second is a system of non-linear Hamilton-Jacobi equations. Initially, we demonstrate an existence and uniqueness result of a regular solution using a comparison principle and a fixed point argument for the GCZ model. Next, we establish a time-based global existence result for the GB model, based on notions of discontinuous viscosity solutions and a new estimate of total solution variation, as well as finite velocity propagation of the governed equations. This result is extended also to the cas of general Hamilton-Jacobi equation systems. Finally, we propose a semi-explicit numerical scheme allowing the discretization of the GB model. Based on the theoretical study, we prove that the discrete solution converges toward the continuous solution, as well as an estimate of error between the continuous solution and the numerical solution has been established. Simulations showing the robustness of the numerical scheme are also presented
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48

Pena, Luciana Prado Mouta. "Análise de um método para equação de convecção formulado à luz da mecânica dos meios contínuos a advecção de anomalias oceânicas e meteorológicas." Universidade do Estado do Rio de Janeiro, 2006. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=872.

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No presente trabalho estudamos e analisamos o método do Tubo de Trajetórias, um algoritmo conservativo, explícito, simples, fisicamente intuitivo, semi-Lagrangiano para equação de convecção. Mostramos que o método é incondicionalmente estável, essencialmente não-dispersivo, convergente e acurado de ordem 2 no tempo e no espaço. Soluções numéricas de sistemas e equações diferenciais ordinárias são testadas no contexto do método do Tubo de Trajetórias, com difíceis problemas clássicos. Aplicações são consideradas no âmbito do transporte oceânico e na advecção de frentes atmosféricas. A fim de testar as propriedades conservativas do método estudado, uma estimativa do erro de balanço de massa é usado aqui. Comparações com outras metodologias mostram a superioridade do método do Tubo de Trajetórias.
In the present work we studied and analyzed the Trajectories Tube method, a conservative, explicit, simple, physically intuitive, semi-Lagrangian algorithm for the convection equation. Kinematical aspects of the mechanics of continuous media are essentially the tools used for formulation and feasibility analysis. We showed that this method is unconditionally stable, essentially nondispersive, convergent and accurate of order two in time and space. Computational experiments with non-isochoric and isochoric motions show that the studied method can be used in compressible and incompressible flow. Numerical solutions of systems of ordinary differential equations (necessary conditions for acomplishment of the scheme) are tested in the Trajectories Tube method context, with classical difficult examples. Applications are considered in the ambit of oceanic transport and advection of atmospheric fronts, including the tracer problem within a Stommel gyre and the computation of the Dowell frontogenesis. Comparisions with other methodologies show the superiority of the Trajectories Tube method.
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49

Scheid, Jean-François. "Étude théorique et numérique de l'évolution morphologique d'interfaces." Paris 11, 1994. http://www.theses.fr/1994PA112027.

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L'objet de cette thèse est d'étudier un processus de dissolution-croissance où une phase solide est en contact avec une phase liquide. Ce phénomène s'accompagne d'un changement de géométrie de l'interface entre ces deux phases. Les lois physiques qui gouvernent ce processus conduisent mathématiquement à une loi de déplacement pour l'interface où la courbure moyenne apparait de façon non linéaire et à une équation parabolique pour la concentration de l'espèce chimique passée en solution. Il s'agit d'un problème de Stefan avec tension superficielle en dimension deux d'espace. Au chapitre 1, nous commençons par décrire le contexte physique et présenter la dérivation mathématique des équations pour la concentration et le déplacement de l'interface. Nous considérons au chapitre 2 le cas où l'interface est paramétrée sous la forme y=f(x,t). Nous montrons l'existence locale en temps d'une solution du problème dans des espaces de type Hölder en utilisant une méthode de point fixe. L’unicité est établie pour une classe de fonctions plus large que celle pour laquelle le résultat d'existence a été démontré. Nous étudions ensuite au chapitre 3 le cas où l'interface est une courbe simple fermée. L’interface est paramétrée par l'abscisse curviligne de l'interface initiale. Nous montrons alors l'existence locale en temps d'une solution de ce problème, en utilisant une méthode de point fixe. Au chapitre 4 nous présentons une méthode de résolution numérique. La difficulté majeure provient de ce que le domaine spatial varie au cours du temps, selon le déplacement de l'interface. Nous utilisons une méthode d'éléments finis pour la discrétisation de l'équation parabolique pour la concentration. Deux algorithmes s'appuyant sur des techniques différentes de calcul du terme de courbure ont été mis en œuvre pour calculer le déplacement de l'interface.
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50

Courtès, Clémentine. "Analyse numérique de systèmes hyperboliques-dispersifs." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLS467/document.

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Le but de cette thèse est d’étudier certaines équations aux dérivées partielles hyperboliques-dispersives. Une part importante est consacrée à l’analyse numérique et plus particulièrement à la convergence de schémas aux différences finies pour l’équation de Korteweg-de Vries et les systèmes abcd de Boussinesq. L’étude numérique suit les étapes classiques de consistance et de stabilité. Nous transposons au niveau discret la propriété de stabilité fort-faible des lois de conservations hyperboliques. Nous déterminons l’ordre de convergence des schémas et le quantifions en fonction de la régularité de Sobolev de la donnée initiale. Si nécessaire, nous régularisons la donnée initiale afin de toujours assurer les estimations de consistance. Une étape d’optimisation est alors nécessaire entre cette régularisation et l’ordre de convergence du schéma. Une seconde partie est consacrée à l’existence d’ondes progressives pour l’équation de Korteweg de Vries-Kuramoto-Sivashinsky. Par des méthodes classiques de systèmes dynamiques : système augmenté, fonction de Lyapunov, intégrale de Melnikov, par exemple, nous démontrons l’existence d’ondes oscillantes de petite amplitude
The aim of this thesis is to study some hyperbolic-dispersive partial differential equations. A significant part is devoted to the numerical analysis and more precisely to the convergence of some finite difference schemes for the Korteweg-de Vries equation and abcd systems of Boussinesq. The numerical study follows the classical steps of consistency and stability. The main idea is to transpose at the discrete level the weak-strong stability property for hyperbolic conservation laws. We determine the convergence rate and we quantify it according to the Sobolev regularity of the initial datum. If necessary, we regularize the initial datum for the consistency estimates to be always valid. An optimization step is thus necessary between this regularization and the convergence rate of the scheme. A second part is devoted to the existence of traveling waves for the Korteweg-de Vries-Kuramoto-Sivashinsky equation. By classical methods of dynamical systems : extended systems, Lyapunov function, Melnikov integral, for instance, we prove the existence of oscillating small amplitude traveling waves
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