The capillary surface formed within a symmetric annular tube is analyzed. Assuming identical contact angles along each boundary, we consider surfaces u(x,y) that satisfy the capillary problem on an annular region. Several qualitative properties of u are determined and in particular, the behaviour of u is examined in the limiting cases of the annular domain approaching a disk as well as a thin ring. The iterative method of Siegel is also applied to the boundary value problem and convergence is demonstrated under conditions which include a contact angle of zero. Moreover, some geometries still yield interleaving iterates, allowing for upper and lower bounds to be placed on the boundary values of u. However, the interleaving properties no longer hold universally and for other geometries, another more complex behaviour is described. Finally, a numerical method is designed to approximate the iterative scheme.

The capillary surface formed within a symmetric annular tube is analyzed. Assuming identical contact angles along each boundary, we consider surfaces u(x,y) that satisfy the capillary problem on an annular region. Several qualitative properties of u are determined and in particular, the behaviour of u is examined in the limiting cases of the annular domain approaching a disk as well as a thin ring. The iterative method of Siegel is also applied to the boundary value problem and convergence is demonstrated under conditions which include a contact angle of zero. Moreover, some geometries still yield interleaving iterates, allowing for upper and lower bounds to be placed on the boundary values of u. However, the interleaving properties no longer hold universally and for other geometries, another more complex behaviour is described. Finally, a numerical method is designed to approximate the iterative scheme.

Ordinary and partial differential equations are often derived as a first approximation to model a real-world situation, where the state of the system depends not only on the present time, but also on the history of the system. In this situation, a higher level of realism can be achieved by incorporating distributed delays in the mathematical models described by differential equations which results in delay Volterra integro-differential equations (denoted DVIDEs). Although DVIDEs serve as indispensable tools for modelling real systems, we still lack efficient and reliable software to approximate the solution of systems of DVIDEs. This thesis is concerned with designing, analyzing and implementing an efficient method to approximate the solution of a general system of neutral Volterra integro-differential equations with time-dependent delay arguments using a continuous Runge-Kutta (CRK) method. We introduce an adaptive stepsize selection strategy resulting in an approximate solution whose associated defect (residual) satisfies certain properties that allow us to monitor the global error reliably and efficiently. We prove the classic and optimal convergence of the numerical approximation to the exact solution. In addition, a companion system of equations is introduced in order to estimate the mathematical conditioning of the problem. A side effect of introducing this companion system is that it provides an effective estimate of the global error of the approximate solution, at a modest increase in cost. We have implemented our approach as an experimental Fortran 90 code capable of handling various kinds of DVIDEs with non-vanishing, vanishing, and infinite delay arguments.

Ordinary and partial differential equations are often derived as a first approximation to model a real-world situation, where the state of the system depends not only on the present time, but also on the history of the system. In this situation, a higher level of realism can be achieved by incorporating distributed delays in the mathematical models described by differential equations which results in delay Volterra integro-differential equations (denoted DVIDEs). Although DVIDEs serve as indispensable tools for modelling real systems, we still lack efficient and reliable software to approximate the solution of systems of DVIDEs. This thesis is concerned with designing, analyzing and implementing an efficient method to approximate the solution of a general system of neutral Volterra integro-differential equations with time-dependent delay arguments using a continuous Runge-Kutta (CRK) method. We introduce an adaptive stepsize selection strategy resulting in an approximate solution whose associated defect (residual) satisfies certain properties that allow us to monitor the global error reliably and efficiently. We prove the classic and optimal convergence of the numerical approximation to the exact solution. In addition, a companion system of equations is introduced in order to estimate the mathematical conditioning of the problem. A side effect of introducing this companion system is that it provides an effective estimate of the global error of the approximate solution, at a modest increase in cost. We have implemented our approach as an experimental Fortran 90 code capable of handling various kinds of DVIDEs with non-vanishing, vanishing, and infinite delay arguments.

We study a new method for the strong approximate solution of stochastic differential equations using coupling and we prove order one error bounds for the new scheme in Lp space assuming the invertibility of the diffusion matrix. We introduce and implement two couplings called the exact and approximate coupling for this scheme obtaining good agreement with the theoretical bound. Also we describe a method for non-invertibility case (Combined method) and we investigate its convergence order which will give O(h3/4 √log(h)j) under some conditions. Moreover we compare the computational results for the combined method with its theoretical error bound and we have obtained a good agreement between them. In the last part of this thesis we work out the performance of the multilevel Monte Carlo method using the new scheme with the exact coupling and we compare the results with the trivial coupling for the same scheme.

Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro / Neste trabalho, será considerado um problema de controle ótimo quadrático para a equação do calor em domínios retangulares com condição de fronteira do tipo Dirichlet é nos quais, a função de controle (dependente apenas no tempo) constitui um termo de fonte. Uma caracterização da solução ótima é obtida na forma de uma equação linear em um espaço de funções reais definidas no intervalo de tempo considerado. Em seguida, utiliza-se uma sequência de projeções em subespaços de dimensão finita para obter aproximações para o controle ótimo, o cada uma das quais pode ser gerada por um sistema linear de dimensão finita. A sequência de soluções aproximadas assim obtidas converge para a solução ótima do problema original. Finalmente, são apresentados resultados numéricos para domínios espaciais de dimensão 1. / In this work, a quadratic optimal control problem will be considered for the heat equation in rectangular domains with Dirichlet type boundary conditions in which the control function (depending only on time) constitutes a source term. A characterization of the solution is obtained in the form of a linear equation in a real function space defined in a considered time interval. Then, a sequence of projections in finite dimensional subspaces is used to obtain approximations for the optimal control, each of them can be generated by a finite dimension linear system. The sequence of approximate solutions obtained in this way converges to an optimal solution of the original problem. Finally, numerical results are presented for spatial domains of 1 dimension.

Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro / Neste trabalho, será considerado um problema de controle ótimo quadrático para a equação do calor em domínios retangulares com condição de fronteira do tipo Dirichlet é nos quais, a função de controle (dependente apenas no tempo) constitui um termo de fonte. Uma caracterização da solução ótima é obtida na forma de uma equação linear em um espaço de funções reais definidas no intervalo de tempo considerado. Em seguida, utiliza-se uma sequência de projeções em subespaços de dimensão finita para obter aproximações para o controle ótimo, o cada uma das quais pode ser gerada por um sistema linear de dimensão finita. A sequência de soluções aproximadas assim obtidas converge para a solução ótima do problema original. Finalmente, são apresentados resultados numéricos para domínios espaciais de dimensão 1. / In this work, a quadratic optimal control problem will be considered for the heat equation in rectangular domains with Dirichlet type boundary conditions in which the control function (depending only on time) constitutes a source term. A characterization of the solution is obtained in the form of a linear equation in a real function space defined in a considered time interval. Then, a sequence of projections in finite dimensional subspaces is used to obtain approximations for the optimal control, each of them can be generated by a finite dimension linear system. The sequence of approximate solutions obtained in this way converges to an optimal solution of the original problem. Finally, numerical results are presented for spatial domains of 1 dimension.

In this essay I defend the notion of deductive explanation mainly against two types of putative counterexamples: those found in genuinely indeterministic systems and those found in complex dynamic systems. Using Railton's notions of explanatory information and ideal explanatory text, deductivism is defended in an indeterministic setting. Furthermore, an argument against non-deductivism that hinges on peculiarities of probabilistic causality is presented. The use of the notion of an ideal explanatory text gives rise to problems in accounting for explanations in complex dynamic systems, regardless of whether they are deterministic or not. These problems are considered in the essay and a solution is suggested. This solution forces the deductivist to abandon the requirement that an explanation consists of a deductive argument, but it is argued that the core of deductivism is saved in so far as we, for full explanations, can still adhere to the fundamental requirement: If A explains B, then A is inconsistent with anything inconsistent with B.

Les travaux de cette thèse portent sur des méthodes de volumes finis sur maillages quelconque pour la discrétisation de problèmes d'évolution non linéaires modélisant le transport de contaminants en milieu poreux et les écoulements diphasiques.Au Chapitre 1, nous étudions une famille de schémas numériques pour la discrétisation d'une équation parabolique dégénérée de convection-reaction-diffusion modélisant le transport de contaminants dans un milieu poreux qui peut être hétérogène et anisotrope. La discrétisation du terme de diffusion est basée sur une famille de méthodes qui regroupe les schémas de volumes finis hybrides, de différences finies mimétiques et de volumes finis mixtes. Le terme de convection est traité à l'aide d'une famille de méthodes qui s'appuient sur les inconnues hybrides associées aux interfaces du maillage. Cette famille contient à la fois les schémas centré et amont. Les schémas que nous étudions permettent une discrétisation localement conservative des termes d'ordre un et d'ordre deux sur des maillages arbitraires en dimensions d'espace deux et trois. Nous démontrons qu'il existe une solution unique du problème discret qui converge vers la solution du problème continu et nous présentons des résultats numériques en dimensions d'espace deux et trois, en nous appuyant sur des maillages adaptatifs.Au Chapitre 2, nous proposons un schéma de volumes finis hybrides pour la discrétisation d'un problème d'écoulement diphasique incompressible et immiscible en milieu poreux. On suppose que ce problème a la forme d'une équation parabolique dégénérée de convection-diffusion en saturation couplée à une équation uniformément elliptique en pression. On considère un schéma implicite en temps, où les flux diffusifs sont discrétisés par la méthode des volumes finis hybride, ce qui permet de pouvoir traiter le cas d'un tenseur de perméabilité anisotrope et hétérogène sur un maillage très général, et l'on s'appuie sur un schéma de Godunov pour la discrétisation des flux convectifs, qui peuvent être non monotones et discontinus par rapport aux variables spatiales. On démontre l'existence d'une solution discrète, dont une sous-suite converge vers une solution faible du problème continu. On présente finalement des cas test bidimensionnels.Le Chapitre 3 porte sur un problème d'écoulement diphasique, dans lequel la courbe de pression capillaire admet des discontinuité spatiales. Plus précisément on suppose que l'écoulement prend place dans deux régions du sol aux propriétés très différentes, et l'on suppose que la loi de pression capillaire est discontinue en espace à la frontière entre les deux régions, si bien que la saturation de l'huile et la pression globale sont discontinues à travers cette frontière avec des conditions de raccord non linéaires à l'interface. On discrétise le problème à l'aide d'un schéma, qui coïncide avec un schéma de volumes finis standard dans chacune des deux régions, et on démontre la convergence d'une solution approchée vers une solution faible du problème continu. Les test numériques présentés à la fin du chapitre montrent que le schéma permet de reproduire le phénomène de piégeage de la phase huile. / In Chapter 1 we study a family of finite volume schemes for the numerical solution of degenerate parabolic convection-reaction-diffusion equations modeling contaminant transport in porous media. The discretization of possibly anisotropic and heterogeneous diffusion terms is based upon a family of numerical schemes, which include the hybrid finite volume scheme, the mimetic finite difference scheme and the mixed finite volume scheme. One discretizes the convection term by means of a family of schemes which makes use of the discrete unknowns associated to the mesh interfaces, and contains as special cases an upwind scheme and a centered scheme. The numerical schemes which we study are locally conservative and allow computations on general multi-dimensional meshes. We prove that the unique discrete solution converges to the unique weak solution of the continuous problem. We also investigate the solvability of the linearized problem obtained during Newton iterations. Finally we present a number of numerical results in space dimensions two and three using nonconforming adaptive meshes and show experimental orders of convergence for upwind and centered discretizations of the convection term.In Chapter 2 we propose a finite volume method on general meshes for the numerical simulation of an incompressible and immiscible two-phase flow in porous media. We consider the case that it can be written as a coupled system involving a degenerate parabolic convection-diffusion equation for the saturation together with a uniformly elliptic equation for the global pressure. The numerical scheme, which is implicit in time, allows computations in the case of a heterogeneous and anisotropic permeability tensor. The convective fluxes, which are non monotone with respect to the unknown saturation and discontinuous with respect to the space variables, are discretized by means of a special Godunov scheme. We prove the existence of a discrete solution which converges, along a subsequence, to a solution of the continuous problem. We present a number of numerical results in space dimension two, which confirm the efficiency of the numerical method.Chapter 3 is devoted to the study of a two-phase flow problem in the case that the capillary pressure curve is discontinuous with respect to the space variable. More precisely we assume that the porous medium is composed of two different rocks, so that the capillary pressure is discontinuous across the interface between the rocks. As a consequence the oil saturation and the global pressure are discontinuous across the interface with nonlinear transmission conditions. We discretize the problem by means of a numerical scheme which reduces to a standard finite volume scheme in each sub-domain and prove the convergence of a sequence of approximate solutions towards a weak solution of the continuous problem. The numerical tests show that the scheme can reproduce the oil trapping phenomenon.