This thesis is concerned with experimental and computational studies on pulse detonation engine (PDE) that has been envisioned as a new concept engine. These engines use the high pressure generated by detonation wave for propulsion. The cycle efficiency of PDE is either higher in comparison to conventional jet engines or at least has similar high performance with much greater simplicity in terms of components. The first part of the work consists of an experimental study of the performance of PDE under choked flame and partial fill conditions. Detonations used in classical PDEs create conditions of Mach numbers of 4-6 and choked flames create conditions in which flame achieves Mach numbers near-half of detonation wave. While classical concepts on PDE's utilize deflagration-to-detonation transition and are more intensively studied, the working of PDE under choked regime has received inadequate attention in the literature and much remains to be explored. Most of the earlier studies claim transition to detonation as success in the working of the PDE and non-transition as failure. After exploring both these regimes, the current work brings out that impulse obtained from the wave traveling near the choked flame velocity conditions is comparable to detonation regime. This is consistent with the understanding from the literature that CJ detonation may not be the optimum condition for maximum specific impulse. The present study examines the details of working of PDE close to the choked regime for different experimental conditions, in comparison with other aspects of PDEs. The study also examines transmission of fast flames from small diameter pipe into larger ducts. This approach in the smaller pipe for flame acceleration also leading to decrease in the time and length of transition process. The second part of the study aims at elucidating the features of deflagration-to-detonation transition with direct numerical simulation (DNS) accounting for and the choice of full chemistry and DNS is based on two features: (a) the induction time estimation at the conditions of varying high pressure and temperature behind the shock can only be obtained through the use of full chemistry, and (b) the complex effects of fine scale of turbulence that have sometimes been argued to influence the acceleration phase in the DDT cannot be captured otherwise. Turbulence in the early stages causes flame wrinkling and helps flame acceleration process. The study of flame propagation showed that the wrinkling of flame has major effect on the final transition phase as flame accelerates through the channel. Further, flame becomes corrugated prior to transition. This feature was investigated using non-uniform initial conditions. Under these conditions the pressure waves emanating from corrugated flame interact with the shock moving ahead and transition occurs in between the flame and the forward propagating shock wave. The primary contributions of this thesis are: (a) Elucidating the phenomenology of choked flames, demonstrating that under partial fill conditions, the specific impulse can be superior to detonations and hence, allowing for the possibility of choked flames as a more appropriate choice for propulsive purposes instead of full detonations, (b) The use of smaller tube to enhance the flame acceleration and transition to detonation. The comparison with earlier experiments clearly shows the enhancements achieved using this method, and (c) The importance of the interaction between pressure waves emanating from the flame front with the shock wave which leads to formation of hot spots finally transitioning to detonation wave.

Le fonctionnement d'un neurone, unité fondamentale du système nerveux, intéresse de nombreuses disciplines scientifiques. Il existe ainsi des modèles mathématiques qui décrivent leur comportement par des systèmes d'EDO ou d'EDP. Plusieurs de ces modèles peuvent ensuite être couplés afin de pouvoir étudier le comportement de réseaux, systèmes complexes au sein desquels émergent des propriétés. Ce travail présente, dans un premier temps, les principaux mécanismes régissant ce fonctionnement pour en comprendre la modélisation. Plusieurs modèles sont alors présentés, jusqu'à celui de FitzHugh-Nagumo (FHN), qui présente une dynamique très intéressante.C'est sur l'étude théorique mais également numérique de la dynamique asymptotique et transitoire du modèle de FHN en EDO, que se concentre la seconde partie de cette thèse. A partir de cette étude, des réseaux d'interactions d'EDO sont construits en couplant les systèmes dynamiques précédemment étudiés. L'étude du phénomène de synchronisation identique au sein de ces réseaux montre l'existence de propriétés émergentes pouvant être caractérisées par exemple par des lois de puissance. Dans une troisième partie, on se concentre sur l'étude du système de FHN dans sa version EDP. Comme la partie précédente, des réseaux d'interactions d'EDP sont étudiés. On entreprend dans cette partie une étude théorique et numérique. Dans la partie théorique, on montre l'existence de l'attracteur global dans l'espace L2(Ω)nd et on donne des conditions suffisantes de synchronisation. Dans la partie numérique, on illustre le phénomène de synchronisation ainsi que l'émergence de lois générales telles que les lois puissances ou encore la formation de patterns, et on étudie l'effet de l'ajout de la dimension spatiale sur la synchronisation. / The neuron, a fundamental unit in the nervous system, is a point of interest in many scientific disciplines. Thus, there are some mathematical models that describe their behavior by ODE or PDE systems. Many of these models can then be coupled in order to study the behavior of networks, complex systems in which the properties emerge. Firstly, this work presents the main mechanisms governing the neuron behaviour in order to understand the different models. Several models are then presented, including the FitzHugh-Nagumo one, which has a interesting dynamic. The theoretical and numerical study of the asymptotic and transitory dynamics of the aforementioned model is then proposed in the second part of this thesis. From this study, the interaction networks of ODE are built by coupling previously dynamic systems. The study of identical synchronization phenomenon in these networks shows the existence of emergent properties that can be characterized by power laws. In the third part, we focus on the study of the PDE system of FHN. As the previous part, the interaction networks of PDE are studied. We have in this section a theoretical and numerical study. In the theoretical part, we show the existence of the global attractor on the space L2(Ω)nd and give the sufficient conditions for identical synchronization. In the numerical part, we illustrate the synchronization phenomenon, also the general laws of emergence such as the power laws or the patterns formation. The diffusion effect on the synchronization is studied.

[pt] O objetivo principal desta dissertação é estudar a desigualdade de Harnack e as estimativas de Holder, para um operador elíptico de segunda ordem, na forma não divergente e na forma divergente, respectivamente, sendo os coeficientes funções mensuráveis e limitadas em um domínio ômega contido em Rn. / [en] The main objective of this dissertation is to study Harnack s inequality and Holder s estimates for a second-order elliptic operator, written in the non-divergent form and in the divergent form, respectively, where the coefficient functions are measurable and bounded functions in a domain omega contained in Rn.

From the viewpoint of the adaptive solution of partial differential equations a finit e element method on hierarchical triangular meshes is developed permitting hanging nodes arising from nonuniform hierarchical refinement. Construction, extension and restriction of the nonuniform hierarchical basis and the accompanying mesh are described by graphs. The corresponding FE basis is generated by hierarchical transformation. The characteristic feature of the generalizable concept is the combination of the conforming hierarchical basis for easily defining and changing the FE space with an accompanying nonconforming FE basis for the easy assembly of a FE equations system. For an elliptic model the conforming FEM problem is solved by an iterative method applied to this nonconforming FEM equations system and modified by projection into the subspace of conforming basis functions. The iterative method used is the Yserentant- or BPX-preconditioned conjugate gradient algorithm. On a MIMD computer system the parallelization by domain decomposition is easy and efficient to organize both for the generation and solution of the equations system and for the change of basis and mesh.

We prove some sufficient conditions for obtaining convergence rates in regularization of linear ill-posed problems in a Hilbert space setting and show that these conditions are directly related with the conditional stability in several concrete inverse problems for partial differential equations.

Title: Estimation of the algebraic error and stopping criteria in numerical solution of partial differential equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor of the master thesis: Zdeněk Strakoš Abstract: After introduction of the model problem and its properties we describe the Conjugate Gradient Method (CG). We present the estimates of the energy norm of the error and a heuristic for the adaptive refinement of the estimate. The difference in the local behaviour of the discretization and the algebraic error is illustrated by numerical experiments using the given model problem. A posteriori estimates for the discretization and the total error that take into account the inexact solution of the algebraic system are then discussed. In order to get a useful perspective, we briefly recall the multigrid method. Then the Cascadic Conjugate Gradient Method of Deuflhard (CCG) is presented. Using the estimates for the error presented in the preceding parts of the thesis, the new stopping criteria for CCG are proposed. The CCG method with the new stopping criteria is then tested. Keywords: numerical PDE, discretization error, algebraic error, error es- timates, locality of the error, adaptivity

Most of PDE-based restoration models and their numerical realizations show a common drawback: loss of fine structures. In particular, they often introduce an unnecessary numerical dissipation on regions where the image content changes rapidly such as on edges and textures. This thesis studies the magnitude data/imagery of magnetic resonance imaging (MRI) which follows Rician distribution. It analyzes statistically that the noise in the magnitude MRI data is approximately Gaussian of mean zero and of the same variance as in the frequency-domain measurements. Based on the analysis, we introduce a novel partial differential equation (PDE)-based denoising model which can restore fine structures satisfactorily and simultaneously sharpen edges as needed. For an efficient simulation we adopt an incomplete Crank-Nicolson (CN) time-stepping procedure along with the alternating direction implicit (ADI) method. The algorithm is analyzed for stability. It has been numerically verified that the new model can reduce the noise satisfactorily, outperforming the conventional PDE-based restoration models in 3-4 alternating direction iterations, with the residual (the difference between the original image and the restored image) being nearly edgeree. It has also been verified that the model can perform edge-enhancement effectively during the denoising of the magnitude MRI imagery. Numerical examples are provided to support the claim.

<p>Oxygen (O2) plays a crucial role in the energy metabolism of complex multicellular life on earth. Due to the small and finite energy stores in the body, fine-tuned changes within the body are required to meet metabolic demand during skeletal muscle contractions, such as during exercise and activities of daily living. The skeletal muscle microcirculation is one of the last steps in the O2 transport pathway from the lungs to muscle cells and represents the largest surface area for O2 and substrate exchange. When skeletal muscle O2 uptake increases during contractions to meet metabolic demand, there must be an increase in muscle O2 delivery. To achieve these elevations in O2 delivery, vessel (arteriole) diameter in the microcirculation is increased, known as vasodilation. This process in the skeletal muscle microcirculation is regulated by several factors, such as neurohumoral, mechanical, endothelial, paracrine, and metabolic influences, which are imperative in properly regulating O2 delivery at rest and during muscular contractions. Two vasodilatory pathways of interest in this dissertation are the cyclooxygenase (COX) and nitric oxide (NO) vasodilatory pathways.</p> <p>The primary aim of my dissertation studies was to determine the mechanisms that modulate skeletal muscle oxygenation in health and to define the impact of a potentially effective intervention, whole-body chronic heat therapy (HT), to treat heart failure with preserved ejection fraction (HFpEF). In Chapter 2, we report that acute selective COX-2 inhibition had no effect on resting or exercising skeletal muscle microvascular oxygenation, pulmonary oxygen uptake, or exercise tolerance in healthy young humans. In Chapter 3, we report that NO, via phosphodiesterase type 5 inhibition, regulates myocyte O2 transport at rest and during recovery from muscle contractions in healthy young rats. In Chapter 4, we show that whole-body chronic HT promotes central and peripheral adaptations, which impact positively exercise tolerance in a pre-clinical rat model of HFpEF. Specifically, whole-body chronic HT had beneficial influences on exercise tolerance, skeletal muscle oxygenation from rest to contractions (driven, at least in part, by enhanced NO bioavailability), body composition, and cardiac function. Chapter 5 is a summary of the results and limitations of the projects presented in Chapters 2-4, with a brief discussion of potential future research directions. </p>

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In this Thesis we explore, both theoretically and numerically, splitting strategies for a hyperbolic system of one-dimensional (1D) blood flow equations with a passive scalar transport equation. Our analysis involves a two-step framework that includes splitting at the level of partial differential equations (PDEs) and numerical methods for discretizing the ensuing problems. This study is inspired by the original flux splitting approach of Toro and Vázquez-Cendón (2012) originally developed for the conservative Euler equations of compressible gas dynamics. In this approach the flux vector in the conservative case, and the system matrix in the non-conservative one, are split into advection and pressure terms: in this way, two systems of partial differential equations are obtained, the advection system and the pressure system. From the mathematical as well as numerical point of view, a basic problem to be solved is the special Cauchy problem called the Riemann problem. This latter provides an analytical solution to evaluate the performance of the numerical methods and, in our approach, it is of primary importance to build the presented numerical schemes. In the first part of the Thesis a detailed theoretical analysis is presented, involving the exact solution of the Riemann problem for the 1D blood flow equations, depicted for a general constant momentum correction coefficient and a tube law that allows to describe both arteries and veins with continuous or discontinuous mechanical and geometrical properties and an advection equation for a passive scalar transport. In literature, this topic has been already studied only for a momentum correction coefficient equal to one, that is related to the prescribed velocity profile and in this case corresponds to a flat one, i.e. an inviscid fluid. In the case of discontinuous properties, only the subsonic regime is considered. In addition we propose a procedure to compute the obtained exact solution and finally we validate it numerically, by comparing exact solutions to those obtained with well-known, numerical schemes on a carefully designed set of test problems. Furthermore, an analogous theoretical analysis and resolution algorithm are presented for the advection system and the pressure system arising from the splitting at the level of PDEs of the complete system of 1D blood flow equations. It is worth noting that the pressure system, in case of veins, presents a loss of genuine non-linearity resulting in the formation of rarefactions, shocks and compound waves, these latter being a composition of rarefactions and shocks. In the second part of the Thesis we present novel finite volume-type, flux splitting-based, numerical schemes for the conservative 1D blood flow equations and splitting-based numerical schemes for the non-conservative 1D blood flow equations that incorporate an advection equation for a passive scalar transport, considering tube laws that allow to model blood flow in arteries and veins and take into account a general constant momentum correction coefficient. A detailed efficiency analysis is performed in order to showcase the advantages of the proposed methodologies in comparison to standard approaches.