Phase-field models are emerging as a promising strategy to simulate interfacial phenomena. Rather than tracking interfaces explicitly as done in sharp interface descriptions, these models use a diffuse order parameter to monitor interfaces implicitly. This implicit description, as well as solid physical and mathematical footings, allow phase-field models to overcome problems found by predecessors. Nonetheless, the method has significant drawbacks. The phase-field framework relies on the solution of high-order, nonlinear partial differential equations. Solving these equations entails a considerable computational cost, so finding efficient strategies to handle them is important. Also, standard discretization strategies can many times lead to incorrect solutions. This happens because, for numerical solutions to phase-field equations to be valid, physical conditions such as mass conservation and free energy monotonicity need to be guaranteed. In this work, we focus on the development of thermodynamically consistent algorithms for time integration of phase-field models. The first part of this thesis focuses on an energy-stable numerical strategy developed for the phase-field crystal equation. This model was put forward to model microstructure evolution. The algorithm developed conserves, guarantees energy stability and is second order accurate in time. The second part of the thesis presents two numerical schemes that generalize literature regarding energy-stable methods for conserved and non-conserved phase-field models. The time discretization strategies can conserve mass if needed, are energy-stable, and second order accurate in time. We also develop an adaptive time-stepping strategy, which can be applied to any second-order accurate scheme. This time-adaptive strategy relies on a backward approximation to give an accurate error estimator. The spatial discretization, in both parts, relies on a mixed finite element formulation and isogeometric analysis. The codes are available online and implemented in PetIGA, a high-performance isogeometric analysis framework.

This thesis explores a range of stability techniques applied to fluid structures that develop in various constant density flows. In particular, the stability of nonlinear structures which develop in rotating plane Couette flow is analyzed using Floquet theory, which allows the global stability of an important secondary nonlinear structure called a Taylor vortex to be determined. From this the distinct tertiary states which emerge as Taylor vortices break down are characterized and their bifurcation behaviour is studied. Also, non-modal stability analyses are conducted in rotating plane Couette flow and annular Poiseuille-Couette flow. In each case the growth mechanisms and the form of the perturbations responsible for the maximum linear energy amplification are discussed. Finally, the non-modal behaviour of the Papkovitch-Fadle operator is treated and its relevance to spatially developing disturbances in Stokes channel flow is examined. The mechanisms and the rates of convergence of the linear spatial energy amplification are investigated and contrasted with temporal energy amplification.

In this work we present some of the general theory of shock waves and their stability properties. We examine the concepts of nonlinear stability and spectral stability, noting that for certain classes of equations the study of nonlinear stability is reduced to the analysis of the spectra of the linearized eigenvalue problem. A useful tool in the study of spectral stability is the Evans function, an analytic function whose zeros correspond to the eigenvalues of the linearized eigenvalue problem. We discuss techniques for numerical Evans function computation that ensure analyticity, allowing standard winding number arguments and rootfinding methods to be used to locate eigenvalues. The Evans function is then used to study the spectra of the high Lewis number combustion system, tracking eigenvalues in the right-half plane.

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L'objectif de cette thèse est d'analyser l'influence des non linéarités, du comportement rhéologique des fluides rhéofluidifiants, sur les conditions de stabilité et de transition vers la turbulence. Dans un premier temps, une analyse linéaire de stabilité avec une approche modale a été réalisée. Les résultats obtenus mettent clairement en évidence l'effet stabilisant de la rhéofluidification. Ensuite, une analyse faiblement non linéaire de stabilité a été menée en vue d'examiner l'influence de la perturbation de la viscosité sur la stabilité vis à vis de perturbations d'amplitude finie. L'analyse de la contribution des termes non linéaires d'inertie et visqueux montre que, contrairement aux termes d'inertie, les termes non linéaires visqueux ont tendance à accélérer l'écoulement et favoriser une bifurcation sur-critique. Les effets rhéofluidifiants tendent à réduire la dissipation visqueuse. Finalement, une analyse fortement non linéaire de stabilité a été conduite en utilisant les techniques de suivi de branches de solutions par des méthodes de continuation. Pour pouvoir traiter les termes visqueux fortement non linéaires, un code de calcul pseudo-spectral a été développé. Des solutions non linéaires d'équilibre ont été obtenues et caractérisées pour différentes valeurs des paramètres rhéologiques / The aim of this study is to understand the influence of the nonlinear rheological behaviour of the shear-thinning fluids on the flow stability and transition to turbulence. First, a linear stability analysis using modal approach was carried out. Results clearly highlight the stabilizing effect of shear-thinning. Then, as a first approach to take into account nonlinear effects of viscosity perturbation on the flow stability, a weakly nonlinear stability analysis is performed in the neighbourhood of the critical conditions. Results indicate that shear-thinning reduces the viscous dissipation and, in contrast to inertial terms, the nonlinear viscous terms tend to accelerate the flow and act in favour of supercritical bifurcation. Finally, a nonlinear stability analysis is done by following solution branches in the parameter space using continuation techniques. To deal with highly nonlinear viscous terms, a pseudo-spectral code is developed. Nonlinear equilibrium solutions was found and characterized for various values of the rheological parameters

The objective of this research is to develop, characterize, and demonstrate novel parametric architectures capable of wideband operation while maintaining high gain and stability. To begin the study, phase-incoherent upconverting parametric amplifiers will be explored by first developing a set of analytical models describing their achievable gain and efficiency. These models will provide a set of design tools to optimize and evaluate prototype circuit boards. The prototype boards will then be used to demonstrate their achievable gain, bandwidth, efficiency, and stability. Further investigation of the analytical models and data collected from the prototype boards will conclude bandwidth and gain limitations and end the investigation into phase-incoherent upconverting parametric amplifiers in lieu of negative-resistance parametric amplifiers. Traditionally, there were two versions of negative-resistance parametric amplifiers available: degenerate and non-degenerate. Both modes of operation are considered single-frequency amplifiers because both the input and output frequencies occur at the source frequency. Degenerate parametric amplifiers offer more power gain than their non-degenerate counterpart and do not require additional circuitry for idler currents. As a result, a phase-coherent degenerate parametric amplifier printed circuit board prototype will be built to investigate achievable gain, bandwidth, and stability. Analytical models will be developed to describe the gain and efficiency of phase-coherent degenerate parametric amplifiers. The presence of a negative resistance suggests the possibility of instability under certain operating conditions, therefore, an in-depth stability study of phase-coherent degenerate parametric amplifiers will be performed. The observation of upconversion gain in phase-coherent degenerate parametric amplifiers will spark investigation into a previously unknown parametric architecture: phase-coherent upconverting parametric amplifiers. Using the phase-coherent degenerate parametric amplifier prototype board, stable phase-coherent upconversion with gain will be demonstrated from the source input frequency to its third harmonic. An analytical model describing the large-signal transducer gain of phase-coherent upconverting parametric amplifiers from the first to the third harmonic of the source input will be derived and validated using the prototype board and simulations.

With the algorithm’s suitability for exploiting current petascale and next-generation exascale supercomputers, stable and structure-preserving properties are necessary to develop predictive computational tools. In this dissertation, summation-by-parts (SBP) operators and a new relaxation Runge–Kutta (RRK) scheme are used to construct mimetic and structure-preserving full discretization for non-reactive compressible computational fluid dynamics (CFD) and reaction-diffusion models. In the first chapter, we provide the necessary background and a literature survey that forms the basis of this dissertation. Next, we provide a short overview of entropy stability for general conservation laws. The second chapter covers the analysis of the Eulerian model for compressible and heat-conducting flows. We provide the necessary background of the new system of parabolic partial differential equation (PDE). Then, we present the entropy stability analysis of the model at the continuous level. Subsequently, using the SBP, we construct an entropy-stable discretization of any order for unstructured grids with tensor-product elements. The third chapter discusses the implementation of RRK methods. We start by reviewing the RRK scheme constructed to guarantee conservation or stability with respect to any inner-product norm. Then, we present the extension and generalization of RRK schemes to general convex functionals and their application to compressible fluid flow problems. The final chapter demonstrates the far-reaching capabilities of the SBP operators and RRK schemes presenting the development of a novel fully discrete Lyapunov stable discretization for reaction models with spatial diffusion. Finally, we conclude this dissertation with an overview of our achievements and future research directions.