A presente dissertação apresenta uma análise da presença de componentes estocásticas na equação de advecção-difusão, e como estas influenciam a estabilidade da solução. Para tal, a equação de advecção-difusão determinística com fonte contínua idealizada é resolvida através da transformada de Fourier. Adiante, a equação determinística é combinada com componentes estocásticas na velocidade do vento, comprimento de rugosidade e coeficiente de difusão turbulenta vertical. Além disso, é considerada uma permeabilidade parcial nos contornos verticais, de modo que parte do poluente ultrapassa a camada limite atmosférica ou o solo, e outra parte reflete e retorna `a atmosfera. Os resultados obtidos foram validados com os dados do experimento de Hanford. / The present work presents an analysis of the presence of stochastic components in the advection-diffusion equation and how they influence the stability of the solution. For this purpose, the deterministic advection-diffusion equation with idealized continuous source is solved by Fourier transform. Further, the deterministic equation is combined with stochastic components in the wind speed, the roughness and the vertical eddy diffusion coefficient. Moreover, partial permeability is considered in the vertical contours, in the sense that part of the pollutant leaks out of the atmospheric boundary layer or into the soil, and a part is reflected back into the atmosphere. Results were validated with the Hanford experimental data.

La résolution numérique du système d'équations pleinement compressibles en vue de son utilisation pour des applications en Prévision Numérique du Temps (PNT) soulève de nombreuses questions. L'une d'elles porte sur le choix des schémas de discrétisation temporelle à mettre en oeuvre afin de résoudre ce système de la manière la plus efficace possible, pour permettre la continuelle amélioration qualitative des prévisions. Jusqu'alors, les schémas de discrétisation temporelle basés sur des techniques semi-implicites (SI) étaient les plus couramment employés PNT, compte tenu de leur robustesse et de leur grande propriété de stabilité. Mais avec l'émergence des machines massivement parallèles à mémoire distribuée, l'efficacité de ces techniques est actuellement remise en question, car leur confortable plage de stabilité est obtenue au prix de l'inversion d'un problème elliptique tri-dimensionnel très gourmand en communications. Ce travail thèse vise à explorer d'autres méthodes de discrétisation temporelle, en remplacemant des méthodes SI, s'appuyant sur des approches de type Horizontalement Explicite et Verticalement Implicite (HEVI). D'une part, ces approches s'affranchissent de la contrainte numérique imposée sur le pas de temps par la propagation verticales des ondes rapides supportées par le système, grâce au traitement implicite des processus verticaux. D'autre part, elles exploitent le paradigme de programmation voulant que chaque colonne verticale du modèle numérique soit traitée par un unique processeur. Ainsi, le traitement implicite de cette direction n'engendre aucunes communications entre les processeurs. Cependant, bien que ces ap- proches HEVI apparaissent comme une solution attractive, rien ne garanti que leurs efficacités puissent être aussi compétitives que celles des sché- mas SI. Pour ce faire, ces schémas HEVI doivent permettre l'utilisation de pas de temps raisonnables pour une application en PNT. L'objectif de ce travail de thèse est d'élaborer un schéma de discrétisation temporelle HEVI le plus efficace possible pour une utilisation en PNT, c'est à dire, un schéma qui autorise le plus long pas de temps possible. Dans cette optique, deux voies ont été explorées : la première, issue des méthodes à pas de temps fractionné, a permis de revisiter et d'améliorer un schéma de discrétisation temporelle déjà proposé mais dont l'examen n'a jamais été approfondi dans la littérature ; il s'agit du schéma d'avance temporelle saute-moutons trapézo\"idal. Il a été mis en évidence que l'ajout d'un simple filtre temporel d'usage commun en PNT, améliore grandement la stabilité de ce schéma, lui permettant ainsi à moindre coût de rivaliser en terme de stabilité avec le schéma Runge-Kutta explicite d'ordre 3. La seconde voie, plus récente, c'est avérée la plus prometteuse. Elle repose sur l'utilisation des méthodes Runge-Kutta Implicite-Explicite (RK-IMEX) HEVI. Au cours l'étude, il a été tout d'abord mis en évidence certains problèmes de stabi- lité des schémas initialement suggérés dans la littérature en présence des processus d'advection. Puis, une nouvelle classe de schéma RK-IMEX HEVI s'appuyant sur un traitement temporel spécifique des termes d'ajustement horizontaux a été proposé / The use non-hydrostatic fully compressible modelling system in the perspective of Numerical Weather Prediction (NWP) raises many challenging questions, among which the choice time discretization scheme. It is commonly acknowledge that the ideal time marching algorithms to integrate the fully compressible system should both overcome the stability constraint imposed on time-step by the fast propagating waves supported by the system, and be scalable enough for efficiently computing on massively parallel computer machine. The assumed poor scalability property of Semi-implicit (SI) time schemes, currently favoured in NWP, is quite a drawback as they require global communications to solve a full three-dimensional elliptic problem. Because it is considered as the best compromise between stability, accuracy and scalability the properties of various classes of Horizontally Explicit Vertically Implicit (HEVI) schemes have been deeply explore in this work in a view of solving the fully system in mass-based coordinate. This class of time discretization approach eliminates all the problems linked to the implicit treatment of horizontal high-frequency forcings by coupling multi-step or multi-stage explicit methods for the horizontal propagation of fast waves to an implicit scheme for the treatment of vertically prop- agating elastic disturbances. The limitation in time-steps compared to SI schemes would be compensated by a much more economical algorithm per time-step. However, it is not firmly established that the efficiency of such a HEVI schemes could compete with one of the semi-implicit schemes. The main objective of this Phd thesis work is to elaborate an efficient HEVI time scheme allowing usable time-step for NWP applications. For this purpose, the so-called explicit time-splitting technique and the recently suggested Runge-Kutta IMEX (RK-IMEX) schemes have been explored un- der HEVI approach. Firstly, the superiority in term of stability of the RK-IMEX methods in respect with the time-splitting approach has been con- firmed. However, in presence of advection processes some unstable numerical behaviour of these schemes has been pointed out. To circumvent this problem a new class of RK-IMEX HEVI schemes has been proposed. This new class of HEVI time schemes reveals to be very attractive since they provide both good stability and accuracy properties. Secondly, in a side aspect of the HEVI approach, the stability impact of the temporal treatment of the terrain following coordinate non-linear metric terms has been demonstrated. Numerical analyses on simplified framework indicate that there might be a benefit to deal with these specific terms in the implicit part of the HEVI schemes. All the theoretical studies have been confirmed by nu- merical testing through the use of a Cartesian vertical plane fully compressible model cast in a mass-based coordinate.

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This thesis provides a mathematical framework for the development of efficient control strategies that satisfy the charters of Integrated Pest Management (IPM) which aims to maintain pest population at a low impact level. This mathematical framework is based on a dynamical system approach and comprises the construction of mathematical models, their theoretical study, the development of adequate schemes for numerical solutions and reliable procedures for parameter identification. The first output of this thesis is the construction of trap-insect spatio-temporal models formulated via advection-diffusion-reaction processes. These models were used to simulate numerically trapping to compare with field data. As a result, practical protocols were identified to estimate pest-population size and distribution as well as its dispersal capacity and parameter values related to the attractiveness of the traps. The second major output of this thesis is the prediction of the impact of a specific control method: mating disruption using a female pheromone and trapping. A compartmental model, formulated via a system of ordinary differential equations, was built based on biological and mating behaviour knowledge of the pest. The theoretical analysis of the model yields threshold values for the dosage of the pheromone above which extinction of the population is ensured. The practical relevance of the results obtained in this thesis shows that mathematical modelling is an essential supplement to experiments in optimizing control strategies. / Thesis (PhD)--University of Pretoria, 2016. / Mathematics and Applied Mathematics / PhD / Unrestricted

<div>Volumetric cloud generation and rendering algorithms are well-developed to meet the need for a realistic sky performance in animation or games. However, it is challenging to create a stylized or designed animation for volumetric clouds using physics-based generation and simulation methods in real-time.</div><div>The problem raised by the research is the current volumetric cloud animation controlling methods are not art-directable. Making a piece of volumetric cloud move in a specific way can be difficult when using only a physics-based simulation method. The purpose of the study is to implement an animating method for volumetric clouds and with art-directable controllers. Using this method, a designer can easily control the cloud's motion in a reliable way. The program will achieve interactive performance using parallel processing with CUDA. Users will be able to animate the cloud by input a few vectors inside the cloud volume. </div><div>After reviewing the literature related to the real-time simulation method of clouds, texture advection algorithms, fluid simulation, and other processes to achieve the results, the thesis offers a feasible design of the algorithm and experiments to test the hypotheses. The study uses noise textures and fractional Brownian motion (fBm) to generate volumetric clouds and render the clouds by the ray marching technique. The program will render user input vectors and a three-dimension interpolation vector field with OpenGL. By adding or changing input vectors, the user will gain a divergence minimization interpolation field. The cloud volume could be animated by the texture advection technique based on the interpolation vector field in real-time. By inputting several vectors, the user could plausibly animate the volume cloud in an art-directable way.</div>

The goal of this study is to characterize the mixing that occurs in a microfluidic chip. To characterize the mixing, the minimum length to complete mixing and evolution of mixing will be investigated. There are two types of mixing that occur within a microfluidic channel, diffusion and advection. In the beginning of the microfluidic chip, diffusion is the dominant form of mixing, and in the later portion of the microfluidic chip advection is the dominant form of mixing. The type of design used for this experiment was a zig zag geometry microfluidic chip with channel dimensions of 60 µm X 500 µm X 522 mm. The minimum length for complete mixing was 361 ± 3.475 µm at a flow rate of 25 mL/hr. The mixing was measured using optical light microscopy. For all flow rates less than 20 mL/hr the flow rate was too low to mix the two fluids. The pressure produced by the 30 mL/hr flow rate caused the microfluidic chip to fail.

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