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Molkenthin, Nora. "Advection-diffusion-networks." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät I, 2014. http://dx.doi.org/10.18452/17064.
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Das globale Klimasystem ist ein ausgesprochen komplexes und hochgradig nichtlineares System mit einer Vielzahl von Einflüssen und Interaktionen zwischen Variablen und Parametern. Komplementär zu der Beschreibung des Systems mit globalen Klimamodellen, kann Klima anhand der Interaktionsstruktur des Gesamtsystems durch Netzwerke beschrieben werden. Statt Details so genau wie möglich zu modellieren, werden hier Zeitreihendaten verwendet um zugrundeliegende Strukturen zu finden. Die Herausforderung liegt dann in der Interpretation dieser Strukturen. Um mich der Frage nach der Interpretation von Netzwerkmaßen zu nähern, suche ich nach einem allgemeinen Zusammenhang zwischen Eigenschaften der Netzwerktopologie und Eigenschaften des zugrundeliegenden physikalischen Systems. Dafür werden im Wesentlichen zwei Methoden entwickelt, die auf der Analyse von Temperaturentwicklungen gemäß der Advektions-Diffusions-Gleichung (ADE) basieren. Für die erste Methode wird die ADE mit offenen Randbedingungen und δ-peak Anfangsbedingungen gelöst. Die resultierenden lokalen Temperaturprofile werden verwendet um eine Korrelationsfunktion und damit ein Netzwerk zu definieren. Diese Netzwerke werden analysiert und mit Klimanetzen aus Daten verglichen. Die zweite Methode basiert auf der Diskretisierung der stochastischen ADE. Die resultierende lineare, stochastische Rekursionsgleichung wird verwendet um eine Korrelationsmatrix zu definieren, die nur von der Übergangsmatrix und der Varianz des stochastischen Störungsterms abhängt. Ich konstruiere gewichtete und ungewichtete Netzwerke für vier verschiedene Fälle und schlage Netzwerkmaße vor, die zwischen diesen Systemen zu unterscheiden helfen, wenn nur das Netzwerk und die Knotenpositionen gegeben sind. Die präsentierten Rekonstruktionsmethoden generieren Netzwerke, die konzeptionell und strukturell Klimanetzwerken ähneln und können somit als "proof of concept" der Methode der Klimanetzwerke, sowie als Interpretationshilfe betrachtet werden.The earth’s climate is an extraordinarily complex, highly non-linear system with a multitude of influences and interactions between a very large number of variables and parameters. Complementary to the description of the system using global climate models, in recent years, a description based on the system’s interaction structure has been developed. Rather than modelling the system in as much detail as possible, here time series data is used to identify underlying large scale structures. The challenge then lies in the interpretation of these structures. In this thesis I approach the question of the interpretation of network measures from a general perspective, in order to derive a correspondence between properties of the network topology and properties of the underlying physical system. To this end I develop two methods of network construction from a velocity field, using the advection-diffusion-equation (ADE) for temperature-dissipation in the system. For the first method, the ADE is solved for δ-peak-shaped initial and open boundary conditions. The resulting local temperature profiles are used to define a correlation function and thereby a network. Those networks are analysed and compared to climate networks from data. Despite the simplicity of the model, it captures some of the most salient features of climate networks. The second network construction method relies on a discretisation of the ADE with a stochastic term. I construct weighted and unweighted networks for four different cases and suggest network measures, that can be used to distinguish between the different systems, based on the topology of the network and the node locations. The reconstruction methods presented in this thesis successfully model many features, found in climate networks from well-understood physical mechanisms. This can be regarded as a justification of the use of climate networks, as well as a tool for their interpretation.
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Dubois, Olivier. "Optimized Schwarz methods for the advection-diffusion equation." Thesis, McGill University, 2003. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=19701.
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The optimized Schwarz methods were recently introduced to enhance the convergence of the classical Schwarz iteration, by replacing the Dirichlet transmission conditions with different conditions obtained through an optimization of the convergence rate. This is formulated as a min-max problem. These new methods are well-studied for elliptic second order symmetric equations. The purpose of this work is to compute optimized Robin transmission conditions for the advection-diffusion equation in two dimensions, by finding the solution of the min-max problem. The asymptotic expansion, for small mesh size h, of the resulting convergence rate is found: it shows a weak dependence on h, if the overlap is 0(h) or no overlap is used. Numerical experiments illustrate the improved convergence of these optimized methods compared to other Schwarz methods, and also justify the continuous Fourier analysis performed on a simple model problem only. The theoretical asymptotic performance is also verified numerically.
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Knaub, Karl R. "On the asymptotic behavior of internal layer solutions of advection-diffusion-reaction equations /." Thesis, Connect to this title online; UW restricted, 2001. http://hdl.handle.net/1773/6772.
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Davies, Kevin L. "Declarative modeling of coupled advection and diffusion as applied to fuel cells." Diss., Georgia Institute of Technology, 2014. http://hdl.handle.net/1853/51814.
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The goal of this research is to realize the advantages of declarative modeling for complex physical systems that involve both advection and diffusion to varying degrees in multiple domains. This occurs, for example, in chemical devices such as fuel cells. The declarative or equation-based modeling approach can provide computational advantages and is compatible with physics-based, object-oriented representations. However, there is no generally accepted method of representing coupled advection and diffusion in a declarative modeling framework.
This work develops, justifies, and implements a new upstream discretization scheme for mixed advective and diffusive flows that is well-suited for declarative models. The discretization scheme yields a gradual transition from pure diffusion to pure advection without switching events or nonlinear systems of equations. Transport equations are established in a manner that ensures the conservation of material, momentum, and energy at each interface and in each control volume. The approach is multi-dimensional and resolved down to the species level, with conservation equations for each species in each phase. The framework is applicable to solids, liquids, gases, and charged particles. Interactions among species are described as exchange processes which are diffusive if the interaction is inert or advective if it involves chemical reactions or phase change.
The equations are implemented in a highly modular and reconfigurable manner using the Modelica language. A wide range of examples are demonstrated—from basic models of electrical conduction and evaporation to a comprehensive model of a proton exchange membrane fuel cell (PEMFC). Several versions of the PEMFC model are simulated under various conditions including polarization tests and a cyclical electrical load. The model is shown to describe processes such as electro-osmotic drag and liquid pore saturation. It can be scaled in complexity from 4000 to 32,000 equations, resulting in a simulation times from 0.2 to 19 s depending on the level of detail. The most complex example is a seven-layer cell with six segments along the length of the channel. The model library is thoroughly documented and made available as a free, open-source software package.
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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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Chakravarty, Lopamudra. "Scalable Hybrid Schwarz Domain Decomposition Algorithms to Solve Advection-Diffusion Problems." Kent State University / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=kent1523325804305835.
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Fu, Xiaoming. "Reaction-diffusion Equations with Nonlinear and Nonlocal Advection Applied to Cell Co-culture." Thesis, Bordeaux, 2019. http://www.theses.fr/2019BORD0216/document.
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Cette thèse est consacrée à l’étude d’une classe d’équations de réaction-diffusion avec advection non-locale. La motivation vient du mouvement cellulaire avec le phénomène de ségrégation observé dans des expérimentations de co-culture cellulaire. La première partie de la thèse développe principalement le cadre théorique de notre modèle, à savoir le caractère bien posé du problème et le comportement asymptotique des solutions dans les cas d'une ou plusieurs espèces.Dans le Chapitre 1, nous montrons qu'une équation scalaire avec un noyau non-local ayant la forme d'une fonction étagée, peut induire des bifurcations de Turing et de Turing-Hopf avec le nombre d’ondes dominant aussi grand que souhaité. Nous montrons que les propriétés de bifurcation de l'état stable homogène sont intimement liées aux coefficients de Fourier du noyau non-local.Dans le Chapitre 2, nous étudions un modèle d'advection non-local à deux espèces avec inhibition de contact lorsque la viscosité est égale à zéro. En employant la notion de solution intégrée le long des caractéristiques, nous pouvons rigoureusement démontrer le caractère bien posé du problème ainsi que la propriété de ségrégation d'un tel système. Par ailleurs, dans le cadre de la théorie des mesures de Young, nous étudions le comportement asymptotique des solutions. D'un point de vue numérique, nous constatons que sous l'effet de la ségrégation, le modèle d'advection non-locale admet un principe d'exclusion.Dans le dernier Chapitre de la thèse, nous nous intéressons à l'application de nos modèles aux expérimentations de co-culture cellulaire. Pour cela, nous choisissons un modèle hyperbolique de Keller-Segel sur un domaine borné. En utilisant les données expérimentales, nous simulons un processus de croissance cellulaire durant 6 jours dans une boîte de pétri circulaire et nous discutons de l’impact de la propriété de ségrégation et des distributions initiales sur les proportions de la population finaleThis thesis is devoted to the study for a class of reaction-diffusion equations with nonlocal advection. The motivation comes from the cell movement with segregation phenomenon observed in cell co-culture experiments. The first part of the thesis mainly develops the theoretical framework of our model, namely the well-posedness and asymptotic behavior of solutions in both single-species and multi-species cases.In Chapter 1, we show a single scalar equation with a step function kernel may display Turing and Turing-Hopf bifurcations with the dominant wavenumber as large as we want. We find the bifurcation properties of the homogeneous steady state is closed related to the Fourier coefficients of the nonlocal kernel.In Chapter 2, we study a two-species nonlocal advection model with contact inhibition when the viscosity equals zero. By employing the notion of the solution integrated along the characteristics, we rigorously prove the well-posedness and segregation property of such a hyperbolic nonlocal advection system. Besides, under the framework of Young measure theory, we investigate the asymptotic behavior of solutions. From a numerical perspective, we find that under the effect of segregation, the nonlocal advection model admits a competitive exclusion principle.In the last Chapter, we are interested in applying our models to a cell co-culturing experiment. To that aim, we choose a hyperbolic Keller-Segel model on a bounded domain. By utilizing the experimental data, we simulate a 6-day process of cell growth in a circular petri dish and discuss the impact of both the segregation property and initial distributions on the finial population proportions
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Martin, Kristin Terese. "Limitations of the Advection-Diffusion Equation for Modeling Tephra Fallout: 1992 Eruption of Cerro Negro Volcano, Nicaragua." [Tampa, Fla.] : University of South Florida, 2004. http://purl.fcla.edu/fcla/etd/SFE0000581.
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Wang, Xiuquan. "Parameter Estimation in the Advection Diffusion Reaction Model With Mean Occupancy Time and Boundary Flux Approaches." OpenSIUC, 2014. https://opensiuc.lib.siu.edu/dissertations/976.
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In this dissertation, we examine an advection diffusion model for insects inhabiting a spatially heterogeneous environment and moving toward a more favorable environment. We first study the effects of adding a term describing drift or advection toward a favorable environment to diffusion models for population dynamics. The diffusion model is a basic linear two-dimensional diffusion equation describing local dispersal of species. The mathematical advection terms are taken to be Fickian and describe directed movement of the population toward the favorable environment. For this model, the landscape is composed of one homogeneous habitat patch embedded in a spatially heterogeneous environment and the boundary of the habitat inhabited by the population acts as a lethal edge. We also derived the mean occupancy time and the boundary flux of the habitat patch. The diffusion rate and advection parameters of the advection diffusion model are estimated based on mean occupancy time and boundary flux. We then introduce two methods for the identification of these coefficients in the model as well as the capture rate. These two new methods have some advantages over other methods of estimating those parameters, including reduced computational cost and ease of use in the field. We further examine the statistical properties of new methods through simulation, and discuss how mean occupancy time and boundary flux could be estimated in field experiments.
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Flegg, Jennifer Anne. "Mathematical modelling of chronic wound healing." Thesis, Queensland University of Technology, 2009. https://eprints.qut.edu.au/40164/1/Jennifer_Flegg_Thesis.pdf.
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Chronicwounds fail to proceed through an orderly process to produce anatomic and functional integrity and are a significant socioeconomic problem. There is much debate about the best way to treat these wounds. In this thesis we review earlier mathematical models of angiogenesis and wound healing. Many of these models assume a chemotactic response of endothelial cells, the primary cell type involved in angiogenesis. Modelling this chemotactic response leads to a system of advection-dominated partial differential equations and we review numerical methods to solve these equations and argue that the finite volume method with flux limiting is best-suited to these problems. One treatment of chronic wounds that is shrouded with controversy is hyperbaric oxygen therapy (HBOT). There is currently no conclusive data showing that HBOT can assist chronic wound healing, but there has been some clinical success. In this thesis we use several mathematical models of wound healing to investigate the use of hyperbaric oxygen therapy to assist the healing process - a novel threespecies model and a more complex six-species model. The second model accounts formore of the biological phenomena but does not lend itself tomathematical analysis. Bothmodels are then used tomake predictions about the efficacy of hyperbaric oxygen therapy and the optimal treatment protocol. Based on our modelling, we are able to make several predictions including that intermittent HBOT will assist chronic wound healing while normobaric oxygen is ineffective in treating such wounds, treatment should continue until healing is complete and finding the right protocol for an individual patient is crucial if HBOT is to be effective. Analysis of the models allows us to derive constraints for the range of HBOT protocols that will stimulate healing, which enables us to predict which patients are more likely to have a positive response to HBOT and thus has the potential to assist in improving both the success rate and thus the cost-effectiveness of this therapy.
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Loeck, Jaqueline Fischer. "Efeitos estocásticos em modelos determinísticos para dispersão de poluentes na camada limite atmosférica." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2014. http://hdl.handle.net/10183/131025.
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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.
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Loeck, Jaqueline Fischer. "Modelo operacional para dispersão de poluentes na camada limite atmosférica com contornos parcialmente reflexivos." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2018. http://hdl.handle.net/10183/183286.
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O presente trabalho propõe um novo modelo para dispersão de poluentes na atmosfera, tal modelo foi idealizado no trabalho de dissertação da autora e continuou-se seu desenvolvimento nesta pesquisa. O modelo é baseado na solução semi-analítica da equação de advecção-difusão para emissão contínua, com resolução através do método de separação de variáveis e da transformada de Fourier. As condições de contorno são tratadas como infinitas reflexões do poluente no solo e no topo da camada limite atmosférica. Adiante, estas reflexões são utilizadas de modo parcial, na tentativa de considerar fenômenos da dispersão que não podem ser explicitados no modelo determinístico, de forma que os contornos podem ser entendidos como estocásticos, ou seja, pode-se interpretar os contornos como uma amostragem de uma distribuição. Além disso, é realizada uma otimização nos contornos parcialmente reflexivos, com o objetivo de desenvolver uma metodologia de otimização e determinar os valores ótimos para a reflexão parcial. Os resultados obtidos foram, primeiramente, comparados com os experimentos de Copenhagen e Hanford. Posteriormente, comparou-se o modelo com dados de concentração coletados em uma fábrica de celulose, a CMPC Celulose Riograndense. Simulou-se, também, a dispersão de poluentes emitidos por uma usina termelétrica no Brasil, que faz parte do programa de pesquisa e desenvolvimento tecnológico do setor de energia elétrica da Agência Nacional de Energia Elétrica (ANEEL).The present work proposes a new model for pollutant dispersion in the atmosphere, this model was idealized in the dissertation work of the author and continued its development in this research. The model is based on the semi-analytic solution of the advectiondiffusion equation for continuous emission, with resolution through the method of separation of variables and the Fourier transform. The boundary conditions are treated as infinite reflections of the pollutant in the soil and at the top of the atmospheric boundary layer. These reflections are used in a partial way in the attempt to consider phenomena of dispersion that can not be explained in the deterministic model, so that the boundaries can be understood as stochastic, that is, one can interpret the boundaries as a sampling of a distribution. In addition, an optimization is performed in the partially reflective boundaries, with the purpose of developing an optimization methodology and determining the optimal values for the partial reflection. The results obtained were firstly compared with the experiments of Copenhagen and Hanford. Subsequently, the model was compared with concentration data collected at a cellulose production plant. The dispersion of pollutants emitted by a thermoelectric plant in Brazil was also simulated, which is part of the research and technological development program of the electric energy sector of the National Electric Energy Agency (ANEEL).
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Dufourd, Claire Chantal. "Spatio-temporal mathematical models of insect trapping : analysis, parameter estimation and applications to control." Thesis, University of Pretoria, 2016. http://hdl.handle.net/2263/58471.
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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
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Caunce, James Frederick Physical Environmental & Mathematical Sciences Australian Defence Force Academy UNSW. "Mathematical modelling of wool scouring." Awarded by:University of New South Wales - Australian Defence Force Academy. School of Physical, Environmental and Mathematical Sciences, 2007. http://handle.unsw.edu.au/1959.4/38650.
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Wool scouring is the first stage of wool processing, where unwanted contaminants are removed from freshly shorn wool. In most scouring machines wool is fed as a continuous mat through a series of water-filled scour and rinse bowls which are periodically drained. The purpose of this project is to mathematically model the scour bowl with the aim of improving efficiency. In this thesis four novel models of contaminant concentration within a scour bowl are developed. These are used to investigate the relationships between the operating parameters of the machine and the concentration of contamination within the scour bowl. The models use the advection-diffusion equation to simulate the settling and mixing of contamination. In the first model considered here, the scour bowl is simulated numerically using finite difference methods. Previous models of the scouring process only considered the average steady-state concentration of contamination within the entire scour bowl. This is the first wool scouring model to look at the bowl in two dimensions and to give time dependent results, hence allowing the effect of different drainage patterns to be studied. The second model looks at the important region at the top of the bowl - where the wool and water mix. The governing equations are solved analytically by averaging the concentration vertically assuming the wool layer is thin. Asymptotic analysis on this model reveals some of the fundamental behaviour of the system. The third model considers the same region by solving the governing equations through separation of variables. A fourth, fully two-dimensional, time dependent model was developed and solved using a finite element method. A model of the swelling of grease on the wool fibres is also considered since some grease can only be removed from the fibre once swollen. The swelling is modelled as a Stefan problem, a nonlinear diffusion equation with two moving boundaries, in cylindrical coordinates. Both approximate, analytical and a numerical solutions are found.
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Chen, Yiping. "Numerical modelling of solute transport processes using higher order accurate finite difference schemes : numerical treatment of flooding and drying in tidal flow simulations and higher order accurate finite difference modelling of the advection diffusion equation for solute transport predictions." Thesis, University of Bradford, 1992. http://hdl.handle.net/10454/4344.
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The modelling of the processes of advection and dispersion-diffusion is the most crucial factor in solute transport simulations. It is generally appreciated that the first order upwind difference scheme gives rise to excessive numerical diffusion, whereas the conventional second order central difference scheme exhibits severe oscillations for advection dominated transport, especially in regions of high solute gradients or discontinuities. Higher order schemes have therefore become increasingly used for improved accuracy and for reducing grid scale oscillations. Two such schemes are the QUICK (Quadratic Upwind Interpolation for Convective Kinematics) and TOASOD (Third Order Advection Second Order Diffusion) schemes, which are similar in formulation but different in accuracy, with the two schemes being second and third order accurate in space respectively for finite difference models. These two schemes can be written in various finite difference forms for transient solute transport models, with the different representations having different numerical properties and computational efficiencies. Although these two schemes are advectively (or convectively) stable, it has been shown that the originally proposed explicit QUICK and TOASOD schemes become numerically unstable for the case of pure advection. The stability constraints have been established for each scheme representation based upon the von Neumann stability analysis. All the derived schemes have been tested for various initial solute distributions and for a number of continuous discharge cases, with both constant and time varying velocity fields. The 1-D QUICKEST (QUICK with Estimated Streaming Term) scheme is third order accurate both in time and space. It has been shown analytically and numerically that a previously derived quasi 2-D explicit QUICKEST scheme, with a reduced accuracy in time, is unstable for the case of pure advection. The modified 2-D explicit QUICKEST, ADI-TOASOD and ADI-QUICK schemes have been developed herein and proved to be numerically stable, with the bility sta- region of each derived 2-D scheme having also been established. All these derived 2-D schemesh ave been tested in a 2-D domain for various initial solute distributions with both uniform and rotational flow fields. They were further tested for a number of 2-D continuous discharge cases, with the corresponding exact solutions having also been derived herein. All the numerical tests in both the 1-D and 2-D cases were compared with the corresponding exact solutions and the results obtained using various other difference schemes, with the higher order schemes generally producing more accurate predictions, except for the characteristic based schemes which failed to conserve mass for the 2-D rotational flow tests. The ADI-TOASOD scheme has also been applied to two water quality studies in the U. K., simulating nitrate and faecal coliform distributions respectively, with the results showing a marked improvement in comparison with the results obtained by the second order central difference scheme. Details are also given of a refined numerical representation of flooding and drying of tidal flood plains for hydrodynamic modelling, with the results showing considerable improvements in comparison with a number of existing models and in good agreement with the field measured data in a natural harbour study.
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Gisch, Debora Lidia. "Simulação da dispersão de poluentes na camada limite planetária : um modelo determinístico-estocástico." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2018. http://hdl.handle.net/10183/182254.
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Questões ambientais estão no centro das discussões nas últimas décadas. A poluição atmosférica, causada pela expansão pós-revolução industrial fez surgir a necessidade de aprender a descrever, usando modelos matemáticos, esse fenômeno. Com esse conhecimento pode-se propor soluções que mitiguem a poluição e os danos colaterais causados ao ambiente. A dispersão de poluentes modelada por soluções analíticas, a partir das equações de advecção-difusão oferecem um conhecimento sobre cada componente que constrói a equação, característica inexistente em outras abordagens, como a numérica. Entretanto ela era incapaz de descrever propriedades que se referem à turbulência, as estruturas coerentes, causadas por componentes não-lineares suprimidas por construção das equações governantes do modelo. Este trabalho estudou uma forma de recuperar características associadas à turbulência através de uma componente fundamental em estruturas coerentes, a fase. Essa é incluída no modelo que passa a descrever manifestações da turbulência em processos de dispersão através de flutuações de pequena escala na concentração da solução do modelo sesquilinear, que é determinístico-estocástico. No decorrer do trabalho há um estudo através de variações de parâmetros para compreender os efeitos da fase no modelo. Ele também foi aplicado ao experimento de Copenhagen e a dois cenários reais com a intenção de compreender o modelo frente à variáveis micrometeorológicas assim como aprimorá-lo para simular a dispersão de poluentes oriundos de fontes de forma realística.Environmental issues have been at the center of discussions in the last few decades. Atmospheric pollution, caused by post-industrial revolution, has increased the necessity to describe, using mathematical models, this phenomenon. With this knowledge is possible to propose solutions mitigating the pollution and collateral damages caused in the environment. The pollutant dispersion modeled by analytical solutions, from advection-diffusion equations, offers a knowledge about each component that constructs the equation, a characteristic that does not exist in other approaches, such as numerical. However it was unable to describe properties that refer to turbulence, coherent structures, caused by nonlinear components suppressed by constructing the model governing equations. This work studied a way to recover characteristics associated with turbulence through a fundamental component in coherent structures, the phase. This is included in the model which describes manifestations of turbulence in the dispersion process through the presence of small-scale concentration fluctuations in the sesquilinear model, which is deterministicstochastic. In the course of this work there is a study through variations of parameters to understand the phase effects in the model. It was also applied to Copenhagen experiment and to two real scenarios with the intention of understanding the model regarding micrometeorological variables as well as improving it to simulate the pollutant dispersion from sources in a realistic way.
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Pereira, Matheus Fernando 1987. "Estudo numérico da equação da difusão unidimensional." [s.n.], 2014. http://repositorio.unicamp.br/jspui/handle/REPOSIP/267720.
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Orientadores: Simone Andrea Pozza, Varese Salvador TimóteoDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Tecnologia
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Resumo: Diversas técnicas vêm sendo apresentadas para resolução da equação da difusão, a qual é empregada para estimativas da concentração de poluentes em função do espaço e do tempo, levando-se conta fatores como fonte emissora, condições meteorológicas, características do meio e velocidade em que o poluente é carreado. Neste estudo, foi empregado um algoritmo de passo variável para a resolução da equação da difusão unidimensional e avaliação da influência do parâmetro de heterogeneidade do meio, da velocidade do fluxo e do coeficiente de dispersão na variação da concentração de poluentes em função do espaço e do tempo. As simulações foram realizadas utilizando as mesmas condições iniciais e de contorno adotadas em dois estudos abordados recentemente na literatura, e de acordo com os resultados, verificou-se que características como meios de menor heterogeneidade, baixa velocidade inicial do fluxo e baixo coeficiente de dispersão implicam em menores valores de concentração, facilitando a dispersão de poluentes. O método utilizado é caracterizado pela rápida convergência, simplicidade do código e baixo tempo computacional, podendo ser utilizado como base para resolução da equação da difusão bi e tridimensional
Abstract: Several techniques have been employed for solving advection-diffusion equation, which is used to estimate pollutants concentration as function of time and space, taking account factors such as emission source, meteorological conditions, medium characteristics and the velocity in which pollutant is adduced. In this study, we used an adaptive-step algorithm for solving one-dimensional advection-diffusion equation, and evaluating the influence of medium inhomogeneity parameter, flow velocity and dispersion coefficient in the pollutants concentration variation as function of space and time. Simulations were performed using the same initial and boundary conditions adopted by Kumar et al. (2010) and by Savovic and Djordjevich (2012), and according to the results, it was found that characteristics such as medium of less inhomogeneity, low initial flow velocity and low dispersion coefficient imply in lower concentration and facilitate pollutants dispersion. The method is characterized by rapid convergence, simplicity of the code and low computational time, and it can be used as a basis for solving the two and the three dimensional advection-diffusion equation
Mestrado
Tecnologia e Inovação
Mestre em Tecnologia
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Weymar, Guilherme Jahnecke. "Uma solução da equação multidimensional de advecção-difusão para a simulação da dispersão de contaminantes reativos na camada limite atmosférica." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2016. http://hdl.handle.net/10183/143895.
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Tendo em vista o aumento considerável da poltúção do ar provocado em grande parte pela industrialização e o aumento da emissão de poluentes resultantes da queima de combustíveis fósseis por veículos automotores, o presente trabalho tem como objetivo melhorar a previsão e o entendimento da dispersão turbulenta atmosférica. Para tanto, apresenta-se, pela primeira vez, uma representação analít ica para a equação de advecção-difusão-reação tridimensional transiente, com perfil de vento e coeficientes de difusão tmbulenta dependentes da altura, que modelam a dispersão de poluentes na atmosfera. A solução da equação é obtida pela combinação do método GILTT ( Generalized Integral Laplace Transform Technique) com o método da Decomposição de Adomian modificado. Consideram-se dois casos para a aplicação do modelo: no primeiro modela-se a dispersão de um poluente secundário formado por uma reação fotoquímica e no segundo caso, utiliza-se o modelo para determinar o campo de concentração de um poluente que sofre perdas e ganhos devido a influência da radiação solar. Para poder realizar essas análises propôs-se uma parametrização para o termo de reação fotoquímica. São apresentados os resultados numéricos e estatísticos, comparandose com os dados da campanha experimental da Usina Termelétrica de Candiota e com os dados de medições realizadas pela Fundação Estadual de Proteção Ambiental Henrique Luiz Roessler (FEPAM).In view of the considerable increase of air pollution caused largely by industrialization and the increase of emission pollutants resulting from burning of fossil fuels by motor vehicles, the present work aims to improve the prediction and understanding of atmospheric turbu- lent dispersion. Therefore, is presented, for the rst time, an analytical representation to the transient three-dimensional advection-diffusion-reaction equation, with wind pro le and turbulent diffusion coefficients dependent of height, modeling the dispersion of pollutants in the atmosphere. The solution of the equation is obtained by combining of the GILTT method (Generalized Integral Laplace Transform Technique) with the modi ed Adomian Decomposition method. It is considered two cases for the application of the model: in the rst is modeled the dispersion of a secondary pollutant formed by a photochemical reaction, and in the second case the model is used to determine the concentration eld of a pollutant that suffers losses and gains due to the in uence of solar radiation. To realise these analisis a parameterization for the photochemical reaction term is proposed. Numerical and statistical results are presented, comparing with the experimental campaign data of the thermoelectric plant of Candiota and with data from measurements performed by the \Funda c~ao Estadual de Prote c~ao Ambiental Henrique Luiz Roessler" (FEPAM).
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Hantsch, Andreas. "A lattice Boltzmann equation model for thermal liquid film flow." Doctoral thesis, Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola", 2013. http://nbn-resolving.de/urn:nbn:de:bsz:105-qucosa-130098.
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Liquid film flow is an important flow type in many applications of process engineering. For supporting experiments, theoretical and numerical investigations are required. The present state of the art is to model the liquid film flow with Navier--Stokes-based methods, whereas the lattice Boltzmann method is employed here. The final model has been developed within this treatise by means of a two-phase flow and a heat transfer model, and boundary and initial conditions. All these sub-models have been applied to simple test cases. It could be found that the two-phase model is capable of solving flow phenomena with a large density ratio which has been shown impressively in conjunction with wall boundary conditions. The heat transfer model was tested against spectral method results with a transient non-uniform flow field. It was possible to find optimal parameters for computation. The final model has been applied to steady-state film flow, and showed very good agreement to OpenFOAM simulations. Tests with transient film flow demonstrated that the model is also able to predict these flow phenomenaFlüssigkeitsfilmströmungen kommen in vielen verfahrenstechnischen Prozessen zum Einsatz. Zur Unterstützung von Experimenten sind theoretische und numerische Untersuchungen nötig. Stand der Technik ist es, Navier--Stokes-basierte Modelle zu verwenden, wohingegen hier die Lattice-Boltzmann-Methode verwendet wird. Das finale Modell wurde unter Verwendung eines Zweiphasen- und eines Wärmeübertragungsmodell entwickelt und geeignete Rand- und Anfangsbedingungen formuliert. Alle Untermodelle wurden anhand einfacher Testfälle überprüft. Es konnte herausgefunden werden, dass das Zweiphasenmodell Strömungen großer Dichteunterschiede rechnen kann, was eindrucksvoll im Zusammenhang mit Wandrandbedingungen gezeigt wurde. Das Wärmeübertragungsmodell wurde gegen eine Spektrallösung anhand eines transienten und nichtuniformen Strömungsproblemes getestet. Stationäre Filmströmungen zeigten sehr gute Übereinstimmungen mit OpenFOAM-Lösungen und instationäre Berechungen bewiesen, dass das Model auch solche Strömungen abbilden kann
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Mahfoudhi, Imed. "Problèmes inverses de sources dans des équations de transport à coefficients variables." Phd thesis, INSA de Rouen, 2013. http://tel.archives-ouvertes.fr/tel-00975168.
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Cette thèse porte sur l'étude de quelques questions liées à l'identifiabilité et l'identification d'un problème inverse non-linéaire de source. Il s'agit de l'identification d'une source ponctuelle dépendante du temps constituant le second membre d'une équation de type advection-dispersion-réaction à coefficients variables. Dans le cas monodimensionnel, la souplesse du modèle stationnaire nous a permis de développer des réponses théoriques concernant le nombre des capteurs nécessaires et leurs emplacements permettant d'identifier la source recherchée d'une façon unique. Ces résultats nous ont beaucoup aidés à définir la ligne de conduite à suivre afin d'apporter des réponses similaires pour le modèle transitoire. Quant au modèle bidimensionnel transitoire, en utilisant quelques résultats de nulle contrôlabilité frontière et des mesures de l'état sur la frontière sortie et de son flux sur la frontière entrée du domaine étudié, nous avons établi un théorème d'identifiabilité et une méthode d'identification permettant de localiser les deux coordonnées de la position de la source recherchée comme étant l'unique solution d'un système non-linéaire de deux équations, et de transformer l'identification de sa fonction de débit en la résolution d'un problème de déconvolution. La dernière partie de cette thèse discute la difficulté principale rencontrée dans ce genre de problèmes inverses à savoir la non identifiabilité d'une source dans sa forme abstraite, propose une alternative permettant de surmonter cette difficulté dans le cas particulier où le but est d'identifier le temps limite à partir duquel la source impliquée a cessé d'émettre, et donc ouvre la porte sur de nouveaux horizons.
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Trojan, Alice von. "Finite difference methods for advection and diffusion." Title page, abstract and contents only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phv948.pdf.
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Includes bibliographical references (leaves 158-163). Concerns the development of high-order finite-difference methods on a uniform rectangular grid for advection and diffuse problems with smooth variable coefficients. This technique has been successfully applied to variable-coefficient advection and diffusion problems. Demonstrates that the new schemes may readily be incorporated into multi-dimensional problems by using locally one-dimensional techniques, or that they may be used in process splitting algorithms to solve complicatef time-dependent partial differential equations.
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De, Santis Dante. "Development of a high-order residual distribution method for Navier-Stokes and RANS equations." Phd thesis, Université Sciences et Technologies - Bordeaux I, 2013. http://tel.archives-ouvertes.fr/tel-00946171.
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The construction of compact high-order Residual Distribution schemes for the discretizationof steady multidimensional advection-diffusion problems on unstructuredgrids is presented. Linear and non-linear scheme are considered. A piecewise continuouspolynomial approximation of the solution is adopted and a gradient reconstructionprocedure is used in order to have a continuous representation of both thenumerical solution and its gradient. It is shown that the gradient must be reconstructedwith the same accuracy of the solution, otherwise the formal accuracy ofthe numerical scheme is lost in applications in which diffusive effects prevail overthe advective ones, and when advection and diffusion are equally important. Thenthe method is extended to systems of equations, with particular emphasis on theNavier-Stokes and RANS equations. The accuracy, efficiency, and robustness of theimplicit RD solver is demonstrated using a variety of challenging aerodynamic testproblems.
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Cajas, Guaca Denis 1983. "Impacto ambiental em meios aquáticos : modelagem, aproximação e simulação de um estudo na Baía de Buenaventura-Colômbia." [s.n.], 2015. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307268.
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Orientador: João Frederico da Costa Azevedo MeyerDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Resumo: Esta pesquisa visa descrever e ilustrar mediante a modelagem matemática e simulação computacional a poluição por esgoto que ocorre na Baía de Buenaventura no sudoeste do Pacífico Colombiano, e a influência do poluente no convívio de duas espécies de peixes. Para a dispersão de poluente usaremos o modelo que envolve a equação de Difusão-Advecção, a qual descreve as principais caraterísticas a considerar para o estudo do nosso problema, com suas respectivas condições de fronteira do entorno natural, considerando absorção de poluente nas margens da baía. Para a dinâmica populacional entre as espécies de peixes será usado um sistema não linear clássico do tipo Lotka-Volterra para modelar este problema, com condições de contorno de Neumann. A solução aproximada do modelo é obtida numericamente usando um método de segunda ordem no espaço e no tempo. Para a discretização da variável espacial usamos um método de diferenças finitas de segunda ordem e o método de Crank Nicolson para a discretização da variável temporal. Os resultados mostrados nas simulações computacionais para a concentração de poluente, e para a dinâmica populacional nos permitem julgar melhor o que está acontecendo ou o que pode acontecer, refletindo a necessidade de que os orgãos governamentais implementem mecanismos de mitigação ao problema ambiental para tentar diminuir os efeitos adversos do despejo direto no mar de águas residuais sem tratamento
Abstract: The propose of this research is to describe and illustrate the water pollution by sewage which occurs in Buenaventura Bay, in the southwest of the Colombian Pacific, and the influence of the pollutant in the interaction of two fish species, using mathematical modeling and computer simulation. Pollutant dispersion will be obtain using the model that involves the Diffusion - Advection equation, which describes the main features to be considered for the study of our problem with its respective boundary conditions of the natural environment, considering pollutant absorption in bayside. In order to describe the population dynamics between the fish species the classic Lotka -Volterra nonlinear system with Neumann boundary conditions will be used. The approximate solution of the model is obtained numerically using a second order method on the space and time. In order to discretize the spatial variable we use a second order finite difference method and the Crank Nicolson method for the time discretization. The results obtained in the computer simulations for the pollutant concentration, and the population dynamics allow us to judge what happening or what might happen. Reflecting in this way the necessity for the government agencies to implement mitigation mechanisms of the environmental problem in order to try reduce the adverse effects of dumping untreated sewage water directly into the sea
Mestrado
Matematica Aplicada
Mestra em Matemática Aplicada
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Buske, Daniela. "Solução GILTT bidimensional em geometria cartesiana : simulação da dispersão de poluentes na atmosfera." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2008. http://hdl.handle.net/10183/13448.
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Na presente tese é apresentada uma nova solução analítica para a equação de ad-vecção-difusão bidimensional transiente para simular a dispersão de poluentes na atmosfera. Para tanto, a equação de advecção-difusão é resolvida pela combinação da transformada de Laplace e da técnica GILTT (Generalized Integral Laplace Transform Technique). O fechamento da turbulência para os casos Fickiano e não-Fickiano é considerado. É investigado o problema de modelagem da dispersão de poluentes em condições de ventos fortes e fracos considerando, para o caso de ventos fracos, a difusão longitudinal na equação de advecção-difusão. Além disso, foi incluída no modelo a velocidade vertical e avaliada sua influência considerando-se o campo de velocidades constante e também geradas via LES (Large Eddy Simulation), para poder simular uma camada limite turbulenta mais realística. Os resultados obtidos por essa metodologia são validados com resultados experimentais disponíveis na literatura.In the present thesis it is presented a new analytical solution for the transient two- dimensional advection-diffusion equation to simulate the pollutant dispersion in atmosphere. For that, the advection-diffusion equation is solved combining the Laplace transform and the GILTT (Generalized Integral Laplace Transform Technique) techniques. The turbulence closure for Fickian and non-Fickian cases is considered. It is investigated the problem of modeling the pollutant dispersion in strong and weak winds considering, for the case of low wind conditions, the longitudinal diffusion in the advection-diffusion equation. Moreover, it was considered in the model the vertical velocity and its influence was evaluated considering velocities field constant and also generated by means of LES (Large Eddy Simulation), to simulate a more realistic turbulent boundary layer. The results attained by this methodology are validated with experimental results available in literature.
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Dvořák, Radim. "Fyzikální modelování a simulace." Doctoral thesis, Vysoké učení technické v Brně. Fakulta informačních technologií, 2014. http://www.nusl.cz/ntk/nusl-261245.
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Disertační práce se zabývá modelováním znečištění ovzduší, jeho transportních a disperzních procesů ve spodní části atmosféry a zejména numerickými metodami, které slouží k řešení těchto modelů. Modelování znečištění ovzduší je velmi důležité pro předpověď kontaminace a pomáhá porozumět samotnému procesu a eliminaci následků. Hlavním tématem práce jsou metody pro řešení modelů popsaných parciálními diferenciálními rovnicemi, přesněji advekčně-difúzní rovnicí. Polovina práce je zaměřena na známou metodu přímek a je zde ukázáno, že tato metoda je vhodná k řešení určitých konkrétních problémů. Dále bylo navrženo a otestováno řešení paralelizace metody přímek, jež ukazuje, že metoda má velký potenciál pro akceleraci na současných grafických kartách a tím pádem i zvětšení přesnosti výpočtu. Druhá polovina práce se zabývá poměrně mladou metodou ELLAM a její aplikací pro řešení atmosférických advekčně-difúzních rovnic. Byla otestována konkrétní forma metody ELLAM společně s navrženými adaptacemi. Z výsledků je zřejmé, že v mnoha případech ELLAM překonává současné používané metody.
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Wang, Chao. "Analyse de quelques problèmes elliptiques et paraboliques semi-linéaires." Phd thesis, Université de Cergy Pontoise, 2012. http://tel.archives-ouvertes.fr/tel-00809045.
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Cette thèse est divisée en deux parties. Dans la première partie, on considère le système de réaction-diffusion-advection (Pε), qui est un modèle d'haptotaxie, mécanisme lié à la dissémination de tumeurs cancéreuses. Le résultat principal concerne la convergence de la solution du systeme (Pε) vers la solution d'un problème à frontière libre (P0) qui est bien défini. Dans la seconde partie, on considère une classe générale d'équations elliptiques du type Hénon:−∆u = |x|^{α} f(u) dans Ω ⊂ R^N avec α > -2. On examine deux cas classiques : f(u) = e^u, |u|^{p−1} u et deux autres cas : f(u) = u^{p}_{+} puis f(u) nonlinéarité générale. En étudiant les solutions stables en dehors d'un ensemble compact (en particulier, solutions stables et solutions avec indice de Morse fini) avec différentes méthodes, on obtient des résultats de classification.
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Gidey, Hagos Hailu. "Numerical solution of advection-diffusion and convective Cahn-Hilliard equations." Thesis, University of Pretoria, 2016. http://hdl.handle.net/2263/60805.
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In this PhD thesis, we construct numerical methods to solve problems described by advectiondiffusion
and convective Cahn-Hilliard equations. The advection-diffusion equation models
a variety of physical phenomena in fluid dynamics, heat transfer and mass transfer or alternatively
describing a stochastically-changing system. The convective Cahn-Hilliard equation
is an equation of mathematical physics which describes several physical phenomena such
as spinodal decomposition of phase separating systems in the presence of an external field
and phase transition in binary liquid mixtures (Golovin et al., 2001; Podolny et al., 2005).
In chapter 1, we define some concepts that are required to study some properties of numerical
methods. In chapter 2, three numerical methods have been used to solve two problems
described by 1D advection-diffusion equation with specified initial and boundary conditions.
The methods used are the third order upwind scheme (Dehghan, 2005), fourth order
scheme (Dehghan, 2005) and Non-Standard Finite Difference scheme (NSFD) (Mickens,
1994). Two test problems are considered. The first test problem has steep boundary layers
near the region x = 1 and this is challenging problem as many schemes are plagued by nonphysical
oscillation near steep boundaries. Many methods suffer from computational noise
when modelling the second test problem especially when the coefficient of diffusivity is very
small for instance 0.01. We compute some errors, namely L2 and L1 errors, dissipation
and dispersion errors, total variation and the total mean square error for both problems and compare the computational time when the codes are run on a matlab platform. We then
use the optimization technique devised by Appadu (2013) to find the optimal value of the
time step at a given value of the spatial step which minimizes the dispersion error and this
is validated by some numerical experiments.
In chapter 3, a new finite difference scheme is presented to discretize a 3D advectiondiffusion
equation following the work of Dehghan (2005, 2007). We then use this scheme
and two existing schemes namely Crank-Nicolson and implicit Chapeau function to solve a
3D advection-diffusion equation with given initial and boundary conditions. We compare the
performance of the methods by computing L2- error, L1-error, dispersion error, dissipation
error, total mean square error and some performance indices such as mass distribution ratio,
mass conservation ratio, total mass and R2 which is a measure of total variation in particle
distribution. We also compute the rate of convergence to validate the order of accuracy of
the numerical methods. We then use optimization techniques to improve the results from
the numerical methods.
In chapter 4, we present and analyze four linearized one-level and multilevel (Bousquet et al.,
2014) finite volume methods for the 2D convective Cahn-Hilliard equation with specified
initial condition and periodic boundary conditions. These methods are constructed in such
a way that some properties of the continuous model are preserved. The nonlinear terms
are approximated by a linear expression based on Mickens' rule (Mickens, 1994) of nonlocal
approximations of nonlinear terms. We prove the existence and uniqueness, convergence
and stability of the solution for the numerical schemes formulated. Numerical experiments
for a test problem have been carried out to test the new numerical methods. We compute
L2-error, rate of convergence and computational (CPU) time for some temporal and spatial
step sizes at a given time. For the 1D convective Cahn-Hilliard equation, we present
numerical simulations and compute convergence rates as the analysis is the same with the
analysis of the 2D convective Cahn-Hilliard equation.Thesis (PhD)--University of Pretoria, 2016.
Mathematics and Applied Mathematics
PhD
Unrestricted
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Neal, David R. "Finite difference approximations of advection-diffusion equations for modeling shark populations /." Electronic version (PDF), 2007. http://dl.uncw.edu/etd/2007-3/neald/davidneal.pdf.
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Turk, Onder. "The Finite Element Method Solution Of Reaction-diffusion-advection Equations In Air Pollution." Master's thesis, METU, 2008. http://etd.lib.metu.edu.tr/upload/3/12609987/index.pdf.
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We consider the reaction-diffusion-advection (RDA) equations resulting in air pollution mod-
eling problems. We employ the finite element method (FEM) for solving the RDA equations
in two dimensions. Linear triangular finite elements are used in the discretization of problem
domains. The instabilities occuring in the solution when the standard Galerkin finite element
method is used, in advection or reaction dominated cases, are eliminated by using an adap-
tive stabilized finite element method. In transient problems the unconditionally stable Crank-
Nicolson scheme is used for the temporal discretization. The stabilization is also applied for
reaction or advection dominant case in the time dependent problems.
It is found that the stabilization in FEM makes it possible to solve RDA problems for very
small diffusivity constants. However, for transient RDA problems, although the stabilization
improves the solution for the case of reaction or advection dominance, it is not that pronounced
as in the steady problems. Numerical results are presented in terms of graphics for some test
steady and unsteady RDA problems. Solution of an air pollution model problem is also provided.
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Lin, Xuelei. "Preconditioning techniques for all-at-once linear systems arising from advection diffusion equations." HKBU Institutional Repository, 2020. https://repository.hkbu.edu.hk/etd_oa/803.
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In this thesis, we mainly study preconditioning techniques for all-at-once linear systems arising from discretization of three types of time-dependent advection-diffusion equation: linear diffusion equation, constant-coefficients advection-diffusion equation, time-fractional sub-diffusion equation. The proposed preconditioners are used with Krylov subspace solvers. The preconditioner developed for linear diffusion equation is based on -circulant ap- proximation of temporal discretization. Diagonalizability, clustering of spectrum and identity-plus-low-rank decomposition are derived for the preconditioned matrix. We also show that generalized minimal residual (GMRES) solver for the preconditioned system has a linear convergence rate independent of matrix-size. The preconditioner for constant-coefficients advection-diffusion equation is based on approximating the discretization of advection term with a matrix diagonalizable by sine transform. Eigenvalues of the preconditioned matrix are proven to be lower and upper bounded by positive constants independent of discretization parameters. Moreover, as the preconditioner is based on spatial approximation, it is also applicable to steady-state problem. We show that GMRES for the preconditioned steady-state problem has a linear convergence rate independent of matrix size. The preconditioner for time-fractional sub-diffusion equation is based on approximat- ing the discretization of diffusion term with a matrix diagonalizable by sine transform. We show that the condition number of the preconditioned matrix is bounded by a constant independent of discretization parameters so that the normalized conjugate gradient (NCG) solver for the preconditioned system has a linear convergence rate independent of discretization parameters and matrix size. Fast implementations based on fast Fourier transform (FFT), fast sine transform (FST) or multigrid approximation are proposed for the developed preconditioners. Numerical results are reported to show the performance of the developed preconditioners
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Yang, Qianqian. "Novel analytical and numerical methods for solving fractional dynamical systems." Thesis, Queensland University of Technology, 2010. https://eprints.qut.edu.au/35750/1/Qianqian_Yang_Thesis.pdf.
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During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.
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Liu, Hon Ho. "A finite element formulation and analysis for advection-diffusion and incompressible Navier-Stokes equations." Case Western Reserve University School of Graduate Studies / OhioLINK, 1993. http://rave.ohiolink.edu/etdc/view?acc_num=case1057156956.
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Santana, Alessandro Alves. "Identificação de parâmetros em problemas de advecção-difusão combinando a técnica do operador adjunto e métodos de volumes finitos de alta ordem." Universidade de São Paulo, 2007. http://www.teses.usp.br/teses/disponiveis/45/45132/tde-08012008-151101/.
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O objetivo desse trabalho consiste no estudo de métodos de identificação de parâmetros em problemas envolvendo a equação de advecção-difusão 2D. Essa equação é resolvida utilizando o método dos volumes finitos, sendo empregada métodos de reconstrução de alta ordem em malhas não-estruturadas de triângulos para calcular os fluxos nas faces dos volumes de controle. Como ferramenta de busca dos parâmetros é empregada a técnica baseadas em gradientes, sendo os mesmos calculados utilizando processos baseados em métodos adjuntos.The aim of this work concern to study parameter identification methods on problems involving the advection-diffusion equation in two dimensions. This equation is solved employing the finite volume methods, and high-order reconstruction methods, on triangle unstructured meshes to solve the fluxes across the faces of control volumes. As parameter searching tool is employed technicals based on gradients. The gradients are solved using processes based on adjoint methods.
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Hejazi, Hala Ahmad. "Finite volume methods for simulating anomalous transport." Thesis, Queensland University of Technology, 2015. https://eprints.qut.edu.au/81751/1/Hala%20Ahmad_Hejazi_Thesis.pdf.
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In this thesis a new approach for solving a certain class of anomalous diffusion equations was developed. The theory and algorithms arising from this work will pave the way for more efficient and more accurate solutions of these equations, with applications to science, health and industry. The method of finite volumes was applied to discretise the spatial derivatives, and this was shown to outperform existing methods in several key respects. The stability and convergence of the new method were rigorously established.
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Ayodele, Segun Gideon [Verfasser]. "Lattice Boltzmann Modeling of Advection-Diffusion-Reaction Equations in Non-equilibrium Transport Processes / Segun Gideon Ayodele." Aachen : Shaker, 2013. http://d-nb.info/1050344170/34.
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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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Cholet, Cybèle. "Fonctionnement hydrogéologique et processus de transport dans les aquifères karstiques du Massif du Jura." Thesis, Bourgogne Franche-Comté, 2017. http://www.theses.fr/2017UBFCD012/document.
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La compréhension du fonctionnement des aquifères karstiques est un enjeu considérable au vu des structures complexes de ces réservoirs. La forte hétérogénéité des écoulements induit une grande vulnérabilité de ces milieux et des comportements variés au cours des crues en lien avec différents processus de recharge. Dans le Massif du Jura, les aquifères karstiques constituent la principale ressource en eau potable et posent la question de leur rôle dans la dégradation de la qualité de l'eau observée depuis plusieurs décennies. Cette thèse propose différentes approches complémentaires pour mieux comprendre les dynamiques de crues dans ces aquifères sous diverses conditions hydrologiques. Plusieurs systèmes karstiques du Massif du Jura, présentant des dimensions variables et dominés par des mécanismes de recharges distincts, sont caractérisés à partir de suivis physico-chimiques et hydrochimiques détaillés.Tout d'abord, les différents systèmes sont comparés à l'échelle du cycle hydrologique et à l'échelle saisonnière afin d'identifier les processus de recharge dominants (infiltrations localisées et/ou diffuses) ainsi que les signatures hydrochimiques caractéristiques (arrivées allochtones, autochtones et/ou anthropiques). Une étude comparative de deux systèmes met en avant la forte variabilité saisonnière de la réponse hydrochimique sur un système marqué par une recharge localisée importante. Les différents systèmes sont ensuite analysés à une échelle de temps plus fine afin de mieux comprendre les dynamiques de crues. Une crue intense d'automne a été ainsi comparée à de plus petites crues précédées par des périodes d'étiages importantes et marquées par des signatures hydrochimiques anthropiques significatives. A partir de ces résultats, la méthode EMMA (End-Member Mixing Analysis) est appliquée afin d'établir les principaux pôles hydrochirniques responsables des contributions caractéristiques des différents systèmes. Ensuite, au vu du transport important de matières en suspension au cours des crues dans ces aquifères, une partie de ce travail vise à mieux comprendre le rôle et l'impact de ces matières sur le transport dissous et colloïdal. Les éléments traces métalliques (ETM) sont utilisés afin de caractériser l'origine et la dynamique des transferts. Ils apparaissent alors comme des outils pertinents pour identifier des phénomènes de dépôts et de remobilisation de particules dans le système. Ces dynamiques s'observent à la fois sur le système de Fourbanne marqué par une infiltration localisée importante et sur le petit système du Dahon, caractérisé par une infiltration diffuse.Finalement, afin de mieux comprendre la variabilité spatio-temporelle des interactions qui ont lieu au cours des crues le long du conduit karstique, une nouvelle approche de modélisation est définit. Elle propose l'utilisation des équations de l'onde diffusante et d'advection-diffusion avec la même résolution mathématique (solution analytique d'Hayarni (1951)) en supposant une distribution uniforme des échanges le long du conduit. A partir d'une modélisation inverse, elle permet alors d'identifier et d'estimer les échanges en termes de flux hydriques et de flux massiques entre deux stations de mesure. Cette méthodologie est appliquée sur le système de Fourbanne le long de deux tronçons caractérisant (1) la zone non-saturée et (2) zone non-saturée et saturée. L'analyse de plusieurs crues permet d'observer des dynamiques d'échanges variées sur les deux tronçons. Elle permet ainsi d'établir un schéma de fonctionnement du système soulignant des interactions importantes dans la zone saturée et également le rôle de la zone non-saturée pour le stockage dans le système karstique.Ce travail de thèse propose donc un ensemble d'outils riches et complémentaires pour mieux comprendre les dynamiques de crues et montre l'importance de coupler l'analyse des processus hydrodynamiques et hydrochimiques afin de mieux déchiffrer le fonctionnement de ces aquifèresThe understanding of karst aquifer functioning is a major issue, given the complex structures of these reservoirs. The high heterogeneity of the flows induces a high vulnerability of these media and implies distinct behaviours during floods because of various infiltration processes. In the Jura Mountains, karst aquifers constitute the main source of water drinking supply and raise the question of their role in the degradation of water quality observed for several decades. This work uses complementary approaches to better understand the dynamics of floods in aquifers under various hydrological conditions. Several karst systems of the Jura Mountains, varying in size and characterized by distinct recharge processes, are investigated by detailed physico-chemical and hydrochemical monitoring.First, the different systems are compared at the hydrological cycle scale and at the seasonal scale to identify the dominant recharge processes (localized and/or diffuse infiltrations) as well as the characteristic hydrochemical signatures (allochtonous, autochthonous and/or anthropogenic). A comparative study of two systems with distinct recharge processes highlights the high seasonal variability of the hydrochemical response. The different systems are then analysed on a finer time scale to shed light on flood dynamics. An intense autumn flood was thus compared to smaller floods preceded by periods of significant low flow and marked by significant anthropogenic hydrochemical signatures. The EMMA (End-Member Mixing Analysis) method is applied to these results in order to establish the main hydrochemical end-members responsible for the characteristic contributions of the different systems.Then, considering the important transport of suspended matter during floods in these aquifers, part of this work aims to better understand the role and impact of these materials on dissolved and colloidal transport. Metal trace elements (ETM) are used to characterize the origin and transfer dynamics. These are relevant tools to identify the processes of storage and remobilization of the particles in the system. These dynamics are observed both on the Fourbanne system with an important localized infiltration, and on the small Dahon system, characterized by diffuse infiltration.Finally, in order to shed light on the spatio-temporal variability of the interactions that occur along the karst network during floods, a new modelling approach is defined. It is based upon the use of the diffusive wave and advectiondiffusion equations with the same mathematical resolution (Hayami's analytical solution (1951)) assuming a uniform distribution of the exchanges along the reach. An inverse modelling approach allows to identify and estimate the exchanges in terms of water flows and solute between two measurement stations. This methodology is applied to the Fourbanne system on two sections characterizing (1) the unsaturated zone and (2) unsaturated and saturated zone. The analysis of several floods highlights the different exchange dynamics on the two sections. It thus makes it possible to establish a functioning scheme of the system, bringing to light the important interactions in the saturated zone and also the storage role of the unsaturated zone in the karst system.This work offers a set of rich and complementary tools to better characterize the dynamics of floods and shows the importance of coupling the analysis of the hydrodynamic and hydrochemical processes to better decipher the functioning of these aquifers
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Uys, Lafras. "Coupling kinetic models and advection-diffusion equations to model vascular transport in plants, applied to sucrose accumulation in sugarcane." Thesis, Stellenbosch : University of Stellenbosch, 2009. http://hdl.handle.net/10019.1/1441.
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Thesis (PhD (Biochemistry))--University of Stellenbosch, 2009.ENGLISH ABSTRACT: The sugarcane stalk, besides being the main structural component of the plant, is also the major storage organ for carbohydrates. Sucrose forms the bulk of stored carbohydrates. Previous studies have modelled the sucrose accumulation pathway in the internodal storage parenchyma of sugarcane using kinetic models cast as systems of ordinary differential equations. Typically, results were analysed with methods such as metabolic control analysis. The present study extends those original models within an advection-diffusion-reaction framework, requiring the use of partial differential equations to model sucrose metabolism coupled to phloem translocation. Let N be a stoichiometric matrix, v a vector of reaction rates, s a vector of species concentrations and r the gradient operator. Consider a coupled network of chemical reactions where the species may be advected with velocities, U, or diffuse with coefficients, D, or both. We propose the use of the dynamic system, s + r (Us) + r (Drs) = Nv; for a kinetic model where species can exist in different compartments and can be transported over long distances in a fluid medium, or involved in chemical reactions, or both. Darcy’s law is used to model fluid flow and allows a simplified, phenomenological approach to be applied to translocation in the phloem. Similarly, generic reversible Hill equations are used to model biochemical reaction rates. These are also phenomenological equations, where all the parameters have operationally defined interpretations. Numerical solutions to this formulation are demonstrated with time-courses of two toy models. The first model uses a simple “linear” pathway definition to study the impact of the system geometry on the solutions. Although this is an elementary model, it is able to demonstrate the up-regulation of photosynthesis in response to a change in sink demand. The second model elaborates on the reaction pathway while keeping the same geometry definition as the first. This pathway is designed to be an abstracted model of sucrose metabolism. Finally, a realistic model of sucrose translocation, metabolism and accumulation is presented, spanning eight internodes and four compartments. Most of the parameters and species concentrations used as initial values were obtained from experimental measurements. To analyse the models, a method of sensitivity analysis called the Fourier Amplitude Sensitivity Test (FAST) is employed. FAST calculates the contribution of the possible variation in a parameter to the total variation in the output from the model, i.e. the species concentrations and reaction rates. The model predicted that the most important factors affecting sucrose accumulation are the synthesis and breakdown of sucrose in futile cycles and the rate of cross-membrane transport of sucrose. The models also showed that sucrose moves down a concentration gradient from the leaves to the symplast, where it is transported against a concentration gradient into the vacuole. There was a net gain in carbohydrate accumulation in the realistic model, despite an increase in futile cycling with internode maturity. The model presented provides a very comprehensive description of sucrose accumulation and is a rigorous, quantitative framework for future modelling and experimental design.
AFRIKAANSE OPSOMMING: Benewens sy strukturele belang, is die suikerrietstingel ook die primêre bergingsorgaan vir koolhidrate. Die oorgrote meerderheid van hierdie koolhidrate word as sukrose opgeberg. Studies tot dusver het die metabolisme rondom sukroseberging in die parenchiem van die onderskeie stingellitte as stelsels gewone differensiaalvergelykings gemodelleer. Die resultate is ondermeer met metaboliese kontrole-analise geanaliseer. Hierdie studie brei uit op die oorspronklike modelle, deur gebruik te maak van ’n stromings-diffusie-reaksie-raamwerk. Parsiële differensiaalvergelykings is geformuleer om die metabolisme van sukrose te koppel aan die vloei in die floëem. Gestel N is ’n stoichiometriese matriks, v ’n vektor van reaksiesnelhede, s ’n vektor van spesie-konsentrasies en r die differensiaalvektoroperator. Beskou ’n netwerk van gekoppelde reaksies waar die onderskeie spesies stroom met snelhede U, of diffundeer met koëffisiënte D, of onderhewig is aan beide prosesse. Dit word voorgestel dat die dinamiese stelsel, _s + r (Us) + r (Drs) = Nv; gebruik kan word vir ’n kinetiese model waar spesies in verskeie kompartemente kan voorkom en vervoer kan word oor lang afstande saam met ’n vloeier, of kan deelneem aan chemiese reaksies, of albei. Darcy se wet word gebruik om die vloeier te modeller en maak dit moontlik om ’n eenvoudige, fenomenologiese benadering toe te pas op floëem-vervoer. Eweneens word generiese, omkeerbare Hill-vergelykings gebruik om biochemiese reaksiesnelhede te modelleer. Hierdie vergelykings is ook fenomenologies van aard en beskik oor parameters met ’n duidelike fisiese betekenis. Hierdie omvattende raamwerk is ondermeer gedemonstreer met behulp van numeriese oplossings van twee vereenvoudigde modelle as voorbeelde. Die eerste model het bestaan uit ’n lineêre reaksienetwerk en is gebruik om die geometrie van die stelsel te bestudeer. Alhoewel hierdie ’n eenvoudige model is, kon dit die toename in fotosintese as gevolg van ’n verandering in metaboliese aanvraag verklaar. Die tweede model het uitgebrei op die reaksieskema van die eerste, terwyl dieselfde stelselgeometrie behou is. Hierdie skema is ontwerp as ’n abstrakte weergawe van sukrosemetabolisme. Ten slotte is ’n realistiese model van sukrosevervoer, metabolisme en berging ontwikkel wat agt stingellitte en vier kompartemente omvat. Die meeste parameters en konsentrasies van biochemiese spesies wat as aanvanklike waardes in die model gebruik is, is direk vanaf eksperimentele metings verkry. Die Fourier Amplitude Sensitiwiteits-Toets (FAST) is gebruik om die modelle te analiseer. FAST maak dit moontlik om die bydrae van parameters tot variasie in modeluitsette soos reaksiesnelhede en die konsentrasies van chemiese spesies te bepaal. Die model het voorspel dat sintese en afbraak van sukrose in ’n futiele siklus, asook transmembraan sukrosevervoer, die belangrikste faktore is wat sukrose-berging beïnvloed. Die model het ook getoon dat sukrose saam met ’n konsentrasiegradiënt beweeg vanaf die blare tot by die stingelparenchiem-sitoplasma, van waar dit teen ’n konsentrasiegradiënt na die vogselholte (vakuool) vervoer word. Volgens die realistiese model was daar ’n netto toename in die totale hoeveelheid koolhidrate, ten spyte van ’n toename in die futile siklus van sukrose in die ouer stingellitte. Die model wat in hierdie proefskrif voorgestel word verskaf ’n uitgebreide, omvattende beskrywing van sukroseberging. Voorts stel dit ’n rigiede kwantitatiewe raamwerk daar vir toekomstige modellering en eksperimentele ontwerp.
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Pöschke, Patrick. "Influence of Molecular Diffusion on the Transport of Passive Tracers in 2D Laminar Flows." Doctoral thesis, Humboldt-Universität zu Berlin, 2018. http://dx.doi.org/10.18452/19526.
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In dieser Arbeit betrachten wir das Strömungs-Diffusions-(Reaktions)-Problem für
passive Markerteilchen, die in zweidimensionalen laminaren Strömungsmustern mit
geringem thermischem Rauschen gelöst sind. Der deterministische Fluss umfasst Zellen
in Form von Quadraten oder Katzenaugen. In ihnen tritt Rotationsbewegung auf.
Einige der Strömungen bestehen aus wellenförmigen Bereichen mit gerader Vorwärtsbewegung.
Alle Systeme sind entweder periodisch oder durch Wände begrenzt. Eine
untersuchte Familie von Strömungen interpoliert kontinuierlich zwischen Reihen von
Wirbeln und Scherflüssen. Wir analysieren zahlreiche numerische Simulationen, die
bisherige theoretische Vorhersagen bestätigen und neue Phänomene offenbaren. Ohne
Rauschen sind die Teilchen in einzelnen Bestandteilen des Flusses für immer gefangen.
Durch Hinzufügen von schwachem thermischen Rauschen wird die normale Diffusion
für lange Zeiten stark verstärkt und führt zu verschiedenen Diffusionsarten für
mittlere Zeiten. Mit Continuous-Time-Random-Walk-Modellen leiten wir analytische
Ausdrücke in Übereinstimmung mit den numerischen Ergebnissen her, die je nach
Parametern, Anfangsbedingungen und Alterungszeiten von subdiffusiver bis superballistischer
anomaler Diffusion für mittlere Zeiten reichen. Wir sehen deutlich, dass
einige der früheren Vorhersagen nur für Teilchen gelten, die an der Separatrix des
Flusses starten - der einzige Fall, der in der Vergangenheit ausführlich betrachtet wurde
- und dass das System zu vollkommen anderem Verhalten in anderen Situationen
führen kann, einschließlich einem Schwingenden beim Start im Zentrum einesWirbels
nach einer gewissen Alterungszeit. Darüber hinaus enthüllen die Simulationen, dass
Teilchenreaktionen dort häufiger auftreten, wo sich die Geschwindigkeit der Strömung
stark ändert, was dazu führt, dass langsame Teilchen von schnelleren getroffen werden,
die ihnen folgen.
Die umfangreichen numerischen Simulationen, die für diese Arbeit durchgeführt
wurden, mussten jetzt durchgeführt werden, da wir die Rechenleistung dafür besitzen.In this thesis, we consider the advection-diffusion-(reaction) problem for passive tracer particles suspended in two-dimensional laminar flow patterns with small thermal noise. The deterministic flow comprises cells in the shape of either squares or cat’s eyes. Rotational motion occurs inside them. Some of the flows consist of sinusoidal regions of straight forward motion. All systems are either periodic or are bounded by walls. One examined family of flows continuously interpolates between arrays of eddies and shear flows. We analyse extensive numerical simulations, which confirm previous theoretical predictions as well as reveal new phenomena. Without noise, particles are trapped forever in single building blocks of the flow. Adding small thermal noise, leads to largely enhanced normal diffusion for long times and several kinds of diffusion for intermediate times. Using continuous time random walk models, we derive analytical expressions in accordance with numerical results, ranging from subdiffusive to superballistic anomalous diffusion for intermediate times depending on parameters, initial conditions and aging time. We clearly see, that some of the previous predictions are only true for particles starting at the separatrix of the flow - the only case considered in depth in the past - and that the system might show a vastly different behavior in other situations, including an oscillatory one, when starting in the center of an eddy after a certain aging time. Furthermore, simulations reveal that particle reactions occur more frequently at positions where the velocity of the flow changes the most, resulting in slow particles being hit by faster ones following them. The extensive numerical simulations performed for this thesis had to be done now that we have the computational means to do so. Machines are powerful tools in order to gain a deeper and more detailed insight into the dynamics of many complicated dynamical and stochastic systems.
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Johansen, Jonathan Frederick. "Mathematical modelling of primary alkaline batteries." Thesis, Queensland University of Technology, 2007. https://eprints.qut.edu.au/16412/1/Jonathan_Johansen_Thesis.pdf.
Full textAbstract:
Three mathematical models, two of primary alkaline battery cathode discharge, and one of primary alkaline battery discharge, are developed, presented, solved and investigated in this thesis. The primary aim of this work is to improve our understanding of the complex, interrelated and nonlinear processes that occur within primary alkaline batteries during discharge.
We use perturbation techniques and Laplace transforms to analyse and simplify an existing model of primary alkaline battery cathode under galvanostatic discharge. The process highlights key phenomena, and removes those phenomena that have very little effect on discharge from the model. We find that electrolyte variation within Electrolytic Manganese Dioxide (EMD) particles is negligible, but proton diffusion within EMD crystals is important. The simplification process results in a significant reduction in the number of model equations, and greatly decreases the computational overhead of the numerical simulation software. In addition, the model results based on this simplified framework compare well with available experimental data.
The second model of the primary alkaline battery cathode discharge simulates step potential electrochemical spectroscopy discharges, and is used to improve our understanding of the multi-reaction nature of the reduction of EMD. We find that a single-reaction framework is able to simulate multi-reaction behaviour through the use of a nonlinear ion-ion interaction term.
The third model simulates the full primary alkaline battery system, and accounts for the precipitation of zinc oxide within the separator (and other regions), and subsequent internal short circuit through this phase. It was found that an internal short circuit is created at the beginning of discharge, and this self-discharge may be exacerbated by discharging the cell intermittently. We find that using a thicker separator paper is a very effective way of minimising self-discharge behaviour.
The equations describing the three models are solved numerically in MATLABR, using three pieces of numerical simulation software. They provide a flexible and powerful set of primary alkaline battery discharge prediction tools, that leverage the simplified model framework, allowing them to be easily run on a desktop PC.
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Johansen, Jonathan Frederick. "Mathematical modelling of primary alkaline batteries." Queensland University of Technology, 2007. http://eprints.qut.edu.au/16412/.
Full textAbstract:
Three mathematical models, two of primary alkaline battery cathode discharge, and one of primary alkaline battery discharge, are developed, presented, solved and investigated in this thesis. The primary aim of this work is to improve our understanding of the complex, interrelated and nonlinear processes that occur within primary alkaline batteries during discharge. We use perturbation techniques and Laplace transforms to analyse and simplify an existing model of primary alkaline battery cathode under galvanostatic discharge. The process highlights key phenomena, and removes those phenomena that have very little effect on discharge from the model. We find that electrolyte variation within Electrolytic Manganese Dioxide (EMD) particles is negligible, but proton diffusion within EMD crystals is important. The simplification process results in a significant reduction in the number of model equations, and greatly decreases the computational overhead of the numerical simulation software. In addition, the model results based on this simplified framework compare well with available experimental data. The second model of the primary alkaline battery cathode discharge simulates step potential electrochemical spectroscopy discharges, and is used to improve our understanding of the multi-reaction nature of the reduction of EMD. We find that a single-reaction framework is able to simulate multi-reaction behaviour through the use of a nonlinear ion-ion interaction term. The third model simulates the full primary alkaline battery system, and accounts for the precipitation of zinc oxide within the separator (and other regions), and subsequent internal short circuit through this phase. It was found that an internal short circuit is created at the beginning of discharge, and this self-discharge may be exacerbated by discharging the cell intermittently. We find that using a thicker separator paper is a very effective way of minimising self-discharge behaviour. The equations describing the three models are solved numerically in MATLABR, using three pieces of numerical simulation software. They provide a flexible and powerful set of primary alkaline battery discharge prediction tools, that leverage the simplified model framework, allowing them to be easily run on a desktop PC.
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Oumouni, Mestapha. "Analyse numérique de méthodes performantes pour les EDP stochastiques modélisant l'écoulement et le transport en milieux poreux." Phd thesis, Université Rennes 1, 2013. http://tel.archives-ouvertes.fr/tel-00904512.
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Ce travail présente un développement et une analyse des approches numériques déterministes et probabilistes efficaces pour les équations aux dérivées partielles avec des coefficients et données aléatoires. On s'intéresse au problème d'écoulement stationnaire avec des données aléatoires. Une méthode de projection dans le cas unidimensionnel est présentée, permettant de calculer efficacement la moyenne de la solution. Nous utilisons la méthode de collocation anisotrope des grilles clairsemées. D'abord, un indicateur de l'erreur satisfaisant une borne supérieure de l'erreur est introduit, il permet de calculer les poids d'anisotropie de la méthode. Ensuite, nous démontrons une amélioration de l'erreur a priori de la méthode. Elle confirme l'efficacité de la méthode en comparaison avec celle de Monte Carlo et elle sera utilisée pour accélérer la méthode par l'extrapolation de Richardson. Nous présentons aussi une analyse numérique d'une méthode probabiliste pour quantifier la migration d'un contaminant dans un milieu aléatoire. Nous considérons le problème d'écoulement couplé avec l'équation d'advection-diffusion, où on s'intéresse à la moyenne de l'extension et de la dispersion du soluté. Le modèle d'écoulement est discrétisé par une méthode des éléments finis mixtes, la concentration du soluté est une densité d'une solution d'une équation différentielle stochastique, qui sera discrétisée par un schéma d'Euler. Enfin, nous présentons une formule explicite de la dispersion et des estimations de l'erreur a priori optimales.
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Mildner, Marcus. "Stabilité de l'équation d'advection-diffusion et stabilité de l'équation d'advection pour la solution du problème approché, obtenue par la méthode upwind d'éléments-finis et de volumes-finis avec des éléments de Crouzeix-Raviart." Phd thesis, Université du Littoral Côte d'Opale, 2013. http://tel.archives-ouvertes.fr/tel-00839524.
Full textAbstract:
On considère le problème d'advection-diffusion stationnaire v(∇u, ∇v)+( β*∇u, v) = (f, v) et non stationnaire d/dt (u(t), v) + v(∇u, ∇v)+( β*∇u, v) = (g(t), v), ainsi que le problème d'advection (β*∇u, v) = (f, v) sur un domaine polygonal borné du plan. Le terme de diffusion est approché par des éléments de Crouzeix Raviart et le terme de convection par une méthode upwind sur des volumes barycentriques finis avec un maillage triangulaire. Pour le problème stationnaire d'advection-diffusion, la L²-stabilité (c'est-à-dire indépendante du coefficient de diffusion v) est démontrée pour la solution du problème approché obtenue par cette méthode d'éléments finis et de volumes finis. Pour cela une condition sur la géométrie doit être satisfaite. Des exemples de maillages sont donnés. Toujours avec cette condition géométrique sur le maillage, une inégalité de stabilité (où la discrétisation en temps n'est pas couplée à une condition sur la finesse du maillage) est obtenue pour le cas non-stationnaire. La discrétisation en temps y est faite par un schéma d'Euler implicite. Une majoration de l'erreur, proportionnelle au pas en temps et à la finesse du maillage, est ensuite proposée et exprimée explicitement en fonction des données du problème. Pour le problème d'advection, une approche utilisant la théorie des graphes est utilisée pour obtenir l'existence et l'unicité de la solution, ainsi que le résultat de stabilité. Comme pour la stabilité du problème d'advection-diffusion, une condition géométrique - qui est équivalente pour les points intérieurs du maillage à celle du problème d'advection-diffusion - est nécessaire.
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Miloš, Japundžić. "Uopštena rešenja nekih klasa frakcionih parcijalnih diferencijalnih jednačina." Phd thesis, Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, 2016. https://www.cris.uns.ac.rs/record.jsf?recordId=102114&source=NDLTD&language=en.
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Doktorska disertacija je posvećena rešavanju Košijevog problema odabranih klasa frakcionih diferencijalnih jednačina u okviru Kolomboovih prostora uopštenih funkcija. U prvom delu disertacije razmatrane su nehomogene evolucione jednačine sa prostorno frakcionim diferencijalnim operatorima reda 0 < α < 2 i koeficijentima koji zavise od x i t. Ova klasa jednačina je aproksimativno rešavana, tako što je umesto početne jednačine razmatrana aproksimativna jednačina data preko regularizovanih frakcionih izvoda, odnosno, njihovih regularizovanih množitelja. Za rešavanje smo koristili dobro poznate uopštene uniformno neprekidne polugrupe operatora. U drugom delu disertacije aproksimativno su rešavane nehomogene frakcione evolucione jednačine sa Kaputovimfrakcionim izvodom reda 0 < α < 2, linearnim, zatvorenim i gusto definisanimoperatorom na prostoru Soboljeva celobrojnog reda i koeficijentima koji zaviseod x. Odgovarajuća aproksimativna jednačina sadrži uopšteni operator asociran sa polaznim operatorom, dok su rešenja dobijena primenom, za tu svrhu u disertaciji konstruisanih, uopštenih uniformno neprekidnih operatora rešenja.U oba slučaja ispitivani su uslovi koji obezbeduju egzistenciju i jedinstvenostrešenja Košijevog problema na odgovarajućem Kolomboovom prostoru.Colombeau spaces of generalized functions. In the firs part, we studied inhomogeneous evolution equations with space fractional differential operators of order 0 < α < 2 and variable coefficients depending on x and t. This class of equations is solved approximately, in such a way that instead of the originate equation we considered the corresponding approximate equation given by regularized fractional derivatives, i.e. their regularized multipliers. In the solving procedure we used a well-known generalized uniformly continuous semigroups of operators. In the second part, we solved approximately inhomogeneous fractional evolution equations with Caputo fractional derivative of order 0 < α < 2, linear, closed and densely defined operator in Sobolev space of integer order and variable coefficients depending on x. The corresponding approximate equation is a given by the generalized operator associated to the originate operator, while the solutions are obtained by using generalized uniformly continuous solution operators, introduced and developed for that purpose. In both cases, we provided the conditions that ensure the existence and uniqueness solutions of the Cauchy problem in some Colombeau spaces.
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Moeleker, Piet. "The filtered advection-diffusion equation : Lagrangian methods and modeling." Thesis, 2000. https://thesis.library.caltech.edu/6112/1/Moeleker_p_2000.pdf.
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This research focuses on the incompressible scalar advection-diffusion equation. After applying a Gaussian filter, an infinite series expansion is found for the advection term to obtain a closed equation. Only the first two terms in this expansion are retained yielding the tensor-diffusivity subgrid model. This model can be interpreted as a tensor-diffusivity term which is proportional to the rate-of-strain tensor of the large-scale
filtered velocity field. Due to the negative diffusion in the stretching directions, care needs to be taken in the choice of a numerical method.
The scalar field is decomposed in a collection of anisotropic or axisymmetric Gaussian particles. Equations of motion for the location and the shape/size of the particles are derived using an expansion in Hermite polynomials. A novel, accurate remeshing scheme was found resulting in explicit expressions for the amplitudes of the new set of particles.
A stagnation flow was used for illustrative purposes and validation. Using a 2D time-dependent velocity field yielding chaotic advection, both axisymmetric and
anisotropic particles yield good agreement with filtered direct numerical simulations and compare favorably with the Smagorinsky subgrid model. Computational efficiency
makes axisymmetric particles the preferred choice.
A literature study using a 3D stationary one-parameter chaotic velocity field was used to validate model and particle-method in 3D. For highly chaotic fields good
agreement was obtained with this study. Computations have been performed for 3D forced isotropic periodic turbulence to study scalar mixing. Comparisons with literature are made.
It was shown that when the unfiltered velocity field is known, the most accurate results are obtained by moving particles using this field. It was concluded that a good
subgrid model modifies the equation of motion to get a good approximation to the unfiltered velocity field.
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Alotaibi, Hammad Mayoof M. "Developing multiscale methodologies for computational fluid mechanics." Thesis, 2017. http://hdl.handle.net/2440/114544.
Full textAbstract:
The development of multiscale computational methods is a key research area in mathematics, physics, engineering and computer science. Engineers and scientists often perform detailed microscale computational simulations of a large scale complicated spatio-temporal system. For most problems of practical interest, there are two major complications in simulating the dynamical behaviour on large macroscopic space-time scales. The first is the often prohibitive computational cost when only a microscopic model is available. The second complication is the memory constraints which often make the simulation over the whole domain of interest infeasible. To overcome these obstacles, the equation-free approach was proposed by Keverkidis and colleagues in 2000. This approach is a multiscale method for capturing the behaviour on large scales of some complicated systems using only relatively small bursts of the microscale models. The patch dynamics scheme was proposed as an essential component of the equation-free framework. The patch scheme promises a great saving in computation time by predicting the macroscopic dynamics using detailed microscopic computation only on relatively small widely distributed patches of the spatial domain. This thesis provides mathematical analysis and computational simulation of some basic atom dynamics on small patches. The most significant novel result of this research is that patches with microscale periodic boundary conditions can be used to efficiently predict macroscale properties of interest. This result is important because microscale computations are often easiest with microscale periodic boundary conditions. As a major test of the approach, we analyse, implement and evaluate such a scheme for a computationally intensive atomistic simulation. Chapter 1 of this dissertation introduces the challenge of multiscale problems and highlights some recent developments of multiscale methods for complex systems. Chapter 2 explores atomistic simulations in three-dimensional space. The microscale atomistic simulator is used to predict a macroscale temperature field. This is achieved by performing atomistic simulation on a small triply-periodic patch. The method uses locally averaged properties over small space-time scales to advance and predict relatively large space scale dynamics. Our ultimate aim for this chapter is to explore the macroscopic properties of a system through atomistic simulation in small periodic patches, but as a pilot study this thesis only considers one small patch coupled over the macroscale to boundaries. The computation is implemented only on the periodic patch, while over most of the domain we interpolate in order to predict the macroscale temperature. The thesis develops appropriate control terms to the microscale action regions of the patch. The control is applied to the left and right action regions surrounding a core region. A proportional controller dependent upon the relatively distant boundaries enables reasonably accurate macroscale predictions. The analysis and computational simulations indicate that this innovative patch scheme empowers computation of large scale simulations of microscale systems. Chapter 3 analyses the case of a one-dimensional microscale diffusion system in a single microscale patch to predict the macroscale dynamics over a comparatively large spatial region. The nature of the solutions of the patch scheme is explored when operating with time-varying boundary conditions that mimic coupling with neighbouring, dynamically varying patches. The patch eigenfunctions and their adjoints form a biorthogonal basis to determine the spectral coefficients in formal series solutions. We also explore this patch scheme with time delays in the communication of boundary values. This models a patch when information from the neighbouring patches is subject to communication delays. The delayed patch scheme prediction is compared with a scheme without delays to delineate when such delays are significant. Chapter 4 analyses diffusion dynamics on multiple coupled patches. Centre manifold theory supports the patch scheme. The patch coupling conditions are standard Lagrange interpolation from the macroscale values at the centre of surrounding patches to the boundaries of each patch. The results of this chapter demonstrate the feasibility of the microscale patch scheme to model diffusion over large spatial scales. Chapter 5 extends the analysis to one-dimensional microscale advection-diffusion dynamics in a single patch and for multiple patches. Eigenvalue analysis suggests that a slow manifold exists on the macroscale. Computer algebra constructs the slow manifold model for the advection-diffusion dynamics. The long-time dynamics behaviour of numerical solutions on one patch is compared with the prediction of the slow manifold. Comparisons among the patch dynamics scheme, the microscale model over the complete domain, and published experimental data determines regimes where the patch dynamics accurately predicts the large scale advection-diffusion dynamics.Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2017.
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(5930003), Yu Liu. "Modeling Granular Material Mixing and Segregation Using a Finite Element Method and Advection-Diffusion-Segregation Equation Multi-Scale Model." Thesis, 2019.
Find full textAbstract:
Granular material blending plays an important role in many industries ranging from those that manufacture pharmaceuticals to those producing agrochemicals. The ability to create homogeneous powder blends can be critical to the final product quality. For example, insufficient blending of a pharmaceutical formulation may have serious consequences on product efficacy and safety. Unfortunately, tools for quantitatively predicting particulate blending processes are lacking. Most often, parameters that produce an acceptable degree of blending are determined empirically.
The objective of this work was to develop a validated model for predicting the magnitude and rate of granular material mixing and segregation for binary mixtures of granular material in systems of industrial interest. The model utilizes finite element method simulations to determine the bulk-level granular velocity field, which is then combined with particle-level diffusion and segregation correlations using the advection-diffusion-segregation equation.
An important factor to the success of the finite element method simulation used in the current work is the material constitutive model used to represent the granular flow behavior. In this work, the Mohr-Coulomb elastoplastic (MCEP) model was used. The MCEP model parameters were calibrated both numerically and experimentally and the procedure is described in the current work. Additionally, the particle-level diffusion and segregation correlations are important to the accurate prediction of mixing and segregation rates. The current work derived the diffusion and segregation correlations from published literature and determined a methodology for obtaining the particle diffusion and segregation parameters from experiments.
The utility of this modelling approach is demonstrated by predicting mixing patterns in a rotating drum and Tote blender as well as segregation patterns in a rotating drum and during the discharge of conical hoppers. Indeed, a significant advantage of the current modeling approach compared to previously published models is that arbitrary system geometries can be modeled.
The model predictions were compared with both DEM simulation and experiment results. The model is able to quantitatively predict the magnitude and rate of powder mixing and segregation in two- and three-dimensional geometries and is computationally faster than DEM simulations. Since the numerical approach does not directly model individual particles, this new modeling approach is well suited for predicting mixing and segregation in large industrial-scale systems.
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Schirén, Whokko. "Finite Element Method for 1D Transient Convective Heat Transfer Problems." Thesis, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-76369.
Full textAbstract:
We study heat transfer in one dimension with and without convection, also called advection-diffusion. This is done using the Finite Element Method (FEM) to discretise the mathematical model, i.e. the heat equation. The results are compared to analytic Fourier series solutions. Our main result is that the FEM could be used to better model the heat transfer which allow for more accurate models than today's use of steady state models.
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De, l'Isle François. "Étude des discrétisations superconsistantes et application à la résolution numérique d’équations d’advection-diffusion." Thèse, 2017. http://hdl.handle.net/1866/20697.
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Chiou, Yu-Sheng, and 裘愉生. "Travelling Wave Solutions for Reaction-Diffusion-Advection Equations." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/58061221519913587970.
Full textAbstract:
碩士臺灣大學
數學研究所
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There are two parts in this paper. Part I is concerned with the travelling wave solutions for reaction-diffusion-advection equations . We consider periodic advection and combustion, monostable nonlinear reaction term . We mainly survey the results of existence, uniqueness, and monotonicity of pulsating waves from the paper by Berestycki and Hamel [1]. Part II deals with exact travelling wave solutions of competitive Lotka-Volterra systems of three species.