Dissertations / Theses on the topic 'Numerical Diffusion'
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1
Lunney, Michael E. "Numerical dynamics of reaction-diffusion equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ61659.pdf.
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Abercrombie, Stuart Christopher Benedict. "Numerical simulation of diffusion controlled reactions." Thesis, University of Southampton, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.401748.
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Gryaznov, Denis, Juergen Fleig, and Joachim Maier. "Numerical study of grain boundary diffusion." Universitätsbibliothek Leipzig, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-195828.
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Nagaiah, Chamakuri. "Adaptive numerical simulation of reaction-diffusion systems." [S.l.] : [s.n.], 2007. http://deposit.ddb.de/cgi-bin/dokserv?idn=985277882.
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Patel, Mayur K. "On the false-diffusion problem in the numerical modelling of convection-diffusion processes." Thesis, University of Greenwich, 1986. http://gala.gre.ac.uk/8697/.
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This thesis is concerned with the classification and evaluation of various numerical schemes that are available for computing solutions for fluid-flow problems, and secondly, with the development of an improved numerical discretisation scheme of the finite-volume type for solving steady-state differential equations for recirculating flows with and without sources. In an effort to evaluate the performance of the various numerical schemes available, some standard test cases were used. The relative merits of the schemes were assessed by means of one-dimensional laminar flows and two-dimensional laminar and turbulent flows, with and without sources. Furthermore, Taylor series expansion analysis was also utilised to examine the limitations that were present. The outcome of this first part of the work was a set of conclusions, concerning the accuracy of the numerous schemes tests, vis-a-vis their stability, ease of implementation, and computational costs. It is hoped that these conclusions can be used by `computational fluid-dynamics' practitioners in deciding on an optimum choice of scheme for their particular problem. From the understanding gained during the first part of the study, and in an effort to combine the attributes of a successful discretisation scheme, eg positive coefficients. conservation and the elimination of 'false-diffusion', a new flow-oriented finite-volume numerical scheme was devised and applied to several test cases in order to evaluate its performance. The novel approach in formulating the new CUPID* scheme (for Corner UPw^nDing) underlines the idea of focussing attention at the control-volume corners rather than at the control-volume cell-faces. In two-dimensions, this leads to an eight neighbour influence for the central grid point value, depending on the flow-directions at the corners of the control-volume. In the formulation of the new scheme, false-diffusion is considered from a pragmatic perspective, with emphasis on physics rather than on strict mathematical considerations such as the order of discretisation, etc. The accuracy of the UPSTREAM scheme (for JJPwind in STREAMIines) indicates that although it is formally only first-order accurate, it considerably reduces 'false-diffusion'. Scalar transport calculations (without sources) show that the UPSTREAM scheme predicts bounded solutions which are more accurate than the upwind-difference scheme and the unbounded skew-upstream-difference scheme. Furthermore, for laminar and turbulent flow calculations, improved results are obtained when compared with the performances of the other schemes. The advantage of the UPSTREAM-difference scheme is that all the influence coefficients are always positive and thus the coefficient matrices are suitable for iterative solution procedures. Finally, the stability and convergence characteristics are similar to those of the upwind-difference scheme, eg converged solutions are guaranteed. What cannot be guaranteed, however, is the conservatism of the scheme and it is recommended that future work should be directed towards improving that disadvantage.
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Gryaznov, Denis, Juergen Fleig, and Joachim Maier. "Numerical study of grain boundary diffusion: size effects." Diffusion fundamentals 2 (2005) 49, S. 1-2, 2005. https://ul.qucosa.de/id/qucosa%3A14382.
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Ferguson, R. C. "Numerical techniques for the drift-diffusion semiconductor equations." Thesis, University of Bath, 1996. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.362239.
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Moroney, Benjamin F., Timothy Stait-Gardner, Gang Zheng, and William S. Price. "Numerical analysis of NMR diffusion experiments in complex systems." Universitätsbibliothek Leipzig, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-185579.
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Ryu, Seungoh. "Numerical modeling of the carbonate and the sandstone formations." Universitätsbibliothek Leipzig, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-192171.
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It is of interest in various scientific and industrial contexts to make a reliable estimation of the transport properties of porous media via more accessible probes such as NMR that yield information on static pore geometry and porosity. When the pore geometry is
simple, there are empirical recipes that have long proven reliable in bridging the gap. For heterogeneous systems, such recipes fail to give a consistent prediction and invite case-by-case modifications. This is just one of many indications that the complex pore
geometry erodes the predictive power of empirical laws that work well in simpler situations. Heterogeneity combined with sizeable diffusive coupling in extended pore space further undermines the validity of the MR interpretation based on simple pore geometry. On top of this, possible spatial variation of surface relaxivity may further complicate the interpretation. Resolution of these issues for real life samples requires elaborate simulations in tandem with experimental verifications on the shared pore geometry. We report on a recent progress which allows combined parallel Lattice Boltzmann and random walk simulations to study transport and diffusion properties in various types of pore geometry, from simple 2D micro-fluidic mazes, 3D glass-bead packs and sandstones to more complex carbonates.
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Meral, Gulnihal. "Numerical Solution Of Nonlinear Reaction-diffusion And Wave Equations." Phd thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/3/12610568/index.pdf.
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In this thesis, the two-dimensional initial and boundary value problems (IBVPs)
and the one-dimensional Cauchy problems defined by the nonlinear reaction-
diffusion and wave equations are numerically solved. The dual reciprocity boundary
element method (DRBEM) is used to discretize the IBVPs defined by single
and system of nonlinear reaction-diffusion equations and nonlinear wave equation,
spatially. The advantage of DRBEM for the exterior regions is made use
of for the latter problem. The differential quadrature method (DQM) is used
for the spatial discretization of IBVPs and Cauchy problems defined by the
nonlinear reaction-diffusion and wave equations.
The DRBEM and DQM applications result in first and second order system
of ordinary differential equations in time. These systems are solved with three
different time integration methods, the finite difference method (FDM), the least
squares method (LSM) and the finite element method (FEM) and comparisons
among the methods are made. In the FDM a relaxation parameter is used to
smooth the solution between the consecutive time levels.
It is found that DRBEM+FEM procedure gives better accuracy for the IBVPs
defined by nonlinear reaction-diffusion equation. The DRBEM+LSM procedure
with exponential and rational radial basis functions is found suitable for exterior wave problem.
The same result is also valid when DQM is used for space
discretization instead of DRBEM for Cauchy and IBVPs defined by nonlinear
reaction-diffusion and wave equations.
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Joglekar, Saurabh Gajanan. "Numerical Study of Reaction-Diffusion Systems using Front Tracking." Thesis, State University of New York at Stony Brook, 2018. http://pqdtopen.proquest.com/#viewpdf?dispub=10622549.
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We study the three component Reaction-Diffusion systems with and without precipitation and crystal growth. Focus is on the generic chemical reaction represented by nA + mB → C, where n,m are the stoichiometric coefficients. In case of the reaction-diffusion system without precipitation, we investigate the movement of the center of reaction zone in for equal and unequal diffusivities. We compare the analytical and numerical solutions for equal diffusivities to establish the accuracy of the numerical method. Then we apply the numerical method to provide numerical evidence in support of a conjecture in the case of unequal diffusivities.
Next, we apply the Front Tracking method to study the reaction-diffusion systems with crystal growth in higher spatial dimensions. The effects of different parameters on the crystal growth are investigated.
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Ashrafizaadeh, Mahmud. "Numerical simulation of pulsating buoyancy driven turbulent diffusion flames." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape17/PQDD_0006/NQ30585.pdf.
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Zhu, Hong. "Numerical studies of diffusion in lipid-sterol bilayer membranes." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/MQ64487.pdf.
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Zhu, Hong 1975. "Numerical studies of diffusion in lipid-sterol bilayer membranes." Thesis, McGill University, 2000. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=30776.
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We examine tracer diffusion in lipid bilayer membranes containing either cholesterol or lanosterol, using a random lattice Ising model. Specifically the model is a two-state off lattice tethered network of hard disks which is dynamically triangulated and the interactions between the hard disks are only effective along the tethers linking the disks. The model was already applied to lipid-sterol systems and was successful in reproducing the phase diagrams and related physical properties. In this thesis we apply this model in conjunction with Monte Carlo simulation methods as follows. We calculate the diffusion constant for lipid-cholesterol and lipid-lanosterol bilayer membranes both as function of temperature and sterol concentration in all accessible regions of the relevant phase diagrams. Comparison with experiment and comments on sterol related evolution are included.
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Hill, Adrian T. "Attractors for convection-diffusion equations and their numerical approximation." Thesis, University of Oxford, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.314907.
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Gavaghan, David. "Parallel numerical algorithms for the solution of diffusion problems." Thesis, University of Oxford, 1991. http://ora.ox.ac.uk/objects/uuid:0ff86015-2fe3-4779-9ebd-f0a11a8eea0f.
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The purpose of this thesis is to determine the most effective parallel algorithm for the solution of the parabolic differential equations characteristic of diffusion problems. The primary aim is to apply the chosen algorithm to obtain solutions to the equations governing the operation of membrane-covered oxygen sensors, known as Clark electrodes, which are used for monitoring the oxygen concentration of blood. The boundary conditions of this problem require the development of a singularity correction technique. A brief history of electrochemical sensors leading to the development of the Clark electrode is given, together with the two-dimensional equations and boundary conditions governing its operation. A locally valid series expansion is derived to take care of the boundary singularity, together with a robust method of matching this to the finite difference approximation. Parallel implementations of three representative numerical algorithms applied to a simple model problem are compared by extending Leland's parallel effectiveness model. The chosen parallel algorithm is combined with the singularity correction to obtain a solution to the Clark electrode problem. Numerical experiments show this solution to achieve the required accuracy. Previous one-dimensional models of the Clark electrode are shown to be inadequate before the two-dimensional model is used to examine the variation of operation with design. The understanding gained allows us to demonstrate the advantages of pulse amperometry over steady-state techniques, and to suggest the most appropriate method and design for use in in vivo clinical monitoring.
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Moroney, Benjamin F., Timothy Stait-Gardner, Gang Zheng, and William S. Price. "Numerical analysis of NMR diffusion experiments in complex systems." Diffusion fundamentals 16 (2011) 69, S. 1-3, 2011. https://ul.qucosa.de/id/qucosa%3A13811.
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Strauss, Arne Karsten. "Numerical Analysis of Jump-Diffusion Models for Option Pricing." Thesis, Virginia Tech, 2006. http://hdl.handle.net/10919/33917.
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Jump-diffusion models can under certain assumptions be expressed as partial integro-differential equations (PIDE). Such a PIDE typically involves a convection term and a nonlocal integral like for the here considered models of Merton and Kou. We transform the PIDE to eliminate the convection term, discretize it implicitly using finite differences and the second order backward difference formula (BDF2) on a uniform grid. The arising dense linear system is solved by an iterative method, either a splitting technique or a circulant preconditioned conjugate gradient method. Exploiting the Fast Fourier Transform (FFT) yields the solution in only $O(n\log n)$ operations and just some vectors need to be stored. Second order accuracy is obtained on the whole computational domain for Merton's model whereas for Kou's model first order is obtained on the whole computational domain and second order locally around the strike price. The solution for the PIDE with convection term can oscillate in a neighborhood of the strike price depending on the choice of parameters, whereas the solution obtained from the transformed problem is stabilized.Master of Science
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MORAN, LUIS RENATO MINCHOLA. "NUMERICAL SIMULATION OF WAX DEPOSITION IN PETROLEUM LINES: ASSESSEMENT OF MOLECULAR DIFFUSION AND BROWNIAN DIFFUSION MECHANISMS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2007. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=12097@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROCOORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
Deposição de parafinas é um dos mais críticos problemas operacionais no transporte de óleo cru, nos dutos que operam em ambientes frios. Portanto, uma predição acurada da deposição de parafinas é crucial para o projeto eficiente de linhas submarinas. Infelizmente, a deposição de parafinas é um processo complexo e os mecanismos de deposição ainda não são bem compreendidos. Visando identificar a importância relativa dos diferentes mecanismos de deposição, dois deles foram investigados: Difusão Molecular e Browniana. Para determinar a quantidade de depósito, as equações de conservação de massa, quantidade de movimento linear, energia, concentração da mistura e concentração da parafina fora da solução foram resolvidas numericamente pelo método de volumes finitos. Um sistema de coordenadas móveis não ortogonais que se adapta a interface do depósito da parafina foi empregado. Apesar da obtenção de uma concordância razoável do perfil de depósito, obtido com os mecanismos selecionados no regime laminar, com resultados disponíveis na literatura, uma discrepância significativa foi observada durante o transiente. O emprego do mecanismo de difusão browniana levou a uma pequena melhora na predição da solução nas regiões sub- resfriadas. A influência do regime turbulento como o mecanismo de difusão molecular também foi investigado, empregando o modelo de turbulência para baixo Reynolds K- (Taxa de dissipação viscosa da energia cinética turbulenta).Os resultados obtidos apresentaram coerência física, com uma taxa menor de aumento do depósito com o tempo, pois a região próxima à interface com temperatura abaixo da temperatura de aparecimento de cristais é menor no regime turbulento.
Wax deposition is one of the major critical operational problems in crude oil pipelines operating in cold environments. Therefore, accurate prediction of the wax deposition is crucial for the efficient design of subsea lines. Unfortunately, wax deposition is a complex process for which the mechanisms are still not fully understood. Aiming at the identification of the relative importance of the different deposition mechanisms, two of them were investigated: Molecular and Brownian Diffusion. To determine the amount of deposit, the conservation equations of mass, momentum, energy, concentration of the mixture and wax concentration outside the solution were numerically solved with the finite volume method. A non-orthogonal moving coordinate system that adapts to the wax interface deposit geometry was employed. Although for the laminar regime, the deposition profile predicted with the selected deposition mechanisms presented a reasonable agreement with available literature results for the steady state regime, a significant discrepancy was observed during the transient. The employment of the Brownian diffusion mechanism led to only a small improvement in the transient solution prediction in sub-cooled regions. The influence of the turbulent regime with the Molecular diffusion mechanism was also investigated by employing the Low Reynolds ê−turbulence model. The results obtained were physically coherent, presenting a smaller deposit thickness, since the region with temperature below the wax appearance temperature is smaller in the turbulent regime.
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Hayman, Kenneth John. "Finite-difference methods for the diffusion equation." Title page, table of contents and summary only, 1988. http://web4.library.adelaide.edu.au/theses/09PH/09phh422.pdf.
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Hivert, Hélène. "Etude mathématique et numérique de quelques modèles cinétiques et de leurs asymptotiques : limites de diffusion et de diffusion anormale." Thesis, Rennes 1, 2016. http://www.theses.fr/2016REN1S025/document.
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L'objet de cette thèse est la construction de schémas numériques pour les équations cinétiques dans différents régimes de diffusion anormale. Comme le modèle devient raide en s'approchant du modèle asymptotique, les méthodes numériques standard deviennent coûteuses dans ce régime. Les schémas Asymptotic Preserving ont été introduits pour pallier à cette difficulté. Ils sont en effet stables le long de la transition du régime mésoscopique au régime microscopique. Dans le premier chapitre, nous considérons le cas d'une distribution d'équilibre qui est une fonction à queue lourde et dont le moment d'ordre 2 est infini. Le poids important des grandes vitesses de l'équilibre fait tomber la limite de diffusion usuelle en défaut, et on montre que le modèle asymptotique est une équation de diffusion fractionnaire. En nous basant sur une analyse asymptotique formelle de la convergence vers le modèle limite, nous construisons trois schémas AP pour le problème. La discrétisation en vitesse est discutée afin de prendre en compte correctement les grandes vitesses, et nous montrons que le troisième schéma est en outre uniformément précis au cours de la transition vers le régime microscopique. Dans le chapitre 2, nous étendons ces résultats au cas d'une fréquence de collision dégénérée en 0 qui mène aussi à une équation de diffusion fractionnaire. Nous adaptons ensuite ces méthodes numériques au cas d'une limite de diffusion normale avec scaling en temps anormal dans l'équation cinétique dans le chapitre 3. Dans ce cadre, la lenteur de la convergence vers le modèle asymptotique rend nécessaire une adaptation de l'approche AP des chapitres précédents. Enfin, le chapitre 4 présente un schéma AP pour l'équation cinétique dans le cas heavy-tail du chapitre 1 lorsque l'opérateur de collision est non-localIn this thesis, we construct numerical schemes for kinetic equations in some anomalous diffusion regimes. As the model becomes stiff when reaching the asymptotic model, the standard numerical methods become costly in this regime. Asymptotic Preserving (AP) schemes have been designed to overcome this difficulty. Indeed, they are uniformly stable along the transition from the mesoscopic regime to the microscopic one. In the first chapter, we study the case of a heavy-tailed equilibrium distribution, with infinite second order moment. The importance of the high velocities in the equilibrium makes the classical diffusion limit fail, and one can prove that the asymptotic model is a fractional diffusion equation. We construct three AP schemes for this problem, based on a formal asymptotic analysis of the convergence towards the limit model. The discretization of the velocities is then discussed to take into account the high velocities. Moreover, we prove that the third scheme enjoys the stronger property of being uniformly accurate along the convergence towards the microscopic regime. In chapter 2, we extend these results to the case of a degenerated collision frequency, also leading to a fractional diffusion limit. In chapter 3, these methods are then adapted to the case of a classical diffusion limit with anomalous time scale in the kinetic equation. In this case, an adaptation of the AP approach of the previous chapter is needed, because of the slow convergence rate of the kinetic equation towards the limit model. Eventually, a AP scheme for kinetic equation with heavy-tailed equilibria and non local collision operator is presented in chapter 4
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Valär, Adrian Luzi. "Direct numerical simulation of cellular structures in jet diffusion flames /." Zürich : ETH, 2008. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=17678.
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Gerisch, Alf. "Numerical methods for the simulation of taxis-diffusion-reaction-systems." [S.l. : s.n.], 2001. http://deposit.ddb.de/cgi-bin/dokserv?idn=963208098.
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Evensen, Tom Richard. "Nanoparticles in dilute solution: : A numerical study of rotational diffusion." Doctoral thesis, Norwegian University of Science and Technology, Department of Physics, 2008. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-2237.
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Höök, Lars Josef. "Variance reduction methods for numerical solution of plasma kinetic diffusion." Licentiate thesis, KTH, Fusionsplasmafysik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-91332.
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Performing detailed simulations of plasma kinetic diffusion is a challenging task and currently requires the largest computational facilities in the world. The reason for this is that, the physics in a confined heated plasma occur on a broad range of temporal and spatial scales. It is therefore of interest to improve the computational algorithms together with the development of more powerful computational resources. Kinetic diffusion processes in plasmas are commonly simulated with the Monte Carlo method, where a discrete set of particles are sampled from a distribution function and advanced in a Lagrangian frame according to a set of stochastic differential equations. The Monte Carlo method introduces computational error in the form of statistical random noise produced by a finite number of particles (or markers) N and the error scales as αN−β where β = 1/2 for the standard Monte Carlo method. This requires a large number of simulated particles in order to obtain a sufficiently low numerical noise level. Therefore it is essential to use techniques that reduce the numerical noise. Such methods are commonly called variance reduction methods. In this thesis, we have developed new variance reduction methods with application to plasma kinetic diffusion. The methods are suitable for simulation of RF-heating and transport, but are not limited to these types of problems. We have derived a novel variance reduction method that minimizes the number of required particles from an optimization model. This implicitly reduces the variance when calculating the expected value of the distribution, since for a fixed error the optimization model ensures that a minimal number of particles are needed. Techniques that reduce the noise by improving the order of convergence, have also been considered. Two different methods have been tested on a neutral beam injection scenario. The methods are the scrambled Brownian bridge method and a method here called the sorting and mixing method of L´ecot and Khettabi[1999]. Both methods converge faster than the standard Monte Carlo method for modest number of time steps, but fail to converge correctly for large number of time steps, a range required for detailed plasma kinetic simulations. Different techniques are discussed that have the potential of improving the convergence to this range of time steps.QC 20120314
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Patacchini, Francesco Saverio. "A variational and numerical study of aggregation-diffusion gradient flows." Thesis, Imperial College London, 2017. http://hdl.handle.net/10044/1/50180.
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This thesis is dedicated to the variational and numerical study of a particular class of continuity equations called aggregation-diffusion equations. They model the evolution of a continuum body whose total mass is conserved in time, undergoing up to three distinct phenomena: diffusion, confinement and aggregation. Diffusion describes the motion of the body’s particles from crowded regions of space to sparser ones; confinement results from an external potential field independent of the mass distribution of the body; and aggregation describes the nonlocal particle interaction within the body. Due to this wide range of effects, aggregation-diffusion equations are encountered in a large variety of applications coming from, among many others, porous medium flows, granular flows, crystallisation, biological swarming, bacterial chemotaxis, stellar collapse, and economics. An aggregation-diffusion equation has the very interesting and rich mathematical property of being the gradient flow for some energy functional on the space of probability measures, which formally means that any solution evolves so as to decrease this energy every time as much as possible. In this thesis we exploit this gradient-flow structure of aggregation-diffusion equations in order to derive properties of solutions and approximate them by discrete particles. We focus on two main aspects of aggregation-diffusion gradient flows: the variational analysis of the pure aggregation equation, i.e., the study of minimisers of the energy when only nonlocal aggregation effects are present; and the particle approximation of solutions, especially when only diffusive effects are taken into account. Regarding the former aspect, we prove that minimisers exist, enjoy some regularity, are supported on sets of specific dimensionality, and can be approximated by finitely supported discrete minimisers. Regarding the latter aspect, we illustrate theoretically and numerically that diffusion can be interpreted at the discrete level by a deterministic motion of particles preserving a gradient-flow structure.
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Krumscheid, Sebastian. "Statistical and numerical methods for diffusion processes with multiple scales." Thesis, Imperial College London, 2014. http://hdl.handle.net/10044/1/25114.
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In this thesis we address the problem of data-driven coarse-graining, i.e. the process of inferring simplified models, which describe the evolution of the essential characteristics of a complex system, from available data (e.g. experimental observation or simulation data). Specifically, we consider the case where the coarse-grained model can be formulated as a stochastic differential equation. The main part of this work is concerned with data-driven coarse-graining when the underlying complex system is characterised by processes occurring across two widely separated time scales. It is known that in this setting commonly used statistical techniques fail to obtain reasonable estimators for parameters in the coarse-grained model, due to the multiscale structure of the data. To enable reliable data-driven coarse-graining techniques for diffusion processes with multiple time scales, we develop a novel estimation procedure which decisively relies on combining techniques from mathematical statistics and numerical analysis. We demonstrate, both rigorously and by means of extensive simulations, that this methodology yields accurate approximations of coarse-grained SDE models. In the final part of this work, we then discuss a systematic framework to analyse and predict complex systems using observations. Specifically, we use data-driven techniques to identify simple, yet adequate, coarse-grained models, which in turn allow to study statistical properties that cannot be investigated directly from the time series. The value of this generic framework is exemplified through two seemingly unrelated data sets of real world phenomena.
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Tiwari, Ganesh. "Numerical Analysis of Non-Fickian Diffusion with a General Source." ScholarWorks@UNO, 2013. http://scholarworks.uno.edu/honors_theses/49.
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The inadequacy of Fick’s law to incorporate causality can be overcome by replacing it with the Green–Naghdi type II (GNII) flux relation. Combining the GNII assumption and conservation of mass leads to [see document for equation] where r (x, t) is the density function, S(p) is a source term and c¥ is a positive constant which carries (SI) units of m/sec. A general source term given by [see document for equation] is proposed. Here, the constants y and ps are the rate coefficient and saturation density respectively. The travelling wave solutions and numerical analysis of four special cases of equation (2), namely: Pearl-Verhulst Growth law, Zel’dovich Law, Newmann Law and Stefan- Boltzmann Law are investigated. For both analysis, results are compared with the available literature and extended for other cases. The numerical analysis is carried out by imposing well-studied Initial Boundary Value Problem and implementing a built-in method in the software package Mathematica 9. For Pearl-Verhulst source type, the results are compared to those found in literature [1]. Confirming the validity of built-in method for Pearl-Verhulst law, the generic built-in method is extended to study the transient signal response for similar initial boundary value problems when the source terms are Zel’dovich law, Newmann law and Stefan-Boltzmann law.
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Zimmer, Leonardo. "Numerical study of soot formation in laminar ethylene diffusion flames." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2016. http://hdl.handle.net/10183/150754.
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O objetivo desta tese é o estudo de formação de fuligem em chamas laminares de difusão. Para o modelo de formação de fuligem é escolhido um modelo semi-empírico de duas equações para prever a fração mássica de fuligem e o número de partículas de fuligem. O modelo descreve os processos de nucleação, de crescimento superficial e de oxidação das partículas. Para o modelo de radiação, a perda de calor por radiação térmica (gás e fuligem) é modelada considerando o modelo de gás cinza no limite de chama opticamente fina (OTA - Optically Thin Approximation). São avaliados diferentes modelos de cálculo das propriedades de transporte (detalhado e simplificado). Em relação à cinética química, tanto modelos detalhados quanto reduzidos são utilizados. No presente estudo, é explorada a técnica automática de redução conhecida como Flamelet Generated Manifold (FGM), sendo que esta técnica é capaz de resolver cinética química detalhada com tempos computacionais reduzidos. Para verificar o modelo de formação de fuligem foram realizados uma variedade de experimentos numéricos, desde chamas laminares unidimensionais adiabáticas de etileno em configuração tipo jatos opostos (counterflow) até chamas laminares bidimensionais com perda de calor de etileno em configuração tipo jato (coflow). Para testar a limitação do modelo os acoplamentos de massa e energia entre a fase sólida e a fase gasosa são investigados e quantificados para as chamas contra-corrente Os resultados mostraram que os termos de radiação da fase gasosa e sólida são os termos de maior importancia para as chamas estudas. Os termos de acoplamento adicionais (massa e propriedade termodinâmicas) são geralmente termos de efeitos de segunda ordem, mas a importância destes termos aumenta conforme a quantidade de fuligem aumenta. Como uma recomendação geral o acoplamento com todos os termos deve ser levado em conta somente quando a fração mássica de fuligem, YS, for igual ou superior a 0.008. Na sequência a formação de fuligem foi estudada em chamas bi-dimensionais de etileno em configuração jato laminar usando cinética química detalhada e explorando os efeitos de diferentes modelos de cálculo de propriedades de transporte. Foi encontrado novamente que os termos de radiação da fase gasosa e sólida são os termos de maior importância e uma primeira aproximação para resolver a chama bidimensional de jato laminar de etileno pode ser feita usando o modelo de transporte simplificado. Finalmente, o modelo de fuligem é implementado com a técnica de redução FGM e diferentes formas de armazenar as informações sobre o modelo de fuligem nas tabelas termoquímicas (manifold) são testadas A melhor opção testada neste trabalho é a de resolver todos os flamelets com as fases sólida e gasosa acopladas e armazenar as taxas de reação da fuligem por área de partícula no manifold. Nas simulações bidimensionais estas taxas são então recuperadas para resolver as equações adicionais de formação de fuligem. Os resultados mostraram uma boa concordância qualitativa entre as predições do FGM e da solução detalhada, mas a grande quantidade de fuligem no sistema ainda introduz alguns desafios para a obtenção de bons resultados quantitativos. Entretanto, este trabalho demonstrou o grande potencial do método FGM em predizer a formação de fuligem em chamas multidimensionais de difusão de etileno em tempos computacionais reduzidos.The objective of this thesis is to study soot formation in laminar diffusion flames. For soot modeling, a semi-empirical two equation model is chosen for predicting soot mass fraction and number density. The model describes particle nucleation, surface growth and oxidation. For flame radiation, the radiant heat losses (gas and soot) is modelled by using the grey-gas approximation with Optically Thin Approximation (OTA). Different transport models (detailed or simplified) are evaluated. For the chemical kinetics, detailed and reduced approaches are employed. In the present work, the automatic reduction technique known as Flamelet Generated Manifold (FGM) is being explored. This reduction technique is able to deal with detailed kinetic mechanisms with reduced computational times. To assess the soot formation a variety of numerical experiments were done, from one-dimensional ethylene counterflow adiabatic flames to two-dimensional coflow ethylene flames with heat loss. In order to assess modeling limitations the mass and energy coupling between soot solid particles and gas-phase species are investigated and quantified for counterflow flames. It is found that the gas and soot radiation terms are of primary importance for flame simulations. The additional coupling terms (mass and thermodynamic properties) are generally a second order effect, but their importance increase as the soot amount increases As a general recommendation the full coupling should be taken into account only when the soot mass fraction, YS, is equal to or larger than 0.008. Then the simulation of soot is applied to two-dimensional ethylene co-flow flames with detailed chemical kinetics and explores the effect of different transport models on soot predictions. It is found that the gas and soot radiation terms are also of primary importance for flame simulations and that a first attempt to solve the two-dimensional ethylene co-flow flame can be done using a simplified transport model. Finally an implementation of the soot model with the FGM reduction technique is done and different forms for storing soot information in the manifold is explored. The best option tested in this work is to solve all flamelets with soot and gas-phase species in a coupled manner, and to store the soot rates in terms of specific surface area in the manifold. In the two-dimensional simulations, these soot rates are then retrieved to solve the additional equations for soot modeling. The results showed a good qualitative agreement between FGM solution and the detailed solution, but the high amount of soot in the system still imposes some challenges to obtain good quantitative results. Nevertheless, it was demonstrated the great potential of the method for predicting soot formation in multidimensional ethylene diffusion flames with reduced computational time.
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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.
Full textAbstract:
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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Yu, Qiang. "Numerical simulation of anomalous diffusion with application to medical imaging." Thesis, Queensland University of Technology, 2013. https://eprints.qut.edu.au/62068/1/Qiang_Yu_Thesis.pdf.
Full textAbstract:
The first objective of this project is to develop new efficient numerical methods and supporting error and convergence analysis for solving fractional partial differential equations to study anomalous diffusion in biological tissue such as the human brain. The second objective is to develop a new efficient fractional differential-based approach for texture enhancement in image processing. The results of the thesis highlight that the fractional order analysis captured important features of nuclear magnetic resonance (NMR) relaxation and can be used to improve the quality of medical imaging.
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Montecinos, Gino Ignacio. "Numerical methods for advection-diffusion-reaction equations and medical applications." Doctoral thesis, Università degli studi di Trento, 2014. https://hdl.handle.net/11572/368529.
Full textAbstract:
The purpose of this thesis is twofold, firstly, the study of a relaxation procedure for numerically solving advection-diffusion-reaction equations, and secondly, a medical application. Concerning the first topic, we extend the applicability of the Cattaneo relaxation approach to reformulate time-dependent advection-diffusion-reaction equations, that may include stiff reactive terms, as hyperbolic balance laws with stiff source terms. The resulting systems of hyperbolic balance laws are solved by extending the applicability of existing high-order ADER schemes, including well-balanced and non-conservative schemes. Moreover, we also present a new locally implicit version of the ADER method to solve general hyperbolic balance laws with stiff source terms. The relaxation procedure depends on the choice of a relaxation parameter $\epsilon$. Here we propose a criterion for selecting $\epsilon$ in an optimal manner, relating the order of accuracy $r$ of the numerical scheme used, the mesh size $\Delta x$ and the chosen $\epsilon$. This results in considerably more efficient schemes than some methods with the parabolic restriction reported in the current literature. The resulting present methodology is validated by applying it to a blood flow model for a network of viscoelastic vessels, for which experimental and numerical results are available. Convergence-rates assessment for some selected second-order model equations, is carried out, which also validates the applicability of the criterion to choose the relaxation parameter. The second topic of this thesis concerns the numerical study of the haemodynamics impact of stenoses in the internal jugular veins. This is motivated by the recent discovery of a range of extra cranial venous anomalies, termed Chronic CerbroSpinal Venous Insufficiency (CCSVI) syndrome, and its potential link to neurodegenerative diseases, such as Multiple Sclerosis. The study considers patient specific anatomical configurations obtained from MRI data. Computational results are compared with measured data.
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Montecinos, Gino Ignacio. "Numerical methods for advection-diffusion-reaction equations and medical applications." Doctoral thesis, University of Trento, 2014. http://eprints-phd.biblio.unitn.it/1200/1/PhDthesisMontecinos.pdf.
Full textAbstract:
The purpose of this thesis is twofold, firstly, the study of a relaxation procedure for numerically solving advection-diffusion-reaction equations, and secondly, a medical application. Concerning the first topic, we extend the applicability of the Cattaneo relaxation approach to reformulate time-dependent advection-diffusion-reaction equations, that may include stiff reactive terms, as hyperbolic balance laws with stiff source terms. The resulting systems of hyperbolic balance laws are solved by extending the applicability of existing high-order ADER schemes, including well-balanced and non-conservative schemes. Moreover, we also present a new locally implicit version of the ADER method to solve general hyperbolic balance laws with stiff source terms. The relaxation procedure depends on the choice of a relaxation parameter $\epsilon$. Here we propose a criterion for selecting $\epsilon$ in an optimal manner, relating the order of accuracy $r$ of the numerical scheme used, the mesh size $\Delta x$ and the chosen $\epsilon$. This results in considerably more efficient schemes than some methods with the parabolic restriction reported in the current literature. The resulting present methodology is validated by applying it to a blood flow model for a network of viscoelastic vessels, for which experimental and numerical results are available. Convergence-rates assessment for some selected second-order model equations, is carried out, which also validates the applicability of the criterion to choose the relaxation parameter. The second topic of this thesis concerns the numerical study of the haemodynamics impact of stenoses in the internal jugular veins. This is motivated by the recent discovery of a range of extra cranial venous anomalies, termed Chronic CerbroSpinal Venous Insufficiency (CCSVI) syndrome, and its potential link to neurodegenerative diseases, such as Multiple Sclerosis. The study considers patient specific anatomical configurations obtained from MRI data. Computational results are compared with measured data.
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Ryu, Seungoh. "Numerical modeling of the carbonate and the sandstone formations." Diffusion fundamentals 10 (2009) 17, S. 1-3, 2009. https://ul.qucosa.de/id/qucosa%3A14108.
Full textAbstract:
It is of interest in various scientific and industrial contexts to make a reliable estimation of the transport properties of porous media via more accessible probes such as NMR that yield information on static pore geometry and porosity. When the pore geometry is
simple, there are empirical recipes that have long proven reliable in bridging the gap. For heterogeneous systems, such recipes fail to give a consistent prediction and invite case-by-case modifications. This is just one of many indications that the complex pore
geometry erodes the predictive power of empirical laws that work well in simpler situations. Heterogeneity combined with sizeable diffusive coupling in extended pore space further undermines the validity of the MR interpretation based on simple pore geometry. On top of this, possible spatial variation of surface relaxivity may further complicate the interpretation. Resolution of these issues for real life samples requires elaborate simulations in tandem with experimental verifications on the shared pore geometry. We report on a recent progress which allows combined parallel Lattice Boltzmann and random walk simulations to study transport and diffusion properties in various types of pore geometry, from simple 2D micro-fluidic mazes, 3D glass-bead packs and sandstones to more complex carbonates.
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Eloul, Shaltiel. "Diffusion to electrodes." Thesis, University of Oxford, 2016. https://ora.ox.ac.uk/objects/uuid:88c5f1d0-9f2f-49d5-b46d-6eeb5b7d4bfe.
Full textAbstract:
This thesis develops diffusion models for modern electrochemical experiments involving the transport of particles to electrodes and adsorbing surfaces. In particular, the models are related to the 'impact' method where particles stochastically arrive at an electrode and detected electrochemically. The studies are carried out using numerical simulations and also analytical methods. Chapter 1 is introductory and outlines some fundamental concepts in mass transport and kinetics, and their relation to electrochemical measurements which are of importance for the reader. Chapter 2 describes the numerical methods which are used for electrochemical simulations. Chapter 3 focuses on a specific two dimensional simulation system and the development of a high performance voltammetry simulation. Chapters 4 and 5 study the stochastic impacts of particles at an electrode surface. In Chapter 4, a 'diffusion only' model is developed using a probabilistic study and random walk simulations in order to provide expressions that can be used in so-called `impact' experiments. In Chapter 5, the practical cases of microdisc and microwire electrodes are investigated. Expressions for the number of impacts are developed and the concept of the lower limit of detection in ultra-dilute solutions is introduced. Then, a comparison study between the microwire electrode and the microdisc electrode explores a geometrical effect and its implications for experimental setups. In Chapter 6, a numerical and analytical study is developed to examine the effect of hindered diffusion as a particle moves close to an adsorbing surface. The study identifies the conditions under which this hindered diffusion is signiffcant even in a non-confined space. The study shows that the domination of hindered diffusion is strongly dependant on the sizes of both the particle and the target. The study focuses on a variety of target shapes and allows the number of hits/impacts to be estimated in practical 'impact' experiments. Moreover, a drastic effect on the calculation of the mean first passage time is observed for a sub-micron sized target, showing the importance of this effect not only for electrochemistry but also in biological systems. Chapters 7 and 8 investigate the properties of an adsorbing insulating surface adjacent to an electrode. In Chapter 7, a numerical study of the effect of 'shielding' by the insulating sheath is carried out. The study examines the in uence of this effect on the magnitude of the current in chronoamperometry experiments. Chapter 8 explores the case of reversible adsorption on the insulating surface for voltammetric enhancement by pre-concentration on the sheath surface. The results identify the conditions under which enhancement of the voltammetric signal can be observed. Finally, Chapter 9 looks at geometrical effects on the current response of insulating particles modified with an electroactive surface layer. Numerical models are developed to model the diffusion of charge transfer between electro-active sites on a modified surface of insulating particles. The current-time responses are simulated for particles with the shape of a sphere, a cube/cuboid, and a cylinder on an electrode. The characteristic currenttime responses are calculated for the various shapes. The observations show that the model can be utilised in experiments to determine the coverage or the diffusion coeficient of charge dissipation on modified insulating particles and, in some situations to identify the particle shape.
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Lin, Xue Lei. "Separable preconditioner for time-space fractional diffusion equations." Thesis, University of Macau, 2017. http://umaclib3.umac.mo/record=b3691377.
Full text
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Melean, Y., Kathryn E. Washburn, P. T. Callaghan, and Christoph H. Arns. "A numerical analysis of NMR pore-pore exchange measurements using micro X-ray computed tomography." Universitätsbibliothek Leipzig, 2015. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-192412.
Full textAbstract:
Pore-pore relaxation exchange experiments are a recent development and hold great promise to spectrally derive length scales and connectivity information relevant for transport in porous media. However, for large pores, NMR diffusion-relaxation techniques reach a limit because bulk relaxation becomes dominant. A combination of NMR and Xray-CT techniques could be beneficial and lead to better models for regions of unresolved porosity in CT images, increasing the accuracy of image based calculations of transport
properties. In this study we carry out numerical NMR pore-pore exchange experiments on selected Xray-CT images of sandstones and carbonate rock, while at the same time tracking information about the geometry and topology of the pore space. We use pore partitioning techniques and geometric distance fields to relate T2-T2 relaxation exchange spectra to underlying structural quantities. It is shown that T2-T2 pore-pore exchange measurements at room temperatures for the samples considered likely reflect exchange between pores and throats or pores and roughness.
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Chaudry, Qasim Ali. "Numerical Approximation of Reaction and Diffusion Systems in Complex Cell Geometry." Licentiate thesis, KTH, Numerical Analysis, NA, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-12099.
Full textAbstract:
The mathematical modelling of the reaction and diffusion mechanism of lipophilic toxic compounds in the mammalian cell is a challenging task because of its considerable complexity and variation in the architecture of the cell. The heterogeneity of the cell regarding the enzyme distribution participating in the bio-transformation, makes the modelling even more difficult. In order to reduce the complexity of the model, and to make it less computationally expensive and numerically treatable, Homogenization techniques have been used. The resulting complex system of Partial Differential Equations (PDEs), generated from the model in 2-dimensional axi-symmetric setting is implemented in Comsol Multiphysics. The numerical results obtained from the model show a nice agreement with the in vitro cell experimental results. The model can be extended to more complex reaction systems and also to 3-dimensional space. For the reduction of complexity and computational cost, we have implemented a model of mixed PDEs and Ordinary Differential Equations (ODEs). We call this model as Non-Standard Compartment Model. Then the model is further reduced to a system of ODEs only, which is a Standard Compartment Model. The numerical results of the PDE Model have been qualitatively verified by using the Compartment Modeling approach. The quantitative analysis of the results of the Compartment Model shows that it cannot fully capture the features of metabolic system considered in general. Hence we need a more sophisticated model using PDEs for our homogenized cell model.
Computational Modelling of the Mammalian Cell and Membrane Protein Enzymology
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YAMASHITA, Hiroshi. "Numerical Study on NOx Production of Transitional Fuel Jet Diffusion Flame." The Japan Society of Mechanical Engineers, 2000. http://hdl.handle.net/2237/8999.
Full text
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Chaudhry, Qasim Ali. "Numerical Approximation of Reaction and Diffusion Systems in Complex Cell Geometry." Licentiate thesis, KTH, Numerisk analys, NA, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-12099.
Full textAbstract:
The mathematical modelling of the reaction and diffusion mechanism of lipophilic toxic compounds in the mammalian cell is a challenging task because of its considerable complexity and variation in the architecture of the cell. The heterogeneity of the cell regarding the enzyme distribution participating in the bio-transformation, makes the modelling even more difficult. In order to reduce the complexity of the model, and to make it less computationally expensive and numerically treatable, Homogenization techniques have been used. The resulting complex system of Partial Differential Equations (PDEs), generated from the model in 2-dimensional axi-symmetric setting is implemented in Comsol Multiphysics. The numerical results obtained from the model show a nice agreement with the in vitro cell experimental results. The model can be extended to more complex reaction systems and also to 3-dimensional space. For the reduction of complexity and computational cost, we have implemented a model of mixed PDEs and Ordinary Differential Equations (ODEs). We call this model as Non-Standard Compartment Model. Then the model is further reduced to a system of ODEs only, which is a Standard Compartment Model. The numerical results of the PDE Model have been qualitatively verified by using the Compartment Modeling approach. The quantitative analysis of the results of the Compartment Model shows that it cannot fully capture the features of metabolic system considered in general. Hence we need a more sophisticated model using PDEs for our homogenized cell model.Computational Modelling of the Mammalian Cell and Membrane Protein Enzymology
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Barwari, Bala Farhad. "Asymptotic and numerical solutions of a two-component reaction diffusion system." Thesis, University of Nottingham, 2016. http://eprints.nottingham.ac.uk/37231/.
Full textAbstract:
In this thesis, we study a two-component reaction diffusion system in one and two spatial dimensions, both numerically and asymptotically. The system is related to a nonlocal reaction diffusion equation which has been proposed as a model for a single species that competes with itself for a common resource. In one spatial dimension, we find that this system admits traveling wave solutions that connect the two homogeneous steady states. We also analyse the long-time behaviour of the solutions. Although there exists a lower bound on the speed of travelling wave solutions, we observe that numerical solutions in some regions of parameter space exhibit travelling waves that propagate for an asymptotically long time with speeds below this lower bound. We use asymptotic methods to both verify these numerical results and explain the dynamics of the problem, which include steady, unsteady, spike-periodic travelling and gap-periodic travelling waves. In two spatial dimensions, the numerical solutions of the axisymmetric form of the system are presented. We also establish the existence of a steady axisymmetric solution which takes a form of a circular patch. We then carry out a linear stability analysis of the system. Finally, we perform bifurcation analysis of these patch solutions via a numerical continuation technique and show how these solutions change with respect to variation of one bifurcation parameter.
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Yang, Yuanjie. "Reaction-diffusion equations with time delay, theory, application, and numerical simulation." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/nq23095.pdf.
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McDermott, Sean. "Reaction-diffusion waves on toroidal manifolds : eikonal and concomitant numerical solutions." Thesis, Glasgow Caledonian University, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.395793.
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Nguyen, Hang Tuan. "Numerical investigations of some mathematical models of the diffusion MRI signal." Thesis, Paris 11, 2014. http://www.theses.fr/2014PA112016/document.
Full textAbstract:
Ma thèse porte sur la relation entre la microstructure des tissus et le signal macroscopique d'imagerie par résonance magnétique de diffusion (IRMd). Les estimations des paramètres de tissus provenant de signaux mesurées expérimentalement est très important dans l'IRMd. En dépit d'une histoire de la recherche intensive dans ce domaine depuis longtemps, de nombreux aspects de ce problème inverse restent mal compris. Nous avons proposé et testé une solution approchée à ce problème, dans lequel le signal d'IRMd est d'abord approché par un modèle macroscopique appropriée, puis le paramètres effectifs de ce modèle sont estimés.Nous avons étudié deux modèles macroscopiques du signal d'IRMd. Le premier est le modèle Karger qui suppose une certaine forme de (macroscopique) diffusion de compartiments multiples et les échanges inter-compartiment, mais est soumis à la restriction d'impulsion étroite sur les impulsions de gradient de champ magnétique diffusion codant. Le deuxième est un modèle ODE de plusieurs aimantations compartiment obtenus à partir de l'homogénéisation mathématique de l'équation de Bloch-Torrey, qui n'est pas soumis à la restriction d'impulsion étroite.Tout d'abord, nous avons étudié la validité de ces modèles macroscopiques en comparant le signal d'IRMd proposée par le modèle Karger et le modèle ODE avec le signal d'IRMd de diffusion simulé sur certaines geometries de tissu relativement complexes en résolvant l'équation de Bloch-Torrey en cas de membranes semi-perméables cellule biologique. Nous avons conclu que la validité de ces deux modèles macroscopiques est limitée au cas où la diffusion dans chaque compartiment est effectivement gaussien et où l'échange inter-compartimentale peut être représenté par des termes cinétiques de premier ordre standard.Deuxièmement, en supposant que les conditions ci-dessus sur la diffusion compartimentale et l'échange inter-compartiment sont satisfaits, nous avons résolu le problème des moindres carrés associée à monter les paramètres du modèle Karger et du modèle ODE au signal simulé d'IRMd obtenu en résolvant l'équation de Bloch-Torrey microscopique. Parmi divers paramètres efficaces, nous avons examiné les fractions volumiques des compartiments intra-cellulaires et extra-cellulaires, la perméabilité de la membrane, la taille moyenne des cellules, la distance inter-cellulaire, ainsi que des coefficients de diffusion apparents. Nous avons commencé par étudier la faisabilité de la méthod des moindres carrés pour les deux groupes de geometries de tissu relativement simples. Pour le premier groupe, dans lequel les domaines sont constitués de cellules identiques ou sphériques de taille variable noyées dans l'espace extra-cellulaire, nous avons conclu que problème d'estimation de paramètres peut être résolu robuste, même en présence de bruit. Dans le second groupe, on a considéré les cellules cylindriques parallèles, qui peuvent être couverts par une couche de membrane d'épaisseur, et noyés dans l'espace extra-cellulaire. Dans ce cas, la qualité de l'estimation des paramètres dépendant fortement de la quantité de la structure cellulaire est allongée dans la direction du gradient. Dans la pratique, l'orientation des cellules allongées n'est pas de priori connue, de plus, les tissus biologiques peuvent contenir des structures allongées orientées de manière aléatoire et également en mélange avec d'autres éléments compacts (par exemple, les axones et les cellules gliales). Cette situation a été étudiée numériquement sur notre domaine le plus complexe dans lequel les couches de cellules cylindriques dans différentes directions sont mélangés avec des couches de cellules sphériques. Nous avons vérifié que certains paramètres peuvent encore être estimés assez fidèlement tandis que l'autre reste inaccessible. Dans tous les cas considérés, le modèle ODE a fourni des estimations plus précises que le modèle KargerMy thesis focused on the relationship between the tissue microstructure and the macroscopic dMRI signal. Inferring tissue parameters from experimentally measured signals is very important in diffusion MRI. In spite of a long standing history of intensive research in this field, many aspects of this inverse problem remain poorly understood. We proposed and tested an approximate solution to this problem, in which the dMRI signal is first approximated by an appropriate macrosopic model and then the effective parameters of this model are estimated.We investigated two macroscopic models of the dMRI signal. The first is the Kärger model that assumes a certain form of (macroscopic) multiple compartmental diffusion and intercompartment exchange, but is subject to the narrow pulse restriction on the diffusion-encoding magnetic field gradient pulses. The second is an ODE model of the multiple compartment magnetizations obtained from mathematical homogenization of the Bloch-Torrey equation, that is not subject to the narrow pulse restriction.First, we investigated the validity of these macroscopic models by comparing the dMRI signal given by the Kärger and the ODE models with the dMRI signal simulated on some relatively complex tissue geometries by solving the Bloch-Torrey equation in case of semi-permeable biological cell membranes. We concluded that the validity of both macroscopic models is limited to the case where diffusion in each compartment is effectively Gaussian and where the inter-compartmental exchange can be accounted for by standard first-order kinetic terms.Second, assuming that the above conditions on the compartmental diffusion and intercompartment exchange are satisfied, we solved the least squares problem associated with fitting the Kärger and the ODE model parameters to the simulated dMRI signal obtained by solving the microscopic Bloch-Torrey equation. Among various effective parameters, we considered the volume fractions of the intra-cellular and extra-cellular compartments, membrane permeability, average size of cells, inter-cellular distance, as well as apparent diffusion coefficients. We started by studying the feasibility of the least squares solution for two groups of relatively simple tissue geometries. For the first group, in which domains consist of identical or variably-sized spherical cells embedded in the extra-cellular space, we concluded that parameters estimation problem can be robustly solved, even in the presence of noise. In the second group, we considered parallel cylindrical cells, which may be covered by a thick membrane layer, and embedded in the extra-cellular space. In this case, the quality of parameter estimation strongly depends on how much the cellular structure is elongated in the gradient direction. In practice, the orientation of elongated cells is not known a priori; moreover, biological tissues may contain elongated structures randomly oriented and also mixed with other compact elements (e.g., axons and glial cells). This situation has been numerically investigated on our most complicated domain in which layers of cylindrical cells in various directions are mixed with layers of spherical cells. We checked that certain parameters can still be estimated rather accurately while the other remains inaccessible. In all considered cases, the ODE model provided more accurate estimates than the Kärger model
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Wessel, Richard Allen Jr. "Spectral element method for numerical simulation of unsteady laminar diffusion flames." Case Western Reserve University School of Graduate Studies / OhioLINK, 1993. http://rave.ohiolink.edu/etdc/view?acc_num=case1057076472.
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Simon, Jean-Marc, Signe Kjelstrup, and Dick Bedeaux. "Numerical evidence for the validity of the local equilibrium hypothesis - the n-octane vapor-liquid interface." Universitätsbibliothek Leipzig, 2016. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-195647.
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Wei, Hui Qin. "Preconditioners for solving fractional diffusion equations with discontinuous coefficients." Thesis, University of Macau, 2017. http://umaclib3.umac.mo/record=b3691375.
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Jenkins, Michael John. "Pattern formation through self-organisation in diffusion-driven mechanisms." Thesis, University of Oxford, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.279895.
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Floyd, John-Patrick II. "A numerical investigation of extending diffusion theory codes to solve the generalized diffusion equation in the edge pedestal." Thesis, Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/39607.
Full textAbstract:
The presence of a large pinch velocity in the edge pedestal of high confinement
(H-mode) tokamak plasmas implies that particle transport in the plasma edge must be
treated by a pinch-diffusion theory, rather than a pure diffusion theory. Momentum
balance also requires the inclusion of a pinch term in descriptions of edge particle
transport. A numerical investigation of solving generalized pinch-diffusion theory using
methods extended from the numerical solution methodology of pure diffusion theory has
been carried out. The generalized diffusion equation has been numerically integrated
using the central finite-difference approximation for the diffusion term and three finite
difference approximations of the pinch term, and then solved using Gauss reduction. The
pinch-diffusion relation for the radial particle flux was solved directly and used as a
benchmark for the finite-difference algorithm solutions to the generalized diffusion
equation. Both equations are solved using several mesh spacings, and it is found that a
finer mesh spacing will be required in the edge pedestal, where the inward pinch velocity
is large in H-mode plasmas, than is necessary for similar accuracy further inward where
the pinch velocity diminishes. An expression for the numerical error of various finite-differencing
algorithms is presented.
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Chatterjee, Sakuntala, and Gunter M. Schütz. "Diffusion in a one-dimensional zeolite channel: an analytical and numerical study." Diffusion fundamentals 11 (2009) 17, S. 1-13, 2009. https://ul.qucosa.de/id/qucosa%3A13955.
Full textAbstract:
Czaplewski et al. have demonstrated in an experiment that in the presence of strongly adsorbed hydrocarbon molecules inside a narrow, effectively one-dimensional zeolite channel, the effective desorption temperature of the weakly adsorbed hydrocarbon component is substantially increased. To explain their experimental data qualitatively, we propose a simple lattice gas model involving the diffusion of hardcore particles on a one-dimensional lattice. We present exact calculation and dynamical Monte Carlo simulations to show that taking into account an Arrhenius dependence of the single molecule diffusion coefficient on temperature, one can explain many significant features of the temperature programmed desorption profile observed in experiments.