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In this thesis, we produce a rigorous and quantitative analysis of the errors introduced by finite difference schemes into strong constraint 4D-Variational (4D-Var) data assimilation. Strong constraint 4D-Var data assimilation is a method that solves a particular kind of inverse problem; given a set of observations and a numerical model for a physical system together with a priori information on the initial condition, estimate an improved initial condition for the numerical model, known as the analysis vector. This method has many forms of error affecting the accuracy of the analysis vector, and is derived under the assumption that the numerical model is perfect, when in reality this is not true. Therefore it is important to assess whether this assumption is realistic and if not, how the method should be modified to account for model error. Here we analyse how the errors introduced by finite difference schemes used as the numerical model, affect the accuracy of the analysis vector. Initially the 1D linear advection equation is considered as our physical system. All forms of error, other than those introduced by finite difference schemes, are initially removed. The error introduced by `representative schemes' is considered in terms of numerical dissipation and numerical dispersion. A spectral approach is successfully implemented to analyse the impact on the analysis vector, examining the effects on unresolvable wavenumber components and the l2-norm of the error. Subsequently, a similar also successful analysis is conducted when observation errors are re-introduced to the problem. We then explore how the results can be extended to weak constraint 4D-Var. The 2D linear advection equation is then considered as our physical system, demonstrating how the results from the 1D problem extend to 2D. The linearised shallow water equations extend the problem further, highlighting the difficulties associated with analysing a coupled system of PDEs.

No description available.

This thesis presents a multi-grid scheme applied to the solution of transport equations in turbulent flow associated with heat transfer. The multi-grid scheme is then applied to flow which occurs in the film cooling of turbine blades. The governing equations are discretized on a staggered grid with the hybrid differencing scheme. The momentum and continuity equations are solved by a nonlinear full multi-grid scheme with the SIMPLE algorithm as a relaxation smoother. The turbulence k — Є equations and the thermal energy equation are solved on each grid without multi-grid correction. Observation shows that the multi-grid scheme has a faster convergence rate in solving the Navier-Stokes equations and that the rate is not sensitive to the number of mesh points or the Reynolds number. A significant acceleration of convergence is also produced for the k — Є and the thermal energy equations, even though the multi-grid correction is not applied to these equations. The multi-grid method provides a stable and efficient means for local mesh refinement with only little additional computational and.memory costs. Driven cavity flows at high Reynolds numbers are computed on a number of fine meshes for both the multi-grid scheme and the local mesh-refinement scheme. Two-dimensional film cooling flow is studied using multi-grid processing and significant improvements in the results are obtained. The non-uniformity of the flow at the slot exit and its influence on the film cooling are investigated with the fine grid resolution. A near-wall turbulence model is used. Film cooling results are presented for slot injection with different mass flow ratios. / Science, Faculty of / Mathematics, Department of / Graduate

Induced seismicity (earthquakes caused by injection or extraction of fluids in Earth's subsurface) is a major, new hazard in the United States, the Netherlands, and other countries, with vast economic consequences if not properly managed. Addressing this problem requires development of predictive simulations of how fluid-saturated solids containing frictional faults respond to fluid injection/extraction. Here we present a numerical method for linear poroelasticity with rate-and-state friction faults. A numerical method for approximating the fully coupled linear poroelastic equations is derived using the summation-by-parts-simultaneous-approximation-term (SBP-SAT) framework. Well-posedness is shown for a set of physical boundary conditions in 1D and in 2D. The SBP-SAT technique is used to discretize the governing equations and show semi-discrete stability and the correctness of the implementation is verified by rigorous convergence tests using the method of manufactured solutions, which shows that the expected convergence rates are obtained for a problem with spatially variable material parameters. Mandel's problem and a line source problem are studied, where simulation results and convergence studies show satisfactory numerical properties. Furthermore, two problem setups involving fault dynamics and slip on faults triggered by fluid injection are studied, where the simulation results show that fluid injection can trigger earthquakes, having implications for induced seismicity. In addition, the results show that the scheme used for solving the fully coupled problem, captures dynamics that would not be seen in an uncoupled model. Future improvements involve imposing Dirichlet boundary conditions using a different technique, extending the scheme to handle curvilinear coordinates and three spatial dimensions, as well as improving the high-performance code and extending the study of the fault dynamics.

<p>In this thesis the two-fluid (Eulerian/Eulerian) formulation for dispersed two-phase flow is considered. Closure laws are needed for this type of models. We investigate both empirically based relations, which we refer to as a nongranular model, and relations obtained from kinetic theory of dense gases, which we refer to as a granular model. For the granular model, a granular temperature is introduced, similar to thermodynamic temperature. It is often assumed that the granular energy is in a steady state, such that an algebraic granular model is obtained. </p><p>The inviscid non-granular model in one space dimension is known to be conditionally well-posed. On the other hand, the viscous formulation is locally in time well-posed for smooth initial data, but with a medium to high wave number instability. Linearizing the algebraic granular model around constant data gives similar results. In this study we consider a couple of issues. </p><p>First, we study the long time behavior of the viscous model in one space dimension, where we rely on numerical experiments, both for the non-granular and the algebraic granular model. We try to regularize the problem by adding second order artificial dissipation to the problem. The simulations suggest that it is not possible to obtain point-wise convergence using this regularization. Introducing a new measure, a concept of 1-D bubbles, gives hope for other convergence than point-wise. </p><p>Secondly, we analyse the non-granular formulation in two space dimensions. Similar results concerning well-posedness and instability is obtained as for the non-granular formulation in one space dimension. Investigation of the time scales of the formulation in two space dimension suggests a sever restriction on the time step, such that explicit schemes are impractical. </p><p>Finally, our simulation in one space dimension show that peaks or spikes form in finite time and that the solution is highly oscillatory. We introduce a model problem to study the formation and smoothness of these peaks.</p>

La compréhension du développement des habiletés numériques est un enjeu majeur pour guider les pratiques éducatives et rééducatives des jeunes enfants. Les résultats des enquêtes nationales et internationales sont unanimes à cet égard : le niveau moyen des élèves en mathématiques a chuté depuis 2003 (PISA, 2012). L’hypothèse la plus avancée à l’heure actuelle est celle d’un déficit des correspondances entre les codes numériques et le « sens des nombres » (Dehaene, 1997). Le « sens du nombre » est contenu dans la représentation analogique et non verbale des nombres. Dans le présent travail, nous cherchons à démontrer que l’estimation numérique permet ainsi d’exercer les correspondances entre les codes afin de donner du sens aux représentations symboliques écrites et orales. Malgré l’importance accordée aujourd’hui au processus d’estimation, son rôle dans les apprentissages doit encore être précisé afin d’orienter et de mieux guider les pratiques (ré)éducatives (Dehaene et Cohen, 2001 ; Von Aster et Shalev 2007). A travers trois études expérimentales, nous explorons ainsi le rôle de l’estimation numérique dans les apprentissages d’un point de vue éducatif (auprès d’enfants scolarisés en CP) et d’un point de vue rééducatif (dans le syndrome génétique de la trisomie 21). L’acquisition des différentes représentations et des relations complexes qui s’établissent entre-elles est également analysée et discutée pour mieux préciser les modèles de traitements du nombres et du calcul actuels. Les résultats obtenus corroborent ainsi une hypothèse de spiralité des apprentissages symboliques. / Understanding the development of numerical abilities is a major issue to guide educative and reeducative practices in mathematics in young children. National and international investigations has demonstrated that the mathematical skills of students has decrease since 2003 (PISA, 2012). Nowadays, weak capacities to map numerical representations and therefore a defective « number sense » (Dehaene, 1997 ; 2001) is the most supported hypothesis. The « number sense » is contained into the analogical and non-verbal representation of number. In this work, we consider that numerical estimation is a good way to exercise this mapping and give sense to verbal and written numerical symbols. Indeed, the role of numerical estimation needs to be specified in order to lead the (re)educative practices (Dehaene, 1997 ; 2001 ; Von Aster et Shalev). During three experimental studies, we explore the role of numerical estimation, in an educative and reeducative way in children in first year and children with down’s syndrom. The acquisition of numerical representations and the complex relations they have are also explored and discussed through a spirality hypothesis in order to better specify actual models of number treatment.

by Tam Yue Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 49-52). / Abstracts in English and Chinese. / List of Tables --- p.vi / List of Figures --- p.vii / Chapter 1 --- Solving Systems of ODEs with BVMs --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- Background --- p.4 / Chapter 1.2.1 --- Linear Multistep Formulae --- p.4 / Chapter 1.2.2 --- Preconditioned GMRES Method --- p.6 / Chapter 1.3 --- Strang-Type Preconditioners with BVMs --- p.7 / Chapter 1.3.1 --- Block-BVMs and Their Matrix Forms --- p.8 / Chapter 1.3.2 --- Construction of the Strang-type Preconditioner --- p.10 / Chapter 1.3.3 --- Convergence Rate and Operation Cost --- p.12 / Chapter 1.3.4 --- Numerical Result --- p.13 / Chapter 1.4 --- Strang-Type BCCB Preconditioner --- p.15 / Chapter 1.4.1 --- Construction of BCCB Preconditioners --- p.15 / Chapter 1.4.2 --- Convergence Rate and Operation Cost --- p.17 / Chapter 1.4.3 --- Numerical Result --- p.19 / Chapter 1.5 --- Preconditioned Waveform Relaxation --- p.20 / Chapter 1.5.1 --- Waveform Relaxation --- p.20 / Chapter 1.5.2 --- Invertibility of the Strang-type preconditioners --- p.23 / Chapter 1.5.3 --- Convergence rate and operation cost --- p.24 / Chapter 1.5.4 --- Numerical Result --- p.25 / Chapter 1.6 --- Multigrid Waveform Relaxation --- p.27 / Chapter 1.6.1 --- Multigrid Method --- p.27 / Chapter 1.6.2 --- Numerical Result --- p.28 / Chapter 1.6.3 --- Concluding Remark --- p.30 / Chapter 2 --- Restoration of Chopped and Nodded Images --- p.31 / Chapter 2.1 --- Introduction --- p.31 / Chapter 2.2 --- The Projected Landweber Method --- p.35 / Chapter 2.3 --- Other Numerical Methods --- p.37 / Chapter 2.3.1 --- Tikhonov Regularization --- p.38 / Chapter 2.3.2 --- MRNSD --- p.41 / Chapter 2.3.3 --- Piecewise Polynomial TSVD --- p.43 / Chapter 2.4 --- Numerical Result --- p.46 / Chapter 2.5 --- Concluding Remark --- p.47 / Bibliography --- p.49

In this thesis the two-fluid (Eulerian/Eulerian) formulation for dispersed two-phase flow is considered. Closure laws are needed for this type of models. We investigate both empirically based relations, which we refer to as a nongranular model, and relations obtained from kinetic theory of dense gases, which we refer to as a granular model. For the granular model, a granular temperature is introduced, similar to thermodynamic temperature. It is often assumed that the granular energy is in a steady state, such that an algebraic granular model is obtained. The inviscid non-granular model in one space dimension is known to be conditionally well-posed. On the other hand, the viscous formulation is locally in time well-posed for smooth initial data, but with a medium to high wave number instability. Linearizing the algebraic granular model around constant data gives similar results. In this study we consider a couple of issues. First, we study the long time behavior of the viscous model in one space dimension, where we rely on numerical experiments, both for the non-granular and the algebraic granular model. We try to regularize the problem by adding second order artificial dissipation to the problem. The simulations suggest that it is not possible to obtain point-wise convergence using this regularization. Introducing a new measure, a concept of 1-D bubbles, gives hope for other convergence than point-wise. Secondly, we analyse the non-granular formulation in two space dimensions. Similar results concerning well-posedness and instability is obtained as for the non-granular formulation in one space dimension. Investigation of the time scales of the formulation in two space dimension suggests a sever restriction on the time step, such that explicit schemes are impractical. Finally, our simulation in one space dimension show that peaks or spikes form in finite time and that the solution is highly oscillatory. We introduce a model problem to study the formation and smoothness of these peaks. / QC 20101018

Thesis (M.S.)--Mississippi State University. Department of Computational Engineering. / Title from title screen. Includes bibliographical references.