Study of the impact of early visual deprivation and its following experience with numbers and numerosities on the elaboration of the semantic numerical representation with the same properties to those postulated in sighted people.

This thesis concerns numerical techniques for two phase flowsimulations; the two phases are immiscible and incompressible fluids. Strategies for accurate simulations are suggested. In particular, accurate approximations of the weakly discontinuousvelocity field, the discontinuous pressure, and the surface tension force and a new model for simulations of contact line dynamics are proposed. In two phase flow problems discontinuities arise in the pressure and the gradient of the velocity field due to surface tension forces and differences in the fluids' viscosity. In this thesis, a new finite element method which allows for discontinuities along an interface that can be arbitrarily located with respect to the mesh is presented. Using standard linear finite elements, the method is for an elliptic PDE proven to have optimal convergence order and a system matrix with condition number bounded independently of the position of the interface.The new finite element method is extended to the incompressible Stokes equations for two fluid systemsand enables accurate approximations of the weakly discontinuous velocity field and the discontinuous pressure. An alternative way to handle discontinuities is regularization. In this thesis, consistent regularizations of Dirac delta functions with support on interfaces are proposed. These regularized delta functions make it easy to approximate surface tension forces in level set methods. A new model for simulating contact line dynamics is also proposed. Capillary dominated flows are considered and it is assumed that contact line movement is driven by the deviation of the contact angle from its static value. This idea is used together with the conservative level set method. The need for fluid slip at the boundary is eliminated by providing a diffusive mechanism for contact line movement. Numerical experiments in two space dimensions show that the method is able to qualitatively correctly capture contact line dynamics. / QC 20110503

This thesis concerns the analysis and numerical simulation of wave propagation problems described by systems of linear hyperbolic partial differential equations. A major challenge in wave propagation problems is numerical simulation of high frequency waves. When the wavelength is very small compared to the overall size of the computational domain, we encounter a multiscale problem. Examples include the forward and the inverse seismic wave propagation, radiation and scattering problems in computational electromagnetics and underwater acoustics. In direct numerical simulations, the accuracy of the approximate solution is determined by the number of grid points or elements per wavelength. The computational cost to maintain constant accuracy grows algebraically with the frequency, and for sufficiently high frequency, direct numerical simulations are no longer feasible. Other numerical methods are therefore needed. Asymptotic methods, for instance, are good approximations for very high frequency waves. They are based on constructing asymptotic expansions of the solution. The accuracy increases with increasing frequency for a fixed computational cost. Most asymptotic techniques rely on geometrical optics equations with frequency independent unknowns. There are however two deficiencies in the geometrical optics solution. First, it does not include diffraction effects. Secondly, it breaks down at caustics. Geometrical theory of diffraction provides a technique for adding diffraction effects to the geometrical optics approximation by introducing diffracted rays. In papers 1 and 2 we present a numerical algorithm for computing an important type of diffracted rays known as creeping rays. Another asymptotic model which is valid also at caustics is based on Gaussian beams. In papers 3 and 4, we present an error analysis of Gaussian beams approximation and develop a new numerical algorithm for computing Gaussian beams, respectively. Another challenge in computation of wave propagation problems arises when the system of equations consists of second order hyperbolic equations involving mixed space-time derivatives. Examples include the harmonic formulation of Einstein’s equations and wave equations governing elasticity and acoustics. The classic computational treatment of such second order hyperbolic systems has been based on reducing the systems to first order differential forms. This treatment has however the disadvantage of introducing auxiliary variables with their associated constraints and boundary conditions. In paper 5, we treat the problem in the second order differential form, which has advantages for both computational efficiency and accuracy over the first order formulation. Finally, paper 6 concerns the concept of well-posedness for a class of linear hyperbolic initial boundary value problems which are not boundary stable. The well-posedness is well established for boundary stable hyperbolic systems for which we can obtain sharp estimates of the solution including estimates at boundaries. There are, however, problems which are not boundary stable but are well-posed in a weaker sense, i.e., the problems for which an energy estimate can be obtained in the interior of the domain but not on the boundaries. We analyze a model problem of this type. Possible applications arise in elastic wave equations and Maxwell’s equations describing glancing and surface waves. / QC 20100830

his work is concentrated on efforts to efficiently compute properties of systems, modelled by differential equations, involving multiple scales. Goal oriented adaptivity is the common approach to all the treated problems. Here the goal of a numerical computation is to approximate a functional of the solution to the differential equation and the numerical method is adapted to this task. The thesis consists of four papers. The first three papers concern the convergence of adaptive algorithms for numerical solution of differential equations; based on a posteriori expansions of global errors in the sought functional, the discretisations used in a numerical solution of the differential equiation are adaptively refined. The fourth paper uses expansion of the adaptive modelling error to compute a stochastic differential equation for a phase-field by coarse-graining molecular dynamics. An adaptive algorithm aims to minimise the number of degrees of freedom to make the error in the functional less than a given tolerance. The number of degrees of freedom provides the convergence rate of the adaptive algorithm as the tolerance tends to zero. Provided that the computational work is proportional to the degrees of freedom this gives an estimate of the efficiency of the algorithm. The first paper treats approximation of functionals of solutions to second order elliptic partial differential equations in bounded domains of ℝd, using isoparametric $d$-linear quadrilateral finite elements. For an adaptive algorithm, an error expansion with computable leading order term is derived %. and used in a computable error density, which is proved to converge uniformly as the mesh size tends to zero. For each element an error indicator is defined by the computed error density multiplying the local mesh size to the power of 2+d. The adaptive algorithm is based on successive subdivisions of elements, where it uses the error indicators. It is proved, using the uniform convergence of the error density, that the algorithm either reduces the maximal error indicator with a factor or stops; if it stops, then the error is asymptotically bounded by the tolerance using the optimal number of elements for an adaptive isotropic mesh, up to a problem independent factor. Here the optimal number of elements is proportional to the d/2 power of the Ldd+2 quasi-norm of the error density, whereas a uniform mesh requires a number of elements proportional to the d/2 power of the larger L1 norm of the same error density to obtain the same accuracy. For problems with multiple scales, in particular, these convergence rates may differ much, even though the convergence order may be the same. The second paper presents an adaptive algorithm for Monte Carlo Euler approximation of the expected value E[g(X(τ),\τ)] of a given function g depending on the solution X of an \Ito\ stochastic differential equation and on the first exit time τ from a given domain. An error expansion with computable leading order term for the approximation of E[g(X(T))] with a fixed final time T>0 was given in~[Szepessy, Tempone, and Zouraris, Comm. Pure and Appl. Math., 54, 1169-1214, 2001]. This error expansion is now extended to the case with stopped diffusion. In the extension conditional probabilities are used to estimate the first exit time error, and difference quotients are used to approximate the initial data of the dual solutions. For the stopped diffusion problem the time discretisation error is of order N-1/2 for a method with N uniform time steps. Numerical results show that the adaptive algorithm improves the time discretisation error to the order N-1, with N adaptive time steps. The third paper gives an overview of the application of the adaptive algorithm in the first two papers to ordinary, stochastic, and partial differential equation. The fourth paper investigates the possibility of computing some of the model functions in an Allen--Cahn type phase-field equation from a microscale model, where the material is described by stochastic, Smoluchowski, molecular dynamics. A local average of contributions to the potential energy in the micro model is used to determine the local phase, and a stochastic phase-field model is computed by coarse-graining the molecular dynamics. Molecular dynamics simulations on a two phase system at the melting point are used to compute a double-well reaction term in the Allen--Cahn equation and a diffusion matrix describing the noise in the coarse-grained phase-field. / QC 20100823

In this thesis a new method for the option pricing will be introduced with the help of the Mellin transforms. Firstly, the Mellin transform techniques for options on a single underlying stock is presented. After that basket options will be considered. Finally, an improvement of existing numerical results applied to Mellin transforms for 1-basket and 2-basket American Put Option will be discussed concisely. Our approach does not require either variable transformations or solving diusion equations.

In this thesis the connections between the boundary conditions and the vari- ance of the solution to a stochastic partial differential equation (PDE) are investigated. In particular a hyperbolical system of PDE’s with stochastic initial and boundary data are considered. The problem is shown to be well- posed on a class of boundary conditions through the energy method. Stability is shown by using summation-by-part operators coupled with simultaneous- approximation-terms. By using the energy estimates, the relative variance of the solutions for different boundary conditions are analyzed. It is concluded that some types of boundary conditions yields a lower variance than others. This is verified by numerical computations.

In this dissertation we study multiscale numerical methods for nonlinear parabolic equations, turbulent diffusion problems, and high contrast parabolic equations. We focus on designing and analysis of multiscale methods which can capture the effects of the small scale locally. At first, we study numerical homogenization of nonlinear parabolic equations in periodic cases. We examine the convergence of the numerical homogenization procedure formulated within the framework of the multiscale finite element method. The goal of the second problem is to develop efficient multiscale numerical techniques for solving turbulent diffusion equations governed by celluar flows. The solution near the separatrices can be approximated by the solution of a system of one dimensional heat equations on the graph. We study numerical implementation for this asymptotic approach, and spectral methods and finite difference scheme on exponential grids are used in solving coupled heat equations. The third problem we study is linear parabolic equations in strongly channelized media. We concentrate on showing that the solution depends on the steady state solution smoothly. As for the first problem, we obtain quantitive estimates for the convergence of the correctors and some parts of truncation error. These explicit estimates show us the sources of the resonance errors. We perform numerical implementations for the asymptotic approach in the second problem. We find that finite difference scheme with exponential grids are easy to implement and give us more accurate solutions while spectral methods have difficulties finding the constant states without major reformulation. Under some assumption, we justify rigorously the formal asymptotic expansion using a special coordinate system and asymptotic analysis with respect to high contrast for the third problem.

Using a numerical method to simulate the Metal-Organic Chemical Vapor Deposition (MOCVD). A study of the GaN films were growth on sapphire substrates, and a new design method which The position of carrier gas inlets and outlets, the gas in inlets by a showerhead reactor, the modified susceptor. The purpose of this research is to maintain deposited GaN film thickness variation range by controlling those parameters which may affect the deposition.

The good of this thesis is to apply a numerical model for studying waveform of an internal solitary wave (ISW) on variable seabed topography. The numerical model developed by Lynett and Liu (2002) is adopted for this work but with modification to improve its accuracy, both mathematically and in programming codes. Numerical experiments using the modified model are then performed and the results compared with laboratory experiments of Kuo (2005), in order to validate its accuracy. The mathematical model derived in the present study is based on the assumption that an internal wave is weakly nonlinear and weakly dispersive in an inviscid fluid. The governing equations based on the continuity equation and Euler equations are solved for ISW propagation over variable topography. The input conditions for the numerical experiments include physical parameters related to water depth and geometry of submarine obstacle, such as depth ratio between upper and lower layers (H1/H2), height (hs) and type (triangular ridge and trapezoidal shelf) of obstacles, in addition to the amplitude (ai) of an incident ISW. From the results of numerical experiments, wave amplitude, phase speed, and wave energy of a transmitted ISW are obtained and compared with that of laboratory experiments. (Kuo, 2005) ISW propagation over a single obstacle is affected by a dimensionless parameter called ¡§blockage parameter", £a= (a1+h1)/(h1+h2-hs). Three types of interaction may be classified (weak interaction, moderate interaction, and wave breaking) depending on the value of£a . For an ISW propagating over two consecutive obstacles, the interval between them is significant in reducing its amplitude and energy, as the interval reduces. Moreover, the effect of relative height between two obstacles may also be classified into two types: (i) within the range of weak interaction, energy dissipation is less for a high obstacle first than for it as the second; (ii) within the range of moderate interaction, the energy dissipation is higher for a high obstacle first than for it as the second. Further comparisons have shown that the modified numerical model is in better agreement with the results of laboratory experiments (Kuo, 2005) than the original model of Lynett and Liu (2002). The results obtained from the present numerical experiments for ISW evolution on variable topography is encouraging which could benefit other who may be interested in internal wave propagation for practical applications in oceanography.

Abstract The steady thermocapillary motion in shallow enclosures is studied. Two different configurations, imposed heat flux and differentially heated side walls, are considered. A numerical simulation of the problem in the imposed heat flux case is made. The Pressure Correction Method is used to treat the pressure velocity coupling, in particular, the SIMPLER approximation. The discretization is made using central differences along with an appropriate non-uniform grid.