Recently the phenomena of Bose-Einstein condensates have been observed in laboratories, and the related problems are extensively studied. In this paper we consider the nonlinear Schrödinger equation in the laser beam rotating magnetic field and compute its corresponding energy functional under the mass conservative condition. By separating time and space variables, factoring real part and image part, and discretizing via finite difference method, the original equation can be transformed to a large scale parametrized polynomial systems. We use continuation method to find the solutions that satisfy the mass conservative condition. We will also explore bifurcation points on the curves and other solutions lying on bifurcation branches. The numerical results show that when the rotating angular momentum is small, we can find the solutions by continuation method along some particular curves and these curves are regular. As the angular momentum is increasing, there will be more bifurcation points on curves.

In the past few decades, many people spent a lot of time and used many different ways, which includes analytic method, numerical method, and experimental observations for investigating the flow around circular cylinder problem. Eventually, the purpose of these investigations is to determinate the force acting on the cylinder and which is very useful and important for marine and hydraulic engineering. Essentially, it can be divided into three circumstances, (i) the flow around a fixed cylinder, (ii) the flow around a rotating cylinder, (iii) the flow around a moving cylinder. The first two conditions have already been will discussed. Consequently, besides analyzing the first two conditions and comparing with reference papers, the purpose of this present is discussing the variation of the flow field and the force acting on the cylinder by using finite difference method. Because of the considerable quantity of computation, using parallel computing for this model to speedup the numerical process is also one of the issues of the present.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1982. / Typescript. Vita. Description based on print version record. Includes bibliographical references (leaves 126-131).

The sound generated by high speed trains can be exacerbated by the presence of trackside structures. Tunnels are the principal structures that have a strong influence on the noise produced by trains. A train entering a tunnel causes air to flow in and out of the tunnel portal, forming a monopole source of low frequency sound ["infrasound"] whose wavelength is large compared to the tunnel diameter. For the compact case, when the tunnel diameter is small, incompressible flow theory can be used to compute the Green's function that determines the monopole sound. However, when the infrasound is "shielded" from the far field by a large "flange" at the tunnel portal, the problem of calculating the sound produced in the far field is more complex. In this case, the monopole contribution can be calculated in a first approximation in terms of a modified Compact Green's function, whose properties are determined by the value at the center of a. disk (modelling the flange) of a diffracted potential produced by a thin circular disk. In this thesis this potential is calculated numerically. The scattering of sound by a thin circular disk is investigated using the Finite Difference Method applied to the three dimensional Helmholtz equation subject to appropriate boundary conditions on the disk. The solution is also used to examine the unsteady force acting on the disk.

The fractional Laplacian is a promising mathematical tool due to its ability to capture the anomalous diffusion and model the complex physical phenomenon with long-range interaction, such as fractional quantum mechanics, image processing, jump process, etc. One of the important applications of fractional Laplacian is a turbulence intermittency model of fractional Navier-Stokes equation which is derived from Boltzmann's theory. However, the efficient computation of this model on bounded domains is challenging as highly accurate and efficient numerical methods are not yet available. The bottleneck for efficient computation lies in the low accuracy and high computational cost of discretizing the fractional Laplacian operator. Although many state-of-the-art numerical methods have been proposed and some progress has been made for the existing numerical methods to achieve quasi-optimal complexity, some issues are still fully unresolved: i) Due to nonlocal nature of the fractional Laplacian, the implementation of the algorithm is still complicated and the computational cost for preparation of algorithms is still high, e.g., as pointed out by Acosta et al \cite{AcostaBB17} 'Over 99\% of the CPU time is devoted to assembly routine' for finite element method; ii) Due to the intrinsic singularity of the fractional Laplacian, the convergence orders in the literature are still unsatisfactory for many applications including turbulence intermittency simulations. To reduce the complexity and computational cost, we consider two numerical methods, finite difference and spectral method with quasi-linear complexity, which are summarized as follows. We develop spectral Galerkin methods to accurately solve the fractional advection-diffusion-reaction equations and apply the method to fractional Navier-Stokes equations. In spectral methods on a ball, the evaluation of fractional Laplacian operator can be straightforward thanks to the pseudo-eigen relation. For general smooth computational domains, we propose the use of spectral methods enriched by singular functions which characterize the inherent boundary singularity of the fractional Laplacian. We develop a simple and easy-to-implement fractional centered difference approximation to the fractional Laplacian on a uniform mesh using generating functions. The weights or coefficients of the fractional centered formula can be readily computed using the fast Fourier transform. Together with singularity subtraction, we propose high-order finite difference methods without any graded mesh. With the use of the presented results, it may be possible to solve fractional Navier-Stokes equations, fractional quantum Schrodinger equations, and stochastic fractional equations with high accuracy. All numerical simulations will be accompanied by stability and convergence analysis.

The short rate is central in the context of interest-rate markets as well as broader finance. As such, accurate modelling of this rate is of particular importance in the pricing of interest-rate options, especially during times of high volatility where increased demand is seen for simpler and lower risk investments. Recent interest has moved away from models of a pure continuous nature towards models that can account for discontinuities in the short rate. These are more representative of real world movements where the short rate is seen to jump due to current and scheduled market information. This dissertation examines this phenomenon in the context of a Vasicek short rate model and accounts for random-sized jumps at deterministic times following ideas similar to those introduced by Kim and Wright (2014). Finite difference methods are used successfully to find PDE solutions via backwards diffusion of the option value equation to its initial state. This procedure is implemented computationally and compared to Monte Carlo benchmark methods in order to assess its accuracy. In both non-jump and jump settings the method constructed was able to accurately price the call option specified and proved to be a viable means for pricing interest-rate options when stochastically-sized discontinuities are present at known times between inception and expiry. Furthermore the method showed that the stochastic discontinues in the short rate most notably affect the option price in the region around and just out of the money.

The Burgers equation, and related partial differential equations (PDEs), can be numerically challenging for small values of the viscosity parameter. For example, these equations can develop discontinuous solutions (or solutions with large gradients) from smooth initial data. Aside from numerical stability issues, standard numerical methods can also give rise to spurious oscillations near these discontinuities. In this study, we consider an equivalent form of the Burgers equation given by Constantin and Iyer, whose solution can be written as the expected value of a stochastic differential equation. This equivalence is used to develop a numerical method for approximating solutions to Burgers equation. Our preliminary analysis of the algorithm reveals that it is a natural generalization of the method of characteristics and that it produces approximate solutions that actually improve as the viscosity parameter vanishes. We present three examples that compare our algorithm to a recently published reference method as well as the vanishing viscosity/entropy solution for decreasing values of the viscosity. / Master of Science / Burgers equation is a Partial Differential Equation (PDE) used to model how fluids evolve in time based on some initial condition and viscosity parameter. This viscosity parameter helps describe how the energy in a fluid dissipates. When studying partial differential equations, it is often hard to find a closed form solution to the problem, so we often approximate the solution with numerical methods. As our viscosity parameter approaches 0, many numerical methods develop problems and may no longer accurately compute the solution. Using random variables, we develop an approximation algorithm and test our numerical method on various types of initial conditions with small viscosity coefficients.

This dissertation deals with variance reduction techniques (VRTs) for improving the reliability of the estimators of interest through a controlled laboratory-like simulation experiment. This research concentrates on correlation methods of VRTs which include common random numbers, antithetic variates and control variates. The basic idea of these methods is to utilize the linear correlation either between the responses or between the response and control variates in order to reduce the variance of estimators of certain system parameters. Combining these methods, we develop procedures for estimating a system parameter of interest. First, we develop three combined methods utilizing antithetic variates and control variates for improving the estimation of the mean response in a single population model. We explore how these methods may reduce the variance of the estimator of interest. A combined method (Combined Method 1) using antithetic variates for the non-control variate stochastic components and independent streams for the control variates yields better results than by applying methods of either antithetic variates or control variates individually for several selected models. Second, we develop variance reduction techniques for improving the estimation of the model parameters in a multipopulation simulation model. We extend Combined Method 1 showing good performance in estimating the mean response of a single population model to the multipopulation context with independent simulation runs across design points. We also develop another extension of Combined Method 1 that incorporates the Schruben-Margolin method to estimate the parameters of a multipopulation model. Under certain conditions, this method is superior to the Schruben-Margolin method. Finally, we propose a new approach (Extended Schruben-Margolin Method) utilizing the control variates under the Schruben-Margolin strategy for improving the estimation in a first-order linear model. Extended Schruben-Margolin Method yields better results than the Schruben-Margolin method in estimating the model parameters of interest. / Ph. D.

Wave propagation is a one of the most studied phenomenons in history due to the variety of applications such as quantum mechanics, electrodynamics and acoustics. In this thesis, the possibilities of improving numerical methods for solving the wave equation will be studied. More specifically, the dispersion relation will be used as a focal point. Generally there is a difference between the dispersion relation in the numerical solution and the analytic solution and the aim will be to decrease this difference and study the consequences. The numerical method that will be used and improved is the finite difference method (FDM). A dispersion relation for the numerical scheme will be derived including parameters from the spatial discretisation. These parameters will be optimised with the gradient descent method while retaining the second order accuracy of the derivative approximation. Performance is tested with numerical examples and the method of optimising for improved dispersion relation is proved to be successful. The optimised second order accurate schemes outperforms the standard second order accurate method in all simulated examples. When comparing the optimised stencil with the equally computationally expensive fourth order accurate method the optimised stencil performs better for sparse grids, especially when the spatial variation in the solution is high. For finer grids the fourth order accurate method quickly achieves smaller errors and is therefore preferable.

A two-dimensional numerical model has been developed to simulate wave-current induced nearshore circulation patterns in beaches and surf zones. The wave model is based on the parabolic wave equation for mild slope beaches. The parabolic equation method has been chosen because it is a viable means of predicting the characteristics of surface waves in slowly varying domains and in its present form dissipation and wave breaking are also included. The two dimensional parabolic mild slope equation was discretised and solved in a fully implicit manner, so stability did not create a major problem. This wave model was then embedded into the existing numerical model DIVAST. The sediment transport formulae from Van Rijn was used to calculate the nearshore sediment transport rate.