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In the present study, a fully nonlinear 2-D finite difference scheme has been developed based on inviscid and incompressible flow in a rectangular tank. The rectangular tank is coupled to a linear elastic-supported structure made up by reinforced concrete. Wave breaking and run-up are not considered in the present numerical model due to the free surface is assumed as a single-value function. The main purpose of this study is to analyze interactions between sloshing forces generated by system vibrations and structure motions. The accuracy of present study is made by comparing to other reported numerical results and the consequence shows well agreement. The present study can be applied for designing various combinations of coupled structure systems for different necessity. The analyses of practical examples are also presented in this study. The present numerical model can provide a quick and accurate way on determining the natural frequencies of connecting fluid-structure system and this is hard to identify through experiment.

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Finite difference numerical methods are developed for the solution system in the biomedical sciences; namely, fox-rabies model. First-order methods and second-order method are developed to solve the fox-rabies equations. The fox-rabies model is extended to one-space dimension to incorporate diffusion. The reaction terms in these systems of partial differential equations contain non-linear expressions. It is seen that the numerical solutions are obtained by solving non-linear algebraic system at each time step, as opposed to solving anon-linear algebraic system which is often required when integrating non-linear partial differential equations. The numerical methods proposed for the solution of the initial-value problem for the fox-rabies model are characterized to be implicit. In each case, however, it seen that the numerical solutions are obtained explicitly. In a series of numerical experiments, in which the ordinary differential equations are solved first of all, it seen that the proposed methods have an identical stability properties to those of the well-known, first-order, Euler method. The proposed methods for the numerical solution of partial differential equations are seen to be economical and reliable. Error analysis for the methods, computer implementation and numerical results are discussed. The stability of the numerical method is analyzed using maximum principle analysis.

Recently a viscoelastic turbulence closure model, based on that of Townsend (1976), for wind-wave interactions by turbulent wind has been proposed by Sajjadi (2001). In that work, the governing equations of mean and turbulence were linearized and solved analytically using an asymptotic method. In this work the equations derived by Sajjadi were solved numerically for the cases of strong turbulence due to wind over surface of a monochromatic water wave. Vortex shedding has been observed at high wind velocities. Also, a layer of vortices separating the main flow of wind from the water surface was observed from the results for high velocities of wind. A finite difference scheme was devised which is second order accurate. The results were compared with another scheme based on the method of superposition coupled with orthonormalization by Scott andWatts (1977). The two schemes agree reasonably well for high velocities while they differ for low velocities. Two test cases were implemented to test the finite difference scheme. The tests show that the finite difference scheme predicts accurate solutions for inhomogeneous equations, while it fails to capture the accurate solution if a non trivial solution exists for homogeneous equations. This is attributed as the reason for the difference in the results.

The increasing complexity of oil and gas reservoirs has led to the need of a better understanding of the processes governing the rock properties. Traditional theoretical and empirical models often fail to predict the behavior of carbonates, tight gas sands and shale gas, for example. An essential part of the necessary investigation is the study of the phenomena occurring at the pore scale. In this direction, the so-called digital rock physics is emerging as a research field that offers the possibility of imaging the rock pore space and simulating the processes therein directly. This report describes our work on developing algorithms to simulate viscous and electric flow through a three dimensional Cartesian representation of the porous space, such as those available through X-ray microtomography. We use finite differences to discretize the governing equations and also propose a new method to enforce the incompressible flow constraint under natural boundary conditions. Parallel computational codes are written targeting performance and computer memory optimization, allowing the use of bigger and more representative samples. Results are reported with an estimate of the error bars in order to help on the simulation appraisal. Tests performed using benchmark samples show good agreement with experimental/theoretical values. Example of application on digital modeling of cement growth and on multiphase fluid distribution are also provided. The final test is done on Bentheimer, Buff Berea and Idaho Brown sandstone samples with available laboratory measurements. Some limitations need to be investigated in future work. First, the computer potential fields show anomalous border effects at the open boundaries. Second, a minor problem arises with the decreased convergence rate for the velocity field due to the increased number of operations, leading to the need of a more sophisticated preconditioner. We intend to expand the algorithms to handle microporosity (e.g. carbonates) and multiphase fluid flow. / text

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This course project is made up of two parts. Part one is an investigation and implementation of pricing of financial derivatives using numerical methods for the solution of partial differential equations. Part two is an introduction of Monte Carlo methods in financial engineering. The name of course is MA573:Computational Methods in Financial Mathematics, spring 2009, given by Professor Marcel Blais.