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In the present study, a two-dimensional numerical model based on a time-independent finite difference method was developed and the model is used to analysis the dynamic interaction among wave, sloshing fluid in tank and the floating tank. The free surface of wave and sloshing fluid in the tank are all assumed to be a single value function and the wave breaking is, therefore, not considered in the study. The numerical model is firstly validated by some bench make studies. Extensive simulation were made to discuss the effects of geometry of the floating tank, the ratio of depth to breadth of fluid in tank, the fundamental freq of floating tank ¡V structure system etc.

Mathematical models of isothermal chemical systems in reactor problems and Turing's theory of morphogenesis with an application in sea-shell patterning are studied. The reaction-diffusion systems describing these models are solved numerically. First- and second-order difference schemes are developed, which are economical and reliable in comparison to classical numerical methods. The linearization process decouples the reaction-diffusion equations thereby allowing the use of different time steps for each differential equation, which may be large due to the excellent stability properties of the methods. The methods avoid having to solve a non-linear algebraic system at each time step. The schemes are suitable for implementation on a parallel machine.

This thesis mainly focuses on analyzing and pricing European swaption via Crank{Nicolson Finite Dierence method. This paper begins with some rather common instruments, denitions and valuations are also provided. MATLAB is the main computer language used throughout this paper, for the numerical examples, the MATLAB codes are also provide in the appendix in order for reader to reproduce the result. Also, the paper extends to price cancellable swap in the end.

High order accurate finite difference methods for hyperbolic and parabolic initial boundary value problems (IBVPs) are considered. Particular focus is on time dependent wave propagating problems in complex domains. Typical applications are acoustic and electromagnetic wave propagation and fluid dynamics. To solve such problems efficiently a strictly stable, high order accurate method is required. Our recipe to obtain such schemes is to: i) Approximate the (first and second) derivatives of the IBVPs with central finite difference operators, that satisfy a summation by parts (SBP) formula. ii) Use specific procedures for implementation of boundary conditions, that preserve the SBP property. iii) Add artificial dissipation. iv) Employ a multi block structure. Stable schemes for weakly nonlinear IBVPs require artificial dissipation to absorb the energy of the unresolved modes. This led to the construction of accurate and efficient artificial dissipation operators of SBP type, that preserve the energy and error estimate of the original problem. To solve problems on complex geometries, the computational domain is broken up into a number of smooth and structured meshes, in a multi block fashion. A stable and high order accurate approximation is obtained by discretizing each subdomain using SBP operators and using the Simultaneous Approximation Term (SAT) procedure for both the (external) boundary and the (internal) interface conditions. Steady and transient aerodynamic calculations around an airfoil were performed, where the first derivative SBP operators and the new artificial dissipation operators were combined to construct high order accurate upwind schemes. The computations showed that for time dependent problems and fine structures, high order methods are necessary to accurately compute the solution, on reasonably fine grids. The construction of high order accurate SBP operators for the second derivative is one of the considerations in this thesis. It was shown that the second derivative operators could be closed with two order less accuracy at the boundaries and still yield design order of accuracy, if an energy estimate could be obtained.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 77-79). Available also in a digital version from Dissertation Abstracts.

This thesis analyses numerical methods used in finding solutions of diferential equations. Numerical methods are viewed as discrete dynamical systems that give useful information on continuous dynamical systems defined by systems of (ordinary) diferential equations. We analyse non-standard finite difference schemes that have no spurious fixed-points compared to the dynamical system under consideration, the linear stability/instability property of the fixed-points being the same for both the discrete and continuous systems. We obtain a sharper condition for the elementary stability of the schemes. For more complex dynamical systems which are dissipative, we design schemes that replicate this property. Furthermore, we investigate the impact of the above analysis on the numerical solution of partial differential equations. We specifically focus on reaction-diffusion equations that arise in many fields of engineering and applied sciences. Often their solutions enjoy the follow- ing essential properties: Stability/instability of the fixed points for the space independent equation, the conservation of energy for the stationary equation, and boundedness and positivity. We design new non-standard finite diference schemes which replicate these properties. Our construction make use of three strategies: the renormalization of the denominator of the discrete derivative, non-local approximation of the nonlinear terms and simple functional relation between step sizes. Numerical results that support the theory are provided. Copyright / Thesis (PhD)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

Yes / Traditional Lattice-Boltzmann modelling of advection–diffusion flow is affected by numerical instability if the advective term becomes dominant over the diffusive (i.e., high-Péclet flow). To overcome the problem, two 3D one-way coupled models are proposed. In a traditional model, a Lattice-Boltzmann Navier–Stokes solver is coupled to a Lattice-Boltzmann advection–diffusion model. In a novel model, the Lattice-Boltzmann Navier–Stokes solver is coupled to an explicit finite-difference algorithm for advection–diffusion. The finite-difference algorithm also includes a novel approach to mitigate the numerical diffusivity connected with the upwind differentiation scheme.

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Column-supported, geosynthetic-reinforced embankments provide effective geotechnical foundations for applications in areas of weak subgrade soils. The system consists of a soil bridging layer with one or more embedded layers of geosynthetic reinforcement supported by driven or deep mixed columnar piles. The geosynthetic promotes load transfer within the bridging layer to the columns, allowing for larger column spacings and varied alignments. This technique is generally used when differential settlements of the embankment or adjacent structures are a concern and to minimize construction time. Recent increase in the popularity of this composite system has generated the need to further investigate its behavior and soil-structure interaction. Current models of geosynthetics are oversimplified and do not represent the true three-dimensional nature of the material. Such simplifications include treating the geosynthetic as a one-dimensional cable as well as neglecting stress concentrations and pile orientations. In this thesis, a complete three-dimensional analysis of the geosynthetic is performed. The geosynthetic was modeled as a thin flexible plate in a single square unit cell of the embankment. The principle of minimum potential energy was then applied, utilizing central finite difference equations. Energy components from vertical loading, soil and column support, as well as bending and membrane stiffness of the geosynthetic are considered. Three pile orentation types were implemented: square piles, circular piles, and square piles rotated 45° to the edges of the unit cell. Each of the pile orientations was analyzed using two distinct parameter sets that are investigated in previously published and ongoing research. Vertical and in-plane deflections, stress resultants, and strains were determined and compared to other geosynthetic models and design guides. Results of each parameter set and pile orientation were also compared to provide design recommendations for geosynthetic-reinforced column-supported embankments. / Master of Science