Environmental flows (e.g. river and atmospheric flows) governed by the shallow water equations (SWE) are usually dominated by the advective mechanism over multiple time-scales. The combination of time dependency and nonlinear advection creates difficulties in the numerical solution of the SWE. A fully-implicit scheme is desirable because a relatively large time step may be used in a simulation. However, nonlinearity in a fully implicit method results in a system of nonlinear equations to be solved at each time step. To address this difficulty, a new method for implicit solution of unsteady nonlinear advection equations is developed in this research. This Time-Centered Split (TCS) method uses a nested application of the midpoint rule to computationally decouple advection terms in a temporally second-order accurate time-marching discretization. The method requires solution of only two sets of linear equations without an outer iteration, and is theoretically applicable to quadratically-nonlinear coupled equations for any number of variables. To explore its characteristics, the TCS algorithm is first applied to onedimensional problems and compared to the conventional nonlinear solution methods. The temporal accuracy and practical stability of the method is confirmed using these 1D examples. It is shown that TCS can computationally linearize unsteady nonlinear advection problems without either 1) outer iteration or 2) calculation of the Jacobian. A family of the TCS method is created in one general form by introducing weighting factors to different terms. We prove both analytically and by examples that the value of the weighting factors does not affect the order of accuracy of the scheme. In addition, the TCS method can not only computationally linearize but also decouple an equation system of coupled variables using special combinations of weighting factors. Hence, the TCS method provides flexibilities and efficiency in applications. / text

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published_or_final_version / Civil Engineering / Doctoral / Doctor of Philosophy

Optimal engineering design specifications are usually derived from an iterative design process. Here, different mathematical programs, each representing a particular problem assumption, are solved in order to gain insight into how and why an ideal design changes as model parameters vary. The mathematical technique used in this process is termed sensitivity analysis. The focus of this study is on techniques for performing such analysis on optimization problems which can be modeled as geometric programs. A dual based computationally attractive numerical procedure was developed to generate the locus of optimal solutions to prototype geometric programs corresponding to a large set of program parameter trajectories. Coefficient variation can include individual or simultaneous changes in any or all cost and exponent values. Sensitivity analysis is accomplished by numerically solving a specially constructed nonlinear initial value differential equation problem. Computational procedures were developed for computing an intitial value point, differential equation construction and solution, primal/dual conversion and problem reconstruction in the event of a primal constraint status change. A computer program written to carry out this scheme was described and used in the design of a batch process chemical plant. Preliminary results show the sensitivity analysis procedure developed in this study is attractive in terms of required computation time and perturbation flexibility of model coefficients.

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In this thesis we study spherically symmetric spacetimes which are static with a perfect fluid source. The Einstein field equations, in a number of equivalent forms, are derived in detail. The physical properties of a relativistic star are briefly reviewed. We specify two particular choices for one of the gravitational potentials. The behaviour of the remaining gravitational potential is governed by a second order differential equation. This equation has solutions in terms of elementary functions for some cases. The differential equation, in other cases, may be expressed as Bessel, confluent hypergeometric and hypergeometric equations. In such instances the solution is given in terms of special functions. A number of solutions to the Einstein field equations are generated. We believe that these solutions may be used to model realistic stars. Many of the solutions found are new and have not been published previously. In some cases our solutions are generalisations of cases considered previously. For some choices of the gravitational potential our solutions are equivalent to well-known results documented in the literature; in these cases we explicitly relate our solutions to those published previously. We have utilised the computer package MATHEMATICA Version 2.0 (Wolfram 1991) to assist with calculations, and to produce figures to describe the gravitational field. In addition, we briefly investigate the approach of specifying an equation of state relating the energy density and the pressure. The solution of the Einstein field equations, for a linear equation of state, is reduced to integrating Abel's equation of the second kind. / Thesis (M.Sc.)-University of Natal, Durban, 1993.

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