Thesis (Ph. D.)--University of Washington, 2008. / Vita. Includes bibliographical references (p. 91-99).

Thesis (M.S.), Oregon Graduate Institute of Science & Technology, 1994.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2004. / Includes bibliographical references (leaves 68-74). Also available in electronic version. Access restricted to campus users.

Thesis (Ph. D.)--University of Washington, 1995. / Vita. Includes bibliographical references (leaves [224]-226).

Thesis (Ph. D.)--University of California, San Diego, 1997. / Vita. Includes bibliographical references (leaves 102-106).

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (leaves 94-96).

In this thesis we develop and analyze a Galerkin-Characteristic method to integrate the potential vorticity equations of a baroclinic ocean. The method proposed is a two stage inductive algorithm. In the first stage the material derivative of the potential vorticity is approximated by combining Galerkin-Characteristic and Particle methods. This yield a computationally efficient algorithm for this stage. Such an algorithm consists of updating the dependent variable at the grid points by cubic spline interpolation at the foot of the characteristic curves of the advective component of the equations. The algorithm is unconditionally stable and conservative for Δt = O(h). The error analysis with respect to L² -norm shows that the algorithm converges with order O(h); however, in the maximum norm it is proved that for sufficiently smooth functions the foot of the characteristic curves are superconvergent points of order O(h⁴ /Δt). The second stage of the algorithm is a projection of the Lagrangian representation of the flow onto the Cartesian space-time Eularian representation coordinated with Crank-Nicholson Finite Elements. The error analysis for this stage with respect to L²-norm shows that the approximation component of the global error is O(h²) for the free-slip boundary condition, and O(h) for the no-slip boundary condition. These estimates represent an improvement with respect to other estimates for the vorticity previously reported in the literature. The evolutionary component of the global error is equal to K(Δt² + h), where K is a constant that depends on the derivatives of the advective quantity along the Characteristic. Since the potential vorticity is a quasi-conservative quantitiy, one can conclude that K is in general small. Numerical experiments illustrate our theoretical results for both stages of the method. / Science, Faculty of / Mathematics, Department of / Graduate

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A new upwind scheme is developed for solving the three-dimensional Euler equations on unstructured tetrahedral meshes. The method yields solution accuracy and efficiency comparable to that currently available from similar structured-grid codes. The key to achieving this result is a novel cell reconstruction process which is based on an analytical formulation for computing solution gradients within tetrahedral cells. Prior methodology requires the application of cumbersome numerical procedures to evaluate surface integrals around the cell volume. The result is that higher-order differences can now be constructed more efficiently to attain computational times per cell comparable to those of structured codes. The underlying philosophy employed in constructing the basic flow solver is to draw on proven structured-grid technology whenever possible in order to reduce risk. Thus, spatial discretization is accomplished by a cell-centered finite-volume formulation using flux-difference splitting. Solutions are advanced in time by a 3- stage Runge-Kutta time-stepping scheme with convergence accelerated to steady state by local time stepping and implicit residual smoothing. The flow solver operates at a speed of 34 microseconds per cell per cycle on a CRAY-2S supercomputer and requires 64 words of memory per cell. Transonic solutions are presented for a broad class of configurations to demonstrate the accuracy, speed, and robustness of the new scheme. Solutions are shown for the ONERA M6 wing, the Boeing 747-200 configuration, a low-wing transport configuration, a high-speed civil transport configuration, and the space shuttle ascent configuration. Computed surface pressure-coefficient distributions on the ONERA M6 wing are compared with structured-grid results as well as experimental data to quantify the accuracy. A further assessment of grid sensitivity and the effect of convergence acceleration parameters is also included for this configuration. The more complex configurations serve to demonstrate the robustness and efficiency of the new method and its potential for performing routine aerodynamic analysis of full aircraft configurations. For example, the basic transonic flow features are well captured on the space shuttle ascent configuration with only 7 megawords of memory and 142 minutes of CRAY-YMP run time. / Ph. D.

A thesis submitted to the faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Doctor of Philosophy, / The inverse problem in the Calculus of Variations involves determining the Lagrangians, if any, associated with a given (system of) differential equation(s). One can classify Lagrangians according to the Lie algebra of symmetries of the Action integral (the Noether algebra). We give a complete classification of first-order Lagrangians defined on the line and produce results pertaining to the dimensionality of the algebra of Noether symmetries and compare and contrast these with similar results on the algebra of Lie symmetries of the corresponding Euler-Lagrange .equations. It is proved that the maximum dimension of the Noether point symmetry algebra of a particle Lagrangian. is five whereas it is known that the maximum dimension Qf the Lie algebra of the corresponding scalar second-order Euler-Lagrange equation is eight. Moreover, we show th'a.t a particle Lagrangian does not admit a maximal four-dimensional Noether point symmeiry algebra and consequently a particle Lagrangian admits the maximal r E {O, 1,2,3, 5}-dimensional Noether point symmetry algebra, It is well .known that an important means of analyzing differential equations lies in the knowledge of the first integrals of the equation. We deliver an algorithm for finding first integrals of partial differential equations and show how some of the symmetry properties of the first integrals help to 'further' reduce the order of the equations and sometimes completely solve the equations. Finally, we discuss some open questions. These include the inverse problem and classification of partial differential equations. ALo, there is the question of the extension of the results to 'higher' dimensions. / Andrew Chakane 2018