Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 185-191) and index.

Thesis (Ph. D.)--University of Washington, 2005. / Vita. Includes bibliographical references (p. 214-234).

Thesis (Ph.D.)--Brown University, 2005. / Vita. Thesis advisor: Chi-Wang Shu. Includes bibliographical references (leaves 83-87). Also available online.

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (p. 101-104).

This paper considers some numerical schemes for the approximate solution of conservation laws and various wavelet methods are reviewed. This is followed by the construction of wavelet spaces based on a polynomial framework for the approximate solution of conservation laws. Construction of a representation of the approximate solution in terms of an entropy satisfying Multiresolution Analysis (MRA) is defined. Finally, a proof of convergence of the approximate solution of conservation laws using the characterization provided by the basis functions in the MRA will be given.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2014. / Symmetries and conservation laws of partial di erential equations (pdes) have been instrumental in giving new approaches for reducing pdes. In this dissertation, we study the symmetries and conservation laws of the two-dimensional Schr odingertype equation and the Benney-Luke equation, we use these quantities in the Double Reduction method which is used as a way to reduce the equations into a workable pdes or even an ordinary di erential equations. The symmetries, conservation laws and multipliers will be determined though di erent approaches. Some of the reductions of the Schr odinger equation produced some famous di erential equations that have been dealt with in detail in many texts.

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Many applications in engineering practice can be described by thehyperbolic partial differential equations (PDEs). Numerical modeling of this typeof equations often involves large gradients or shocks, which makes it achallenging task for conventional numerical methods to accurately simulate suchsystems. Thus developing accurate and efficient shock capturing numericalschemes becomes important for the study of hyperbolic equations.In this dissertation, a detailed study of the numerical methods for linearand nonlinear unsteady hyperbolic equations was carried out. A new finitedifference shock capturing scheme of finite volume style was developed. Thisscheme is based on the high order Pad?? type compact central finite differencemethod with the weighted essentially non-oscillatory (WENO) reconstruction toeliminate non-physical oscillations near the discontinuities while maintain stablesolution in the smooth areas. The unconditionally stable semi-implicit Crank-Nicolson (CN) scheme is used for time integration.The theoretical development was conducted based on one-dimensionalhomogeneous scalar equation and system equations. Discussions were alsoextended to include source terms and to deal with problems of higher dimension.For the treatment of source terms, Strang splitting was used. For multidimensionalequations, the ?? -form Douglas-Gunn alternating direction implicit(ADI) method was employed. To compare the performance of the scheme withENO type interpolation, the current numerical framework was also applied usingENO reconstruction.The numerical schemes were tested on 1-D and 2-D benchmark problems,as well as published experimental results. The simulated results show thecapability of the proposed scheme to resolve discontinuities while maintainingaccuracy in smooth regions. Comparisons with the experimental results validatethe method for dam break problems. It is concluded that the proposed scheme isa useful tool for solving hyperbolic equations in general, and from engineeringapplication perspective it provides a new way of modeling open channel flows.