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<p> The Hammett acidity function, Ho, has been extended from dilute solutions of water in sulfuric acid to the highly acidic HSO3F-SbF5•3SO3 system, a range of more than eight Ho units. The protonation behaviour of eleven aromatic nitro indicators was studied by means of a spectrophotometric technique. These indicators were found to behave in a similar fashion to the previously studied aniline indicators which have been used to define the Ho function from infinitely dilute solutions of acids in water to concentrated sulfuric acid solutions. The superacid systems studied were H2O-H2SO4-SO3, H2SO4-HSO3Cl, H2SO4-HSO3F, H2SO4-HB(HSO4)4, KSO3F-HSO3F, HSO3F-SO3, HSO3F-AsF5, HSO3F-SbF5, and HSO3F-SbF5•3SO3 of which the latter is the most acidic solvent system known.</p> / Thesis / Doctor of Philosophy (PhD)

The numerical solutions of Maxwell’s equations have been the cornerstone of photonic design for over a century. In recent years, the field of photonics has witnessed a surge in interest in inverse design, driven by the potential to engineer nonintuitive photonic structures with remarkable properties. However, the conventional approach to inverse design, which relies on fully discretized numerical simulations, faces significant challenges in terms of computational efficiency and scalability. This thesis delves into an alternative paradigm for inverse design, leveraging the power of semi-analytical methods. Unlike their fully discretized counterparts, semi-analytical methods hold the promise of enabling simulations that are independent of the computational grid size, potentially revolutionizing the design and optimization of photonic structures. To achieve this goal, we put forth a more generalized formalism for semi-analytical methods and have developed a comprehensive differential theory to underpin their operation. This theoretical foundation not only enhances our understanding of these methods but also paves the way for their broader application in the field of photonics. In the final stages of our investigation, we illustrate how the semi-analytical simulation framework can be effectively employed in practical photonic design scenarios. We demonstrate the synergy of semi-analytical methods with ray tracing techniques, showcasing their combined potential in the creation of large-scale optical lens systems and other complex optical devices.

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This study seeks to reduce the cost of numerically solving non-linear partial differential equations by reducing the number of computations without compromising accuracy. This was done by using accurate local time stepping. This algorithm uses local time stepping but compensates for the inconsistencies in the temporal dimension by interpolations and/or extrapolations. Reduction in computations are obtained by time-stepping only a particular region with small time steps. A shock tube problem and a detonation wave were the two test cases considered. The performance of the solution using this algorithm was compared with an algorithm that does not use accurate local time stepping.

Implementations of numerical simulations for solving systems of partial differential equations are often not verified and are falsely assumed to work correctly. As a result, the implementations are prone to coding errors that could degrade the accuracy of the solution. In order to ensure that a code is written correctly, rigorous verification of all parts of the code is necessary. Code verification is the task of ascertaining whether a numerical algorithm is solving the governing equations of the problem correctly. If an exact solution existed for the governing equations then verification would be easier but these solutions are rare because of the non-linearity of common Computational Fluid Dynamics (CFD) problems. In the absence of exact solutions, grid refinement studies are the most commonly used methods to verify codes using simulations on a sequence of grids but even these studies have limitations. The Method of Manufactured Solutions (MMS) is a novel and a recently developed technique that verifies the observed order-ofuracy of the implementation of a numerical algorithm. The method is more general and overcomes many of the limitations of the method of exact solutions and grid refinement studies. The central idea is to modify the governing equations and the boundary conditions by adding forcing functions or source terms in order to drive the discrete solution to a prescribed or ``manufactured' solution chosen a priori. A grid convergence study is performed subsequently to determine the observed orders. Two methods of accuracy assessment are presented here - solution accuracy analysis and residual error analysis. The method based on the error in the spatial residual is computationally less expensive and proved to be a valuable debugging tool. In the present work, the Method of Manufactured Solutions (MMS) is implemented on a compressible flow solver that solves the two-dimensional Euler equations on structured grids and an incompressible code that solves the two-dimensional Navier-Stokes equations on unstructured meshes. Exponential functions are used to ``manufacture' steady solutions to the governing equations. Solution and residual error analyses are presented. The influence of grid non-uniformity on the numerical accuracy is studied.

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