Riemann surfaces frequently possess automorphisms which can be exploited to simplify calculations. However, existing computer software (Maple in particular) is designed for maximum generality and has not yet been extended to make use of available symmetries. In many calculations, the symmetries can be most easily used by choosing a specific basis for H₁(Σ,Z) under which the automorphism group acts neatly. This thesis describes a Maple library, designed to be used in conjunction with the existing algcurves, which allows such a choice to be made. In addition we create a visual tool to simplify the presentation of Riemann surfaces as sheeted covers of C and the creation of homology bases suitable for use in the Maple library. Two applications are considered for these techniques, first Klein's curve and then Bring's. Both of these possess maximal symmetry groups for their genus, and this fact is exploited to obtain neat algebraic homology bases. In the Klein case the basis is novel; Bring's is derived from work in the hyperbolic setting by Riera. In both cases previous hyperbolic work and calculations are related to the algebraic results. Vectors of Riemann constants are calculated for both curves, again exploiting the symmetry. Finally this thesis moves back to simpler cases, and presents a general algorithm to convert results from general genus 2 curves into results based on a symmetric basis if one exists. This is applied to algebraic and numeric examples where we discover an elliptic surface covered in each case.

Currently, the accurate prediction of the impact of an extreme wave on infrastructure located near shore is difficult to assess. There is a lack of established methods to accurately quantify these impacts. Extreme waves, such as tsunamis generate, through breaking, extremely powerful hydraulic bores that impact and significantly damage coastal structures and buildings located close to the shoreline. The damage induced by such hydraulic bores is often due to structural failure. Examples of devastating coastal disasters are the 2004 Indian Ocean Tsunami, 2005 Hurricane Katrina and most recently, the 2011 Tohoku Japan Tsunami. As a result, more advanced research is needed to estimate the magnitude of forces exerted on structures by such bores. This research presents results of a numerical model based on the Smoothed Particle Hydrodynamics (SPH) method which is used to simulate the impact of extreme hydrodynamic forces on shore protection walls. Typically, fluids are modeled numerically based on a Lagrangian approach, an Eulerian approach or a combination of the two. Many of the common problems that arise from using more traditional techniques can be avoided through the use of SPH-based models. Such challenges include the model computational efficiency in terms of complexity of implementation. The SPH method allows water particles to be individually modeled, each with their own characteristics, which then accurately depicts the behavior and properties of the flow field. An open source code, known as SPHysics, was used to run the simulations presented in this thesis. Several cases analysed consist of hydraulic bores impacting a flat vertical wall as well as a sloping seawall. The analysis includes comparisons of the numerical results with published experimental data. The model is shown to accurately reproduce the formation of solitary waves as well as their propagation and breaking. The impacting bore profiles as well as the resulting pressures are also efficiently simulated using the model.

Currently, the accurate prediction of the impact of an extreme wave on infrastructure located near shore is difficult to assess. There is a lack of established methods to accurately quantify these impacts. Extreme waves, such as tsunamis generate, through breaking, extremely powerful hydraulic bores that impact and significantly damage coastal structures and buildings located close to the shoreline. The damage induced by such hydraulic bores is often due to structural failure. Examples of devastating coastal disasters are the 2004 Indian Ocean Tsunami, 2005 Hurricane Katrina and most recently, the 2011 Tohoku Japan Tsunami. As a result, more advanced research is needed to estimate the magnitude of forces exerted on structures by such bores. This research presents results of a numerical model based on the Smoothed Particle Hydrodynamics (SPH) method which is used to simulate the impact of extreme hydrodynamic forces on shore protection walls. Typically, fluids are modeled numerically based on a Lagrangian approach, an Eulerian approach or a combination of the two. Many of the common problems that arise from using more traditional techniques can be avoided through the use of SPH-based models. Such challenges include the model computational efficiency in terms of complexity of implementation. The SPH method allows water particles to be individually modeled, each with their own characteristics, which then accurately depicts the behavior and properties of the flow field. An open source code, known as SPHysics, was used to run the simulations presented in this thesis. Several cases analysed consist of hydraulic bores impacting a flat vertical wall as well as a sloping seawall. The analysis includes comparisons of the numerical results with published experimental data. The model is shown to accurately reproduce the formation of solitary waves as well as their propagation and breaking. The impacting bore profiles as well as the resulting pressures are also efficiently simulated using the model.

Currently, the accurate prediction of the impact of an extreme wave on infrastructure located near shore is difficult to assess. There is a lack of established methods to accurately quantify these impacts. Extreme waves, such as tsunamis generate, through breaking, extremely powerful hydraulic bores that impact and significantly damage coastal structures and buildings located close to the shoreline. The damage induced by such hydraulic bores is often due to structural failure. Examples of devastating coastal disasters are the 2004 Indian Ocean Tsunami, 2005 Hurricane Katrina and most recently, the 2011 Tohoku Japan Tsunami. As a result, more advanced research is needed to estimate the magnitude of forces exerted on structures by such bores. This research presents results of a numerical model based on the Smoothed Particle Hydrodynamics (SPH) method which is used to simulate the impact of extreme hydrodynamic forces on shore protection walls. Typically, fluids are modeled numerically based on a Lagrangian approach, an Eulerian approach or a combination of the two. Many of the common problems that arise from using more traditional techniques can be avoided through the use of SPH-based models. Such challenges include the model computational efficiency in terms of complexity of implementation. The SPH method allows water particles to be individually modeled, each with their own characteristics, which then accurately depicts the behavior and properties of the flow field. An open source code, known as SPHysics, was used to run the simulations presented in this thesis. Several cases analysed consist of hydraulic bores impacting a flat vertical wall as well as a sloping seawall. The analysis includes comparisons of the numerical results with published experimental data. The model is shown to accurately reproduce the formation of solitary waves as well as their propagation and breaking. The impacting bore profiles as well as the resulting pressures are also efficiently simulated using the model.