This report discusses three topics relating to errors of numerical methods and to improvements of numerical approximations. The introduction connects these topics to the secondary mathematics curriculum. The three chapters which follow develop the three selected topics: improving approximations of irrational numbers, error analysis of numerical integration methods, and discretization versus rounding error in Euler’s Method for solving ordinary differential equations. The conclusion describes specific national secondary mathematical standards and classroom activities relevant to numerical approximations and error analysis. / text

The increase in the availability and power of computational resources over the last fifteen years has contributed to the development of many different types of numerical optimization methods and created a large area of research focussed on numerical aerodynamic shape optimization and, more recently, high-fidelity multidisciplinary optimization. Numerical optimization provides dramatic savings when designing new aerodynamic configurations, as it allows the designer to focus more on the development of a well-posed design problem rather than on performing an exhaustive search of the design space via the traditional cut-and-try approach, which is expensive and time-consuming. It also reduces the dependence on the designer’s experience and intuition, which can potentially lead to more optimal designs. Numerical optimization methods are particularly attractive when designing novel, unconventional aircraft for which the designer has no pre-existing studies or experiences from which to draw; these methods have the potential to discover new designs that might never have been arrived at without optimization. This work presents an extension of an efficient gradient-based numerical aerodynamic shape optimization algorithm to enable optimization in turbulent flow. The algorithm includes an integrated geometry parameterization and mesh movement scheme, an efficient parallel Newton-Krylov-Schur algorithm for solving the Reynolds-Averaged Navier-Stokes (RANS) equations, which are fully coupled with the one-equation Spalart-Allmaras turbulence model, and a discrete-adjoint gradient evaluation. In order to develop an efficient methodology for optimization in turbulent flows, the viscous and turbulent terms in the ii governing equations were linearized by hand. Additionally, a set of mesh refinement tools was introduced in order to obtain both an acceptable control volume mesh and a sufficiently refined computational mesh from an initial coarse mesh. A series of drag minimization studies was carried out which show that the algorithm is able to maintain robustness in the mesh movement and flow analysis in the presence of large shape changes, an important requirement for performing exploratory optimizations aiming to discover novel configurations and for multidisciplinary optimization. Additionally, the algorithm is able to find incremental improvements when given well-designed initial planar and nonplanar geometries. A comparison of Euler-based and RANS-based optimizations highlights the importance of considering viscous and turbulent effects. A multi-point optimization demonstrates that the algorithm is able to address practical aerodynamic design problems.

The increase in the availability and power of computational resources over the last fifteen years has contributed to the development of many different types of numerical optimization methods and created a large area of research focussed on numerical aerodynamic shape optimization and, more recently, high-fidelity multidisciplinary optimization. Numerical optimization provides dramatic savings when designing new aerodynamic configurations, as it allows the designer to focus more on the development of a well-posed design problem rather than on performing an exhaustive search of the design space via the traditional cut-and-try approach, which is expensive and time-consuming. It also reduces the dependence on the designer’s experience and intuition, which can potentially lead to more optimal designs. Numerical optimization methods are particularly attractive when designing novel, unconventional aircraft for which the designer has no pre-existing studies or experiences from which to draw; these methods have the potential to discover new designs that might never have been arrived at without optimization. This work presents an extension of an efficient gradient-based numerical aerodynamic shape optimization algorithm to enable optimization in turbulent flow. The algorithm includes an integrated geometry parameterization and mesh movement scheme, an efficient parallel Newton-Krylov-Schur algorithm for solving the Reynolds-Averaged Navier-Stokes (RANS) equations, which are fully coupled with the one-equation Spalart-Allmaras turbulence model, and a discrete-adjoint gradient evaluation. In order to develop an efficient methodology for optimization in turbulent flows, the viscous and turbulent terms in the ii governing equations were linearized by hand. Additionally, a set of mesh refinement tools was introduced in order to obtain both an acceptable control volume mesh and a sufficiently refined computational mesh from an initial coarse mesh. A series of drag minimization studies was carried out which show that the algorithm is able to maintain robustness in the mesh movement and flow analysis in the presence of large shape changes, an important requirement for performing exploratory optimizations aiming to discover novel configurations and for multidisciplinary optimization. Additionally, the algorithm is able to find incremental improvements when given well-designed initial planar and nonplanar geometries. A comparison of Euler-based and RANS-based optimizations highlights the importance of considering viscous and turbulent effects. A multi-point optimization demonstrates that the algorithm is able to address practical aerodynamic design problems.

Sea-ice ecosystem modelling is a novel field of research. In this thesis, the main organism studied is sea-ice algae. A basic introduction to algae and its importance in the aquatic food web is given first. An introduction to modeling and its purposes is presented, and this is followed by a brief description of ice algae models in practice with some physical conditions which influence ecosystem modelling. In the following Chapter, a simple mathematical model to represent the algae population is derived, and analyzed using pseudo spectral numerical methods implemented with MATLAB. The behaviour of the algae population and the boundary layers are discussed by examining the numerical results. Perturbation and asymptotic analysis is used for further analysis of the system using Maple. In the following Chapter a Nutrient Phytoplankton Zooplankton Detritus (or NPZD) model, which is a commonly used type of model in marine ecosystem modelling, is developed based on the framework of Soetaert and Herman. The model is examined under five different experimental setups (herein we mean numerical experiments) and the results are discussed. The NPZD model implemented is compared with a well-studied model in the literature. Our model can be considered somewhat simpler than other models in the literature (though it still has a much larger parameter space than the idealized model discussed in the previous Chapter). Finally we discuss future directions for research.

We study the effects of loop quantum gravity motivated corrections in classical systems. Computational methods are used to simulate black hole formation from the gravitational collapse of a massless scalar field in Painleve-Gullstrand coordinates. Singularities present in the classical case are resolved by a radiation-like phase in the quantum collapse. The evaporation is not complete but leaves behind an outward moving shell of mass that disperses to infinity. We reproduce Choptuik scaling showing the usual behaviour for the curvature scaling, while observing previously unseen behaviour in the mass scaling. The quantum corrections are found to impose a lower limit on black hole mass and generate a new universal power law scaling relationship. In a parallel study, we quantize the Hamiltonian for a particle in the singular $1/r^2$ potential, a form that appears frequently in black hole physics. In addition to conventional Schrodinger methods, the quantization is performed using full and semiclassical polymerization. The various quantization schemes are in excellent agreement for the highly excited states but differ for the low-lying states, and the polymer spectrum is bounded below even when the Schrodinger spectrum is not.

Herein a fully parallel, upwind and flux-split Finite-Volume Time-Domain (FVTD) numerical engine for solving Maxwell's Equations on unstructured grids is developed. The required background theory for solving Maxwell's Equations using FVTD is given in sufficient detail, including a description of both the temporal and spatial approximations used. The details of the local-time stepping strategy of Fumeaux et al. is included. A global mesh-truncation scheme using field integration over a Huygens' surface is also presented. The capabilities of the FVTD algorithm are augmented with thin-wire and subcell circuit models that permit very flexible and accurate simulations of circuit-driven wire structures. Numerical and experimental validation shows that the proposed models have a wide-range of applications. Specifically, it appears that the thin-wire and subcell circuit models may be very well suited to the simulation of radio-frequency coils used in magnetic resonance imaging systems. A parallelization scheme for the volumetric field solver, combined with the local-time stepping, global mesh-truncation and subcell models is developed that theoretically provides both linear time- and memory scaling in a distributed parallel environment. Finally, the FVTD code is converted to the frequency domain and the possibility of using different flux-reconstruction schemes to improve the iterative convergence of the Finite-Volume Frequency-Domain algorithm is investigated.

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We study the effects of loop quantum gravity motivated corrections in classical systems. Computational methods are used to simulate black hole formation from the gravitational collapse of a massless scalar field in Painleve-Gullstrand coordinates. Singularities present in the classical case are resolved by a radiation-like phase in the quantum collapse. The evaporation is not complete but leaves behind an outward moving shell of mass that disperses to infinity. We reproduce Choptuik scaling showing the usual behaviour for the curvature scaling, while observing previously unseen behaviour in the mass scaling. The quantum corrections are found to impose a lower limit on black hole mass and generate a new universal power law scaling relationship. In a parallel study, we quantize the Hamiltonian for a particle in the singular $1/r^2$ potential, a form that appears frequently in black hole physics. In addition to conventional Schrodinger methods, the quantization is performed using full and semiclassical polymerization. The various quantization schemes are in excellent agreement for the highly excited states but differ for the low-lying states, and the polymer spectrum is bounded below even when the Schrodinger spectrum is not.

Herein a fully parallel, upwind and flux-split Finite-Volume Time-Domain (FVTD) numerical engine for solving Maxwell's Equations on unstructured grids is developed. The required background theory for solving Maxwell's Equations using FVTD is given in sufficient detail, including a description of both the temporal and spatial approximations used. The details of the local-time stepping strategy of Fumeaux et al. is included. A global mesh-truncation scheme using field integration over a Huygens' surface is also presented. The capabilities of the FVTD algorithm are augmented with thin-wire and subcell circuit models that permit very flexible and accurate simulations of circuit-driven wire structures. Numerical and experimental validation shows that the proposed models have a wide-range of applications. Specifically, it appears that the thin-wire and subcell circuit models may be very well suited to the simulation of radio-frequency coils used in magnetic resonance imaging systems. A parallelization scheme for the volumetric field solver, combined with the local-time stepping, global mesh-truncation and subcell models is developed that theoretically provides both linear time- and memory scaling in a distributed parallel environment. Finally, the FVTD code is converted to the frequency domain and the possibility of using different flux-reconstruction schemes to improve the iterative convergence of the Finite-Volume Frequency-Domain algorithm is investigated.

In Part I of this thesis we extend the Lanchester Ordinary Differential Equations and construct a new physically meaningful set of partial differential equations with the aim of more realistically representing soldier dynamics in order to enable a deeper understanding of the nature of conflict. Spatial force movement and troop interaction components are represented with both local and non-local terms, using techniques developed in biological aggregation modelling. A highly accurate flux limiter numerical method ensuring positivity and mass conservation is used, addressing the difficulties of inadequate methods used in previous research. We are able to reproduce crucial behaviour such as the emergence of cohesive density profiles and troop regrouping after suffering losses in both one and two dimensions which has not been previously achieved in continuous combat modelling. In Part II, we reproduce for the first time apparently complex cellular automaton behaviour with simple partial differential equations, providing an alternate mechanism through which to analyse this behaviour. Our PDE model easily explains behaviour observed in selected scenarios of the cellular automaton wargame ISAAC without resorting to anthropomorphisation of autonomous 'agents'. The insinuation that agents have a reasoning and planning ability is replaced with a deterministic numerical approximation which encapsulates basic motivational factors and demonstrates a variety of spatial behaviours approximating the mean behaviour of the ISAAC scenarios. All scenarios presented here highlight the dangers associated with attributing intelligent reasoning to behaviour shown, when this can be explained quite simply through the effects of the terms in our equations. A continuum of forces is able to behave in a manner similar to a collection of individual autonomous agents, and shows decentralised self-organisation and adaptation of tactics to suit a variety of combat situations. We illustrate the ability of our model to incorporate new tactics through the example of introducing a density tactic, and suggest areas for further research.