In this thesis, four different numerical techniques are implemented for the purpose of simulating resonant tunnelling diodes (RTDs). The chosen methods were: piecewise constant transfer matrix (TMM-C), piecewise linear transfer matrix (TMM-L), quantum transmitting boundary method (QTBM), and the Crank-Nicolson method (CN). The numerical methods converged compared with the known analytic simulation for plane waves tunnelling through a single barrier. To better represent the semiconductor-based RTDs, the effective mass approximation was adopted with accompanying modifications to the Hamiltonian operators to ensure the continuity of the wave function and its derivative. Using the Tsu-Esaki formula, the current density was calculated as a function of bias voltage for two different RTD devices. The numerically obtained current density was of the same order of magnitude as referenced experimental values but differed significantly enough to require better models if engineering applications are decided. The models in this thesis were able to display resonant tunnelling and a negative differential resistance (NDR), giving them plausible educational value.

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Dictyostelium discoideum (Dd) is a model organism, studied for reasons from cell movement to chemotaxis to human disease control. Creating a computer model of the life cycle of Dd has garnered great interest, one part of which is the Aggregation Stage, where thousands of amoeba gather together to form a slug. Chemotaxis is the mechanism through which this is accomplished. This thesis develops two- and three-dimensional alternating direction implicit code which solves the diffusion equation on an adaptive grid. The calculated values for both two and three dimensions are checked against the actual solution and error results are provided. Comparisons are made between the coarse grid with refinement case and a fine grid without refinement case. Also, a non-negativity condition for two dimensions is derived to give a bound on the three major parameters: the diffusion coefficient and the spatial and time discretizations.

An unsteady, finite-rate, one-dimensional model has been developed for the analysis for gas-turbine combustors. The basis of the model is the one-dimensional, integral form of the conservation equations for multi-species, non-equilibrium, reacting mixtures. Special procedures were devised for the flow-division of the inlet flow into primary- and annular-flows, for both straight- and reverse-flow combustors. This allows the model to handle complete combustor configurations, which at present are beyond the reach of more sophisticated CFD tools. The model was validated with a steady-state analytical solution for a basic problem, and with steady-state results from a production code applied to a production combustor. Additional calculations show the ability of the code to predict blow-out due to rich and lean mixtures, and to predict the response of a combustor to perturbations in operating and boundary conditions. / Ph. D.

Many complexities arise when writing software for computational physics. The choice of underlying data structures, physics model representation, and numerical methods used for the solver all add to the overall complexity of a code and significantly affect the simulation speed and accuracy of the solution. This work has integrated multiple recently developed software tools into a unified framework called ELEMENTS. ELEMENTS contains tools to address the complexities of data representation and numerical methods implementation for computational physics applications. ELEMENTS consists of multiple software packages: Elements, MATAR, Swage, Geometry, and SLAM. MATAR is a performance portability and productivity implementation of data-oriented design that leverages KOKKOS for multi-architecture portability. MATAR's data-oriented design allows for highly efficient memory use through the use of contiguous memory allocation and access for optimal performance. The elements library contains the requisite mathematical functions for a wide range of numerical methods and high order field representation, including the Serendipity basis set that allows for a higher-order solution with fewer degrees of freedom than the more standard tensor product elements. Swage is a novel mesh class capable of representing all of the geometric entities required to implement low and high-order continuous and discontinuous Galerkin methods on unstructured hexahedral meshes as well as connectivity structures between the disparate index spaces. SLAM is a library for linear algebra solvers and tools for linking to external solver packages. Combining these tools allows for the research and development of novel methods for solving problems in computational physics. This work discusses the ELEMENTS package and reviews multiple numerical methods built using ELEMENTS.

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In this thesis we present TREVR (Tree-based Reverse Ray Tracing), a general algo- rithm for computing the radiation field, including absorption, in astrophysical sim- ulations. TREVR is designed to handle large numbers of sources and absorbers; it is based on a tree data structure and is thus suited to codes that use trees for their gravity or hydrodynamics solvers (e.g. Adaptive Mesh Refinement). It achieves com- putational speed while maintaining a specified accuracy via controlled lowering of resolution of both sources and rays from each source. TREVR computes the radiation field in O(N log(N)) time without absorption and O (Nlog(N)log(N)) time with absorption. These claims are substantiated by mathematically predicting and testing the algorithm’s general scaling. The scalings arise from merging sources of radiation according to an opening angle criterion and walking the tree structure to trace a ray to a depth that gives the chosen accuracy for absorption. The absorption-depth refinement criterion is unique to TREVR and is presented here for the first time. We provide a suite of tests demonstrating the algorithm’s ability to accurately compute fluxes, ionization fronts and shadows. Two novel test cases are presented here for the first time as part of this suite. / Thesis / Master of Science (MSc) / In this thesis we present TREVR (Tree-based Reverse Ray Tracing), a general method for computing the effects of of radiation in astrophysical simulations.

The problem of rough surface scattering and propagation over rough terrain in ducting environments has been receiving considerable attention in the literature. One popular method of modeling this problem is the parabolic wave equation (PWE) method. In this method, the Helmholtz wave equation is replaced by a PWE under the assumption of predominant forward propagation and scattering. The resulting PWE subjected to the appropriate boundary condition(s) is then solved, given an initial field distribution, using marching techniques such as the split-step Fourier algorithm. As is obvious from the assumption on which it is based, the accuracy of the PWE approximation deteriorates in situations involving appreciable scattering away from the near-forward direction, i.e. when the terrain under consideration is considerably rough. The backscattered field is neglected in all PWE-based models. An alternative and more rigorous method for modeling the problem under consideration is the boundary integral equation (BIE) method, which is formulated in two steps. The first step involves setting up an integral equation (the magnetic field integral equation, MFIE, or the electric field integral equation EFIE) governing currents induced on the rough surface by the incident field and solving for these currents numerically. The resulting currents are then used in the appropriate radiation integrals to calculate the field scattered by the surface everywhere in space. The BIE method accounts for all orders of multiple scattering on the rough surface and predicts the scattered field in all directions in space (including the backscattering direction) in an exact manner. In homogeneous media, the implementation of the BIE approach is straightforward since the kernel (Green's function or its normal derivative) which appears in the integral equation and the radiation integrals is well known. This is not the case, however, in inhomogeneous media (ducting environments) where the Green's function is not readily known. Due to this fact, there has been no attempt, up to our knowledge, at using the BIE (except under the parabolic approximation) to model the problem under consideration prior to the work presented in this thesis. In this thesis, a closed-form approximation of the Green's function for a two-dimensional ducting environment formed by the presence of a linear-square refractivity profile is derived using the asymptotic methods of stationary phase and steepest descents. This Green's function is then modified to more closely model the one associated with a physical ducting medium, in which the refractivity profile decreases up to a certain height, beyond which it becomes constant. This modified Green's function is then used in the BIE approach to study low grazing angle (LGA) propagation over rough surfaces in the aforementioned ducting environment. The numerical method used to solve the MFIE governing the surface currents is MOMI, which is a very robust and efficient method that does not require matrix storage or inversion. The proposed method is meant as a benchmark for people studying forward propagation over rough surfaces using the parabolic wave equation (PWE). Rough surface scattering results obtained via the PWE/split-step approach are compared to those obtained via the BIE/MOMI approach in ducting environments. These comparisons clearly show the shortcomings of the PWE/split-step approach. / Ph. D.

The Burgers equation, and related partial differential equations (PDEs), can be numerically challenging for small values of the viscosity parameter. For example, these equations can develop discontinuous solutions (or solutions with large gradients) from smooth initial data. Aside from numerical stability issues, standard numerical methods can also give rise to spurious oscillations near these discontinuities. In this study, we consider an equivalent form of the Burgers equation given by Constantin and Iyer, whose solution can be written as the expected value of a stochastic differential equation. This equivalence is used to develop a numerical method for approximating solutions to Burgers equation. Our preliminary analysis of the algorithm reveals that it is a natural generalization of the method of characteristics and that it produces approximate solutions that actually improve as the viscosity parameter vanishes. We present three examples that compare our algorithm to a recently published reference method as well as the vanishing viscosity/entropy solution for decreasing values of the viscosity. / Master of Science / Burgers equation is a Partial Differential Equation (PDE) used to model how fluids evolve in time based on some initial condition and viscosity parameter. This viscosity parameter helps describe how the energy in a fluid dissipates. When studying partial differential equations, it is often hard to find a closed form solution to the problem, so we often approximate the solution with numerical methods. As our viscosity parameter approaches 0, many numerical methods develop problems and may no longer accurately compute the solution. Using random variables, we develop an approximation algorithm and test our numerical method on various types of initial conditions with small viscosity coefficients.