We develop an accurate partial differential equation based methodology that predicts the time-optimal paths of autonomous vehicles navigating in any continuous, strong and dynamic ocean currents, obviating the need for heuristics. The goal is to predict a sequence of steering directions so that vehicles can best utilize or avoid currents to minimize their travel time. Inspired by the level set method, we derive and demonstrate that a modified level set equation governs the time-optimal path in any continuous flow. We show that our algorithm is computationally efficient and apply it to a number of experiments. First, we validate our approach through a simple benchmark application in a Rankine vortex flow for which an analytical solution is available. Next, we apply our methodology to more complex, simulated flow-fields such as unsteady double-gyre flows driven by wind stress and flows behind a circular island. These examples show that time-optimal paths for multiple vehicles can be planned, even in the presence of complex flows in domains with obstacles. Finally, we present, and support through illustrations, several remarks that describe specific features of our methodology.

This thesis develops and applies a procedure to generate high quality 2D meshes for any given ocean region with complex coastlines. The different criteria used in determining mesh element sizes for a given domain are discussed, especially sizing criteria that depend on local properties of the bathymetry and relevant dynamical scales. Two different smoothing techniques, Laplacian conditioning and targeted averaging, were applied to the fields involved in calculating the sizing matrix. The L^2 norm was used to quantify which technique had the greatest preservation of the original field. In both the reduced gradient and gradient cases, targeted averaging had a lower L^2 norm. The sizing matrices were used as inputs for two mesh generators, Distmesh and GMSH, and their meshing results were presented over a set of ocean domains in the Gulf of Maine and Massachusetts Bay region. Further research into the capabilities of each mesh generator are needed to provide a detailed evaluation. Mesh quality issues near coastlines revealed the need for small scale feature size recognition algorithms that could be implemented and studied in the future.