No abstract

National parks often serve as hotspots for environmental crime such as illegal deforestation and animal poaching. Previous attempts to model environmental crime were either discrete and network-based or required very restrictive assumptions on the geometry of the protected region and made heavy use of radial symmetry. We formulate a level set method to track criminals inside a protected region which uses real elevation data to determine speed of travel, does not require any assumptions of symmetry, and can be applied to regions of arbitrary shape. In doing so, we design a Hamilton-Jacobi equation to describe movement of criminals while also incorporating the effects of patrollers who attempt to deter the crime. We discuss the numerical schemes that we use to solve this Hamilton-Jacobi equation. Finally, we apply our method to Yosemite National Park and Kangaroo Island, Australia and design practical patrol strategies with the goal of minimizing the area that is affected by criminal activity.

A grid-less, fully implicit, spectrally accurate algorithm for solving three-dimensional, both stationary and time-dependent, heat conduction problems in slots formed by either fixed or time-dependent boundaries has been developed. The algorithm is based on the concept of immersed boundary conditions (IBC), where the physical domain is immersed within the computational domain and the boundary conditions take the form of internal constraints. The IBC method avoids the need to construct adaptive, time-dependent grids resulting in the reduction of the required computational resources and, at the same time, maintaining accurate information about the location of the boundaries. The algorithm is spectrally accurate in space and capable of delivering first-, second-, third-and fourth-order accuracy in time. Given a potentially large size of the resultant linear algebraic system, various methods that take advantage of the special structure of the coefficient matrix have been explored in search for an efficient solver, including a specialized direct solver as well as serial and parallel iterative solvers. The specialized direct solver has been found to be the most efficient from the viewpoints of the speed of the computations and the memory requirements.IBC METHOD FOR HEAT CONDUCTION PROBLEMS 479 tracking one follows motion of a set of points with the surface represented through interpolation between these points and, in this sense, accurate information about the location of the boundary is always available [2,3]. On the other hand, in volume tracking one does not store information about the location of the boundary but reconstructs its position when needed. The reconstruction is based upon the presence of special markers within a specified region, e.g. MAC Markers and Cell [4], VOF [5,6], and Level Set [7,8]. As a result, the location of the boundary is not known precisely. Adaptive grid methods reconcile the problem of the MB by overlapping one of the grid lines with the boundary. This means that at each time step the entire grid must be recomputed, a costly venture. Mapping methods bypass numerical grid generation but their applicability is limited to special cases only.Lagrangian methods use a coordinate system that moves with the fluid. By doing so, each computational cell contains the same fluid and the location of the boundary is always known accurately. Although Lagrangian methods are well suited to MB problems, they can suffer from mesh tangling and loss of numerical accuracy arising from highly distorted meshes (see Reference [1] for potential remedies).A different approach for solving the MB problems, generally referred to as the immersed boundaries (IB) method, relies on extending the computational domain in such a way that it completely engulfs the physical domain during simulation. The concept, which was first proposed in the context of cardiac mechanics problems by Peskin [9], is very attractive as one can work with a fixed, regular computational domain regardless of the shape of the physical domain so that...