Enriched discrete spaces for time domain wave equations

The second order linear wave equation is simple in representation but its numerical approximation is challenging, especially when the system contains waves of high frequencies. While 10 grid points per wavelength is regarded as the rule of thumb to achieve tolerable approximation with the standard numerical approach, high resolution or high grid density is often required at high frequency which is often computationally demanding. As a contribution to tackling this problem, we consider in this thesis the discretization of the problem in the framework of the space-time discontinuous Galerkin (DG) method while investigating the solution in a finite dimensional space whose building blocks are waves themselves. The motivation for this approach is to reduce the number of degrees of freedom per wavelength as well as to introduce some analytical features of the problem into its numerical approximation. The developed space-time DG method is able to accommodate any polynomial bases. However, the Trefftz based space-time method proves to be efficient even for a system operating at high frequency. Comparison with polynomial spaces of total degree shows that equivalent orders of convergence are obtainable with fewer degrees of freedom. Moreover, the implementation of the Trefftz based method is cheaper as integration is restricted to the space-time mesh skeleton. We also extend our technique to a more complicated wave problem called the telegraph equation or the damped wave equation. The construction of the Trefftz space for this problem is not trivial. However, the exibility of the DG method enables us to use a special technique of propagating polynomial initial data using a wave-like solution (analytical) formula which gives us the required wave-like local solutions for the construction of the space. This thesis contains important a priori analysis as well as the convergence analysis for the developed space-time method, and extensive numerical experiments

On the capacity provisioning on dynamic networks

In this thesis, we consider the development of algorithms suitable for designing evacuation procedures in sparse or remote communities. The works are extensions of sink location problems on dynamic networks, which are motivated by real-life disaster events such as the Tohoku Japanese Tsunami, the Australian wildfire and many more. The available algorithms in this context consider the location of the sinks (safe-havens) with the assumptions that the evacuation by foot is possible, which is reasonable when immediate evacuation is needed in urban settings. However, for remote communities, emergency vehicles may need to be dispatched or situated strategically for an efficient evacuation process. With the assumption removed, our problems transform to the task of allocating capacities on the edges of dynamic networks given a budget capacity c. We first of all consider this problem on a dynamic path network of n vertices with the objective of minimizing the completion time (minmax criterion) given that the position of the sink is known. This leads to an O(nlogn + nlog(c/ξ)) time, where ξ is a refinement or precision parameter for an additional binary search in the worst case scenario. Next, we extend the problem to star topologies. The case where the sink is located at the middle of the star network follows the same approach for the path network. However, when the sink is located on a leaf node, the problem becomes more complicated when the number of links (edges) exceeds three. The second phase of this thesis focuses on allocating capacities on the edges of dynamic path networks with the objective of minimizing the total evacuation time (minsum criterion) given the position of the sink and the budget (fixed) capacity. In general, minsum problems are more difficult than minmax problems in the context of sink location problems. Due to few combinatorial properties discovered together with the possibility of changing objective. function configuration in the course of the optimization process, we consider the development of numerical procedure which involves the use of sequential quadratic programming (SQP). The sequential quadratic programming employed allows the specification of an arbitrary initial capacities and also helps in monitoring the changing configuration of the objective function. We propose to consider these problems on more complex topolgies such as trees and general graph in future.NSERC Discovery Grants program. University of Lethbridge Graduate Research Award. Alberta Innovates Awar