This thesis presents results aiming to enhance and broaden the applicability of the discontinuous Galerkin (''DG'') method in a variety of ways. DG was chosen as a foundation for this work because it yields high-order finite element discretizations with very favorable numerical properties for the treatment of hyperbolic conservation laws.In a first part, I examine progress that can be made on implementation aspects of DG. In adapting the method to mass-market massively parallel computation hardware in the form of graphics processors (''GPUs''), I obtain an increase in computation performance per unit of cost by more than an order of magnitude over conventional processor architectures. Key to this advance is a recipe that adapts DG to a variety of hardware through automated self-tuning. I discuss new parallel programming tools supporting GPU run-time code generation which are instrumental in the DG self-tuning process and contribute to its reaching application floating point throughput greater than 200 GFlops/s on a single GPU and greater than 3 TFlops/s on a 16-GPU cluster in simulations of electromagnetics problems in three dimensions. I further briefly discuss the solver infrastructure that makes this possible.In the second part of the thesis, I introduce a number of new numerical methods whose motivation is partly rooted in the opportunity created by GPU-DG: First, I construct and examine a novel GPU-capable shock detector, which, when used to control an artificial viscosity, helps stabilize DG computations in gas dynamics and a number of other fields. Second, I describe my pursuit of a method that allows the simulation of rarefied plasmas using a DG discretization of the electromagnetic field. Finally, I introduce new explicit multi-rate time integrators for ordinary differential equations with multiple time scales, with a focus on applicability to DG discretizations of time-dependent problems.
Kloeckner, Andreas P.,
"High-Performance High-Order Simulation of Wave and Plasma Phenomena"
(2010).
Applied Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0ST7N2W
