High-Performance High-Order Simulation of Wave and Plasma Phenomena

of “High-Performance High-Order Simulation of Wave and Plasma Phenomena” by Andreas Klöckner, Ph.D., Brown University, May 2010 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.