Performance and Scalability Improvements for Discontinuous Galerkin Solutions to Conservation Laws on Unstructured Grids

This paper presents a computational framework developed to improve both the serial and parallel performance of two dimensional, unstructured, discontinuous Galerkin (DG) solutions to hyperbolic conservation laws. The coding techniques employed factor in advancements trending in HPC technologies. They are designed to maximize loop vectorization, efficiently utilize cache, facilitate straightforward shared memory parallelization, reduce message passing volume, and increase the overlap between computation and communication. With today’s CPU technology and HPC networks rapidly evolving, it is important to quantitatively assess and compare these methodologies with standard paradigms in order to maximize current computational resources. In our benchmark studies, we specifically investigate the shallow water equations to show that the refactored algorithm implementation is able to provide a significant performance increase over the conventional elemental DG code structure in terms of both CPU time and parallel scalability. Our results show that the serial optimizations result in a 28–38 % performance increase. For parallel computations our improvements give rise to a 1.5–2.0 speedup factor for local problem sizes between 10 and 2000 elements per core, regardless of the overall problem size. The computational benchmarks were performed on the Lonestar and Stampede supercomputers at the Texas Advanced Computing Center.