Techniques, Tricks and Algorithms for Efficient GPU-Based Processing of Higher Order Hyperbolic PDEs

TL;DR

This paper demonstrates that optimized GPU algorithms significantly accelerate higher order Godunov schemes across CFD, MHD, and CED applications, often outperforming lower order schemes and CPUs in speed and efficiency.

Contribution

It introduces optimized GPU techniques for higher order hyperbolic PDE solvers, showing superior performance and scalability across diverse scientific applications.

Findings

GPU algorithms achieve substantial speedups for higher order schemes

Higher order schemes can outperform lower order schemes on GPUs

GPUs outperform CPUs in real-world hyperbolic PDE applications

Abstract

GPU computing is expected to play an integral part in all modern Exascale supercomputers. It is also expected that higher order Godunov schemes will make up about a significant fraction of the application mix on such supercomputers. It is, therefore, very important to prepare the community of users of higher order schemes for hyperbolic PDEs for this emerging opportunity. We focus on three broad and high-impact areas where higher order Godunov schemes are used. The first area is computational fluid dynamics (CFD). The second is computational magnetohydrodynamics (MHD) which has an involution constraint that has to be mimetically preserved. The third is computational electrodynamics (CED) which has involution constraints and also extremely stiff source terms. Together, these three diverse uses of higher order Godunov methodology, cover many of the most important applications areas. In all three cases, we show that the optimal use of algorithms, techniques and tricks, along with the use of OpenACC, yields superlative speedups on GPUs! As a bonus, we find a most remarkable and desirable result: some higher order schemes, with their larger operations count per zone, show better speedup than lower order schemes on GPUs. In other words, the GPU is an optimal stratagem for overcoming the higher computational complexities of higher order schemes! Several avenues for future improvement have also been identified. A scalability study is presented for a real-world application using GPUs and comparable numbers of high-end multicore CPUs. It is found that GPUs offer a substantial performance benefit over comparable number of CPUs, especially when all the methods designed in this paper are used.