Parallelizing Explicit and Implicit Extrapolation Methods for Ordinary Differential Equations

TL;DR

This paper introduces parallelized extrapolation methods for solving ODEs that exploit within-method parallelism, achieving significant speedups for implicit methods on multicore CPUs, but not for explicit methods.

Contribution

It presents the first open-source within-method parallelization of extrapolation ODE solvers, demonstrating substantial performance gains for implicit methods on standard hardware.

Findings

Implicit parallel extrapolation methods achieve 2x-4x speedup on multicore CPUs.

Explicit parallel extrapolation methods show no significant improvement over existing methods.

The software is the first widely available open-source implementation of within-method parallel ODE solvers.

Abstract

Numerically solving ordinary differential equations (ODEs) is a naturally serial process and as a result the vast majority of ODE solver software are serial. In this manuscript we developed a set of parallelized ODE solvers using extrapolation methods which exploit "parallelism within the method" so that arbitrary user ODEs can be parallelized. We describe the specific choices made in the implementation of the explicit and implicit extrapolation methods which allow for generating low overhead static schedules to then exploit with optimized multi-threaded implementations. We demonstrate that while the multi-threading gives a noticeable acceleration on both explicit and implicit problems, the explicit parallel extrapolation methods gave no significant improvement over state-of-the-art even with a multi-threading advantage against current optimized high order Runge-Kutta tableaus. However, we demonstrate that the implicit parallel extrapolation methods are able to achieve state-of-the-art performance (2x-4x) on standard multicore x86 CPUs for systems of $<200$ stiff ODEs solved at low tolerance, a typical setup for a vast majority of users of high level language equation solver suites. The resulting method is distributed as the first widely available open source software for within-method parallel acceleration targeting typical modest compute architectures.