SIMD-vectorized implicit symplectic integrators can outperform explicit ones

TL;DR

This paper introduces a SIMD-vectorized implementation of a high-order implicit symplectic integrator that outperforms explicit methods in high-precision Hamiltonian ODE simulations.

Contribution

The work presents a SIMD-vectorized implementation of a 16th-order implicit Runge-Kutta integrator that significantly improves performance over existing explicit symplectic integrators.

Findings

IRKGL16-SIMD outperforms explicit integrators in high-precision tests

SIMD parallelism enhances the performance of the implicit integrator

Numerical experiments demonstrate efficiency gains in Hamiltonian system integrations

Abstract

The main purpose of this work is to present a SIMD-vectorized implementation of the symplectic 16th-order 8-stage implicit Runge-Kutta integrator based on collocation with Gauss-Legendre nodes (IRKGL16-SIMD), and to show that it can outperform state-of-the-art symplectic explicit integrators for high-precision numerical integrations (in double-precision floating-point arithmetic) of non-stiff Hamiltonian ODE systems. Our IRKGL16-SIMD integrator leverages Single Instruction Multiple Data (SIMD) based parallelism (in a way that is transparent to the user) to significantly enhance the performance of the sequential IRKGL16 implementation. We present numerical experiments comparing IRKGL16-SIMD with state-of-the-art high-order explicit symplectic methods for the numerical integration of several Hamiltonian systems in double-precision floating-point arithmetic.