A novel energy-conserving scheme for eight-dimensional systems

TL;DR

This paper introduces a new energy-conserving implicit integration method for eight-dimensional Hamiltonian systems, demonstrating superior long-term energy preservation and accuracy over existing methods despite higher computational costs.

Contribution

The paper presents a novel, exactly energy-conserving implicit integrator for high-dimensional Hamiltonian systems, with a unique discretization approach for the Hamiltonian gradient.

Findings

Excellent energy conservation in simulations of physical systems.

Superior long-term accuracy compared to Runge-Kutta and symplectic methods.

Potential applications in astrophysics and electromagnetic modeling.

Abstract

We design a novel, exactly energy-conserving implicit non-symplectic integration method for an eight-dimensional Hamiltonian system with four degrees of freedom. In our algorithm, each partial derivative of the Hamiltonian with respect to one of phase-space variables is discretized by the average of eight Hamiltonian difference terms. Such a discretization form is a second-order approximation to the Hamiltonian gradient. It is shown numerically via simulations of an FPU-$\beta$ system and a post-Newtonian conservative system of compact binaries with one body spinning that the newly proposed method has extremely good energy-conserving performance, compared to the Runge-Kutta, implicit midpoint symplectic method and extended phase-space explicit symplectic-like integrators. The new method is advantageous over very long times and for large time steps compared to state-of-the-art Runge-Kutta method in the accuracy of numerical solutions. Although such an energy-conserving integrator exhibits a higher computational cost than any one of the other three algorithms, the superior results justify its use for satisfying some specific purposes on the preservation of energies in numerical simulations with much longer times, e.g. obtaining a high enough accuracy of the semimajor axis in a Keplerian problem in the Solar System or accurately grasping the frequency of a gravitational wave from a circular orbit in a post-Newtonian system of compact binaries. The new integrator will be potentially applied to model time-varying external electromagnetic fields or time-dependent spacetimes.