WHFast: A fast and unbiased implementation of a symplectic Wisdom-Holman integrator for long term gravitational simulations (1506.01084v1)

Abstract: We present WHFast, a fast and accurate implementation of a Wisdom-Holman symplectic integrator for long-term orbit integrations of planetary systems. WHFast is significantly faster and conserves energy better than all other Wisdom-Holman integrators tested. We achieve this by significantly improving the Kepler-solver and ensuring numerical stability of coordinate transformations to and from Jacobi coordinates. These refinements allow us to remove the linear secular trend in the energy error that is present in other implementations. For small enough timesteps we achieve Brouwer's law, i.e. the energy error is dominated by an unbiased random walk due to floating-point round-off errors. We implement symplectic correctors up to order eleven that significantly reduce the energy error. We also implement a symplectic tangent map for the variational equations. This allows us to efficiently calculate two widely used chaos indicators the Lyapunov characteristic number (LCN) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). WHFast is freely available as a flexible C package, as a shared library, and as an easy-to-use python module.

• The paper introduces WHFast, a new Wisdom-Holman integrator that significantly boosts computational speed and achieves unbiased energy error trends in long-term gravitational simulations.
• It refines the Kepler-solver and Jacobi coordinate transformations, incorporating high-order symplectic correctors up to the eleventh order to mitigate energy drift.
• The integrator efficiently computes chaos indicators like the Lyapunov characteristic number and MEGNO, making it essential for exploring complex planetary dynamics.

A Fast and Unbiased Wisdom-Holman Integrator for Gravitational Simulations

The paper by Hanno Rein and Daniel Tamayo addresses the development of a new, more efficient implementation of a symplectic Wisdom-Holman integrator. This integrator is designed for long-term simulations of gravitational systems, particularly planetary systems. The paper details substantial improvements in computational speed and accuracy, contributing to advances in celestial mechanics through numerical integration.

Core Contributions

The authors introduce an integrator described as significantly faster and more precise compared to existing Wisdom-Holman integrators. Their principal achievements include the refinement of the Kepler-solver and the enhancement of numerical stability in transformations to and from Jacobi coordinates. Notably, these improvements eliminate the linear secular trend in energy errors evident in prior implementations. For small timesteps, the integrator's energy error conforms to Brouwer's law, which indicates that it follows an unbiased random walk dominated by floating-point round-off errors.

In a novel development, this integrator includes symplectic correctors up to the eleventh order, which drastically minimize energy errors. Furthermore, the implementation of a symplectic tangent map enables the efficient computation of chaos indicators like the Lyapunov characteristic number (LCN) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). These innovations allow researchers to explore chaotic dynamics in planetary systems more effectively.

Numerical Benchmarks and Performance

Through extensive numerical testing, the integrator demonstrates remarkable accuracy in energy conservation and speed in execution compared to other popular integrators such as MERCURY, SWIFT, and RMVS. The relative energy error remains stable over extended periods, highlighting the long-term reliability of the integrator's numerical output even in scenarios of high eccentricity and significant interplanetary interactions. The careful handling of floating-point arithmetic ensures unbiased error accumulation and refrains from introducing systemic biases that can lead to rapid error growth.

Practical and Theoretical Implications

Practically, the integrator's ability to conserve energy over long timescales offers significant utility for simulations involving complex interplanetary dynamics, such as those in the Solar System. The incorporation of high-order symplectic correctors caters to systems with low mass ratios, enhancing the range of systems that can be accurately modeled using this method. The unbiased error growth will help prevent numerical artifacts from skewing the results of long-term dynamical studies.

Theoretically, the improvements pose implications for studies that analyze planetary stability and resonance phenomena, often sensitive to minute computational errors. The provision of chaos indicators with little additional computational cost opens new avenues for investigating the boundary between stability and chaos in dynamical systems.

Future Developments

Looking forward, the integrator sets a precedent in computational techniques for N-body simulations, encouraging further exploration into hybrid integrators, which can seamlessly transition between symplectic and non-symplectic methods during close encounters. The current implementation is freely accessible as a Python module, demonstrating a commitment to reproducibility and accessibility in scientific computing.

Overall, this integrator advances the capability to perform accurate, efficient long-term simulations of planetary systems, delivering practical benefits to researchers across celestial mechanics and beyond. This paper serves as a critical benchmark in the paper and application of numerical integration techniques within astrophysical research.