Abstract:

Orbital dynamics is a fundamental problem in celestial mechanics, yet its governing equations are characterized by irrational terms and denominator-type nonlinearities. Traditional numerical methods, which are locally discrete, may lead to ambiguities near singularities (e.g., x=y=z=0), thereby limiting the numerical stability and accuracy. To address these challenges, we propose a Lie derivative algorithm that constructs discrete iterative schemes based on the Lie series expansion. Unlike local schemes, this approach discretizes the vector field in a global manner, effectively avoiding singular inconsistencies while ensuring stable long-term integration. Numerical experiments demonstrate that, when compared with a high-accuracy reference solution under uniform step-size settings, the proposed approach not only achieves higher accuracy but also improves computational efficiency by up to 47% in the two-body problem and 67% in the circular restricted three-body problem, relative to classical second-order schemes. These results indicate that the Lie derivative algorithm provides an efficient and practical alternative for high-precision orbital dynamics computations.

Key words: geometric numerical algorithm, Lie derivative, high efficiency, three-body problem