An adaptive symplectic integrator for gravitational dynamics

¹ School of Aeronautics and Astronautics, Shenzhen Campus of Sun Yat-sen University, Shenzhen, Guangdong 518107, China
² Shenzhen Key Laboratory of Intelligent Microsatellite Constellation, Shenzhen Campus of Sun Yat-sen University, Shenzhen, Guangdong 518107, China
³ School of Mechanics and Aerospace Engineering, State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, Dalian University of Technology, Dalian, Liaoning 116081, China

^⋆ Corresponding author: liuxd36@mail.sysu.edu.cn

Received: 7 August 2024
Accepted: 3 June 2025

Abstract

This paper presents an adaptive symplectic integrator, SQQ-PTQ, developed on the basis of the fixed-step symplectic integrator SQQ. To mitigate the Runge phenomenon, SQQ-PTQ employs Chebyshev interpolation for approximating the action, enhancing both the precision and stability of the interpolation. In addition, to reduce the computational cost of evaluating interpolation functions, SQQ-PTQ introduces a projection method that improves the efficiency of these computations. A key feature of SQQ-PTQ is its use of the time transformation to implement an adaptive time step. To address the challenge of computing complicated Jacobian matrices attributed to the time transformation, SQQ-PTQ adopts a quasi-Newton method based on Broyden’s method. This strategy accelerates the solution of nonlinear equations, thereby improving the overall computational performance. The effectiveness and robustness of SQQ-PTQ are demonstrated via three numerical experiments. In particular, SQQ-PTQ demonstrates adaptability in handling close-encounter problems. Moreover, during long-term integrations, SQQ-PTQ maintains the energy conservation, further confirming its advantages as a symplectic algorithm.