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arXiv · 2507.14881

An adaptive symplectic integrator for gravitational dynamics

Keqi Ye·Zizhe Cai·Mingji Wang·Kun Yang·Xiaodong Liu
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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.

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Keqi Ye, Zizhe Cai, Mingji Wang, Kun Yang, Xiaodong Liu. 2025-07-20. An adaptive symplectic integrator for gravitational dynamics. https://doi.org/10.1051/0004-6361%2F202451822

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