An adaptive symplectic integrator for gravitational dynamics

TL;DR

This paper introduces SQQ-PTQ, an adaptive symplectic integrator for gravitational dynamics that enhances accuracy, efficiency, and stability through Chebyshev interpolation, a projection method, and a quasi-Newton approach, demonstrated by numerical experiments.

Contribution

The paper develops a novel adaptive symplectic integrator, SQQ-PTQ, combining Chebyshev interpolation, a projection method, and Broyden's quasi-Newton method for improved gravitational simulations.

Findings

Demonstrates high accuracy and stability in numerical experiments.

Effectively handles close-encounter problems in gravitational dynamics.

Maintains energy conservation during long-term integrations.

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.