The use of Kepler solver in numerical integrations of quasi-Keplerian orbits

Abstract

<jats:title>ABSTRACT</jats:title> <jats:p>A Kepler solver is an analytical method used to solve a two-body problem. In this paper, we propose a new correction method by slightly modifying the Kepler solver. The only change to the analytical solutions is that the obtainment of the eccentric anomaly relies on the true anomaly that is associated with a unit radial vector calculated by an integrator. This scheme rigorously conserves all integrals and orbital elements except the mean longitude. However, the Kepler energy, angular momentum vector, and Laplace–Runge–Lenz vector for perturbed Kepler problems are slowly varying quantities. However, their integral invariant relations give the quantities high-precision values that directly govern five slowly varying orbital elements. These elements combined with the eccentric anomaly determine the desired numerical solutions. The newly proposed method can considerably reduce various errors for a post-Newtonian two-body problem compared with an uncorrected integrator, making it suitable for a dissipative two-body problem. Spurious secular changes of some elements or quasi-integrals in the outer Solar system may be caused by short integration times of the fourth-order Runge–Kutta algorithm. However, they can be eliminated in a long integration time of 108 yr by the proposed method, similar to Wisdom–Holman second-order symplectic integrator. The proposed method has an advantage over the symplectic algorithm in the accuracy but gives a larger slope to the phase error growth.</jats:p>

Construction of explicit symplectic integrators in general relativity. IV. Kerr black holes

“…This is an advantage of a symplectic scheme in long-term integrations. In particular, Deng et al (2020) reported that RK4 combined with manifold correction exhibits poorer performance than the second-order symplectic leapfrog in a 10 8 year integration of the outer solar system involving the Sun and four outer planets. In addition, Figures 2 (a) and (c) clearly show that the new fourth-order explicit symplectic integrator S4 is two orders of magnitude higher than RK4 in accuracy.…”

“…However, it would yield a secular drift in energy errors and would provide unreliable numerical results for a long-term integration. The manifold correction of Nacozy (1971) and its extensions (Fukushima 2003;Wu et al 2007;Ma et al 2008;Wang et al 2016;Wang et al 2018;Deng et al 2020) are helpful to compensate for the defect of RK4 by pulling the integrated orbit back to the original integral hypersurface.…”

Construction of explicit symplectic integrators in general relativity. III. Reissner-Nordstrom-(anti)-de Sitter black holes

“…2 The preservation of constants of motion (such as energy integral) in symplectic integrators does not mean that the constants are exactly conserved, but means that errors of the constants are bounded. The energy-preserving integrators (Bacchini et al 2018a, b;Hu et al 2019Hu et al , 2021 and the projection methods (Fukushima 2003a(Fukushima ,b,c, 2004Ma et al 2008;Wang et al 2016Wang et al , 2018Deng et al 2020) are not symplectic, whereas they do exactly conserve energy.…”

Construction of explicit symplectic integrators in general relativity. II. Reissner-Nordstrom black holes

“…The integrals of motion (e.g., energy integral) along the trajectory are nearly conserved for the structure preserving integrators, but their errors increase linearly with time for the standard integration schemes. In addition, the overall phase error only grows linearly with time for the former algorithms, whereas it is normally proportional to the square of the length of the integration interval for the latter schemes (Deng et al 2020). The above-mentioned curved spacetimes can be expressed in terms of Hamiltonian systems, and thus symplectic integrators are naturally chosen.…”