Symplectic integrators have made it possible to study the long-term evolution of planetary systems with direct N-body simulations. In this paper we reassess the accuracy of such simulations by running a convergence test on 20Myr integrations of the Solar System using various symplectic integrators.
We find that the specific choice of metric for determining a simulation's accuracy is important. Only looking at metrics related to integrals of motions such as the energy error can overestimate the accuracy of a method. As one specific example, we show that symplectic correctors do not improve the accuracy of secular frequencies compared to the standard Wisdom-Holman method without symplectic correctors, despite the fact that the energy error is three orders of magnitudes smaller. We present a framework to trace the origin of this apparent paradox to one term in the shadow Hamiltonian. Specifically, we find a term that leads to negligible contributions to the energy error but introduces non-oscillatory errors that result in artificial periastron precession. This term is the dominant error when determining secular frequencies of the system. We show that higher order symplectic methods such as the Wisdom-Holman method with a modified kernel or the SABAC family of integrators perform significantly better in secularly evolving systems because they remove this specific term.
We describe a parallel hybrid symplectic integrator for planetary system integration that runs on a graphics processing unit (GPU). The integrator identifies close approaches between particles and switches from symplectic to Hermite algorithms for particles that require higher resolution integrations. The integrator is approximately as accurate as other hybrid symplectic integrators but is GPU accelerated.
Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by using a high order integration scheme for the duration of the encounter while otherwise using a standard 2nd order Wisdom-Holman scheme, thereby optimizing both speed and accuracy. In this paper we reassess the criteria for choosing the switching function that determines which parts of the Hamiltonian are integrated with the high order integrator. We show that t…
Symplectic integrators are the preferred method of solving conservative $N$-body problems in cosmological, stellar cluster, and planetary system simulations because of their superior error properties and ability to compute orbital stability. Newtonian gravity is scale free, and there is no preferred time or length scale: this is at odds with construction of traditional symplectic integrators, in which there is an explicit timescale in the time-step. Additional timescales have been incorporated into symplectic integration using various techniques, such as hybrid methods and potential decomposit…
Direct N-body simulations and symplectic integrators are effective tools to study the long-term evolution of planetary systems. The Wisdom-Holman (WH) integrator in particular has been used extensively in planetary dynamics as it allows for large timesteps at good accuracy. One can extend the WH method to achieve even higher accuracy using several different approaches. In this paper we survey integrators developed by Wisdom et al. (1996), Laskar & Robutel (2001, and Blanes et al. (2013). Since some of these methods are harder to implement and not as readily available to astronomers compared to…
The symplectic Wisdom-Holman map revolutionized long-term integrations of planetary systems. There is freedom in such methods of how to split the Hamiltonian and which coordinate system to employ, and several options have been proposed in the literature. These choices lead to different integration errors, which we study analytically and numerically. The Wisdom-Holman method in Jacobi coordinates and the method of Hernandez, H16, compare favorably and avoid problems of some of the other maps, such as incorrect center-of-mass position or truncation errors even in the one-planet case. We use H16 …
The numerical modeling of co-existing circumplanetary disks/rings and satellites is particularly challenging because each part of the system requires a very different approach. Disks are generally well represented by a fluid-like dense medium, whose evolution can be calculated by a hydrocode. On the other hand, the orbital evolution of satellites is generally performed using N-body integrators. We have developed a new numerical model that combines a 1-dimensional hydrocode with the N-body integrator SyMBA. The disk evolves due to its viscosity, and resonant torques from satellites. The latter …
We present a new mixed variable symplectic (MVS) integrator for planetary systems, that fully resolve close encounters. The method is based on a time regularisation that allows keeping the stability properties of the symplectic integrators, while also reducing the effective step size whenever two planets encounter. We use a high order MVS scheme such that it is possible to integrate with large time steps far away from close encounters. We show that this algorithm is able to resolve almost exact collisions (i.e with a mutual separation of a fraction of the physical radius) while using the same …
Hill's equations are an approximation that is useful in a number of areas of astrophysics including planetary rings and planetesimal disks. We derive a symplectic method for integrating Hill's equations based on a generalized leapfrog. This method is implemented in the parallel N-body code, PKDGRAV and tested on some simple orbits. The method demonstrates a lack of secular changes in orbital elements, making it a very useful technique for integrating Hill's equations over many dynamical times. Furthermore, the method allows for efficient collision searching using linear extrapolation of partic…
High precision Symplectic Integrators for the Solar System
Astronomie et Systèmes Dynamiques, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC（巴黎天文动力学与系统研究所）；Instituto de Matemática Multidisciplinar, Universitat Politécnica de València（瓦伦西亚理工大学多学科数学研究所）；Institut de Matemàtiques i Aplicacions de Castelló and Departament de Matemàtiques, Universitat Jaume I（卡斯特利翁数学与应用研究所及马德里卡洛斯三世大学数学系）；Konputazio Zientziak eta A.A. saila, Informatika Fakultatea, EHU/UPV（巴斯克大学计算机学院）
An accurate and efficient method dealing with the few-body dynamics is important for simulating collisional N-body systems like star clusters and to follow the formation and evolution of compact binaries. We describe such a method which combines the time-transformed explicit symplectic integrator (Preto & Tremaine 1999; Mikkola & Tanikawa 1999) and the slow-down method (Mikkola & Aarseth 1996). The former conserves the Hamiltonian and the angular momentum for a long-term evolution, while the latter significantly reduces the computational cost for a weakly perturbed binary. In this work, the Ha…
A planetary instability occurring at time $<100$ My after formation of the giant planets in our solar system can be responsible for some characteristics of the inner solar system. However, the actual influence of the instability on the terrestrial planet formation is not well understood. The simulations of terrestrial planet formation are very CPU-expensive, and this limits the exploration of different instability scenarios. To include the effects of the giant planets instability in the simulations of terrestrial planets formation in a feasible way, we approach the problem in two steps. First,…
WHFast: A fast and unbiased implementation of a symplectic Wisdom-Holman integrator for long term gravitational simulations
University of Toronto at Scarborough（多伦多大学士嘉堡校区）；University of Toronto（多伦多大学）；Canadian Institute for Theoretical Astrophysics（加拿大理论天体物理研究所）；Centre for Planetary Sciences（行星科学中心）
Calculating the long term solution of ordinary differential equations, such as those of the $N$-body problem, is central to understanding a wide range of dynamics in astrophysics, from galaxy formation to planetary chaos. Because generally no analytic solution exists to these equations, researchers rely on numerical methods which are prone to various errors. In an effort to mitigate these errors, powerful symplectic integrators have been employed. But symplectic integrators can be severely limited because they are not compatible with adaptive stepping and thus they have difficulty accommodatin…
Compared to other symplectic integrators (the Wisdom and Holman map and its higher order generalizations) that also take advantage of the hierarchical nature of the motion of the planets around the central star, our methods require solving implicit equations at each time-step. We claim that, despite this disadvantage, FCIRK16 is more efficient than explicit symplectic integrators for high precision simulations thanks to: (i) its high order of precision, (ii) its easy parallelization, and (iii) its efficient mixed-precision implementation which reduces the effect of round-off errors. In additio…
It has previously been shown that varying the numerical timestep during a symplectic orbital integration leads to a random walk in energy and angular momentum, destroying the phase space-conserving property of symplectic integrators. Here we show that when altering the timestep symplectic correctors can be used to reduce this error to a negligible level. Furthermore, these correctors can also be employed to avoid a large error introduction when changing the Hamiltonian's partitioning. We have constructed a numerical integrator using this technique that is nearly as accurate as widely used fixe…
Slimplectic Integrators: Variational Integrators for General Nonconservative Systems
McGill University（麦吉尔大学）；Walter Burke Institute for Theoretical Physics, California Institute of Technology（加州理工学院沃尔特·伯克理论物理研究所）；Cornell University（康奈尔大学）
Hamiltonian systems such as the gravitational N-body problem have time-reversal symmetry. However, all numerical N-body integration schemes, including symplectic ones, respect this property only approximately. In this paper, we present the new N-body integrator JANUS, for which we achieve exact time-reversal symmetry by combining integer and floating point arithmetic. JANUS is explicit, formally symplectic and satisfies Liouville's theorem exactly. Its order is even and can be adjusted between two and ten. We discuss the implementation ofJANUS and present tests of its accuracy and speed by per…
In this work we propose a new numerical approach to distinguish between regular and chaotic orbits in Hamiltonian systems, based on the simultaneous integration of both the orbit and the deviation vectors using a symplectic scheme, hereby called {\em global symplectic integrator}. In particular, the proposed method allows us to recover the correct orbits character with very large integration time steps, small energy losses and short CPU times. To illustrate the numerical performances of the global symplectic integrator we will apply it to two well-known and widely studied problems: the Hénon-H…
