scholarly journals High-order regularised symplectic integrator for collisional planetary systems

2019 ◽  
Vol 628 ◽  
pp. A32 ◽  
Author(s):  
Antoine C. Petit ◽  
Jacques Laskar ◽  
Gwenaël Boué ◽  
Mickaël Gastineau
Keyword(s):  
Hybrid Methods ◽  
Planetary Systems ◽  
High Order ◽  
Symplectic Integrators ◽  
Step Size ◽  
Time Step ◽  
Close Encounters ◽  
The Stability ◽  
Strong Scattering

We present a new mixed variable symplectic (MVS) integrator for planetary systems that fully resolves 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 when two planets encounter. We used a high-order MVS scheme so that it was 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 time-step as in a weakly perturbed problem such as the solar system. We demonstrate the long-term behaviour in systems of six super-Earths that experience strong scattering for 50 kyr. We compare our algorithm to hybrid methods such as MERCURY and show that for an equivalent cost, we obtain better energy conservation.

scholarly journals On the accuracy of symplectic integrators for secularly evolving planetary systems

2019 ◽  
Vol 490 (4) ◽  
pp. 5122-5133 ◽  
Author(s):  
Hanno Rein ◽  
Garett Brown ◽  
Daniel Tamayo
Keyword(s):  
Planetary Systems ◽  
Symplectic Integrators ◽  
Apparent Paradox ◽  
Energy Error ◽  
Evolving Systems ◽  
Specific Term ◽  
Term Evolution ◽  
Specific Choice ◽  
Shadow Hamiltonian

ABSTRACT 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 20 Myr 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.

scholarly journals Stability analysis of three exoplanet systems

2020 ◽  
Vol 494 (2) ◽  
pp. 2280-2288
Author(s):  
J P Marshall ◽  
J Horner ◽  
R A Wittenmyer ◽  
J T Clark ◽  
M W Mengel
Keyword(s):  
Time Scales ◽  
Planetary Systems ◽  
Orbital Parameters ◽  
Exoplanetary Systems ◽  
Best Fitting ◽  
The Stability ◽  
Short Time ◽  
Exoplanet Systems ◽  
Best Fit

ABSTRACT The orbital solutions of published multiplanet systems are not necessarily dynamically stable on time-scales comparable to the lifetime of the system as a whole. For this reason, dynamical tests of the architectures of proposed exoplanetary systems are a critical tool to probe the stability and feasibility of the candidate planetary systems, with the potential to point the way towards refined orbital parameters of those planets. Such studies can even help in the identification of additional companions in such systems. Here, we examine the dynamical stability of three planetary systems, orbiting HD 67087, HD 110014, and HD 133131A. We use the published radial velocity measurements of the target stars to determine the best-fitting orbital solutions for these planetary systems using the systemic console. We then employ the N-body integrator mercury to test the stability of a range of orbital solutions lying within 3σ of the nominal best fit for a duration of 100 Myr. From the results of the N-body integrations, we infer the best-fitting orbital parameters using the Bayesian package astroemperor. We find that both HD 110014 and HD 133131A have long-term stable architectures that lie within the 1σ uncertainties of the nominal best fit to their previously determined orbital solutions. However, the HD 67087 system exhibits a strong tendency towards instability on short time-scales. We compare these results to the predictions made from consideration of the angular momentum deficit criterion, and find that its predictions are consistent with our findings.

scholarly journals Fly-by encounters between two planetary systems I: Solar system analogues

2019 ◽  
Vol 488 (1) ◽  
pp. 1366-1376 ◽  
Author(s):  
Daohai Li ◽  
Alexander J Mustill ◽  
Melvyn B Davies
Keyword(s):  
Solar System ◽  
Planetary System ◽  
Planetary Systems ◽  
Open Clusters ◽  
Giant Planets ◽  
Direct Imaging ◽  
Solar Mass ◽  
Close Encounters ◽  
Large Numbers

ABSTRACTStars formed in clusters can encounter other stars at close distances. In typical open clusters in the Solar neighbourhood containing hundreds or thousands of member stars, 10–20 per cent of Solar-mass member stars are expected to encounter another star at distances closer than 100 au. These close encounters strongly perturb the planetary systems, directly causing ejection of planets or their capture by the intruding star, as well as exciting the orbits. Using extensive N-body simulations, we study such fly-by encounters between two Solar system analogues, each with four giant planets from Jupiter to Neptune. We quantify the rates of loss and capture immediately after the encounter, e.g. the Neptune analogue is lost in one in four encounters within 100 au, and captured by the flying-by star in 1 in 12 encounters. We then perform long-term (up to 1 Gyr) simulations investigating the ensuing post-encounter evolution. We show that large numbers of planets are removed from systems due to planet–planet interactions and that captured planets further enhance the system instability. While encounters can initially leave a planetary system containing more planets by inserting additional ones, the long-term instability causes a net reduction in planet number. A captured planet ends up on a retrograde orbit in half of the runs in which it survives for 1Gyr; also, a planet bound to its original host star but flipped during the encounter may survive. Thus, encounters between planetary systems are a channel to create counter-rotating planets, This would happen in around 1 per cent of systems, and such planets are potentially detectable through astrometry or direct imaging.

scholarly journals Improving the accuracy of simulated chaotic N-body orbits using smoothness

2019 ◽  
Vol 490 (3) ◽  
pp. 4175-4182 ◽  
Author(s):  
David M Hernandez
Keyword(s):  
Hybrid Methods ◽  
Hamiltonian Dynamics ◽  
Computational Cost ◽  
Symplectic Integrators ◽  
Jacobi Constant ◽  
Planetary Orbits ◽  
Particle Separations ◽  
Differentiability Class ◽  
Infinite Differentiability

ABSTRACT Symplectic integrators are a foundation to the study of dynamical N-body phenomena, at scales ranging from planetary to cosmological. These integrators preserve the Poincaré invariants of Hamiltonian dynamics. The N-body Hamiltonian has another, perhaps overlooked, symmetry: it is smooth, or, in other words, it has infinite differentiability class order (DCO) for particle separations greater than 0. Popular symplectic integrators, such as hybrid methods or block adaptive stepping methods do not come from smooth Hamiltonians and it is perhaps unclear whether they should. We investigate the importance of this symmetry by considering hybrid integrators, whose DCO can be tuned easily. Hybrid methods are smooth, except at a finite number of phase space points. We study chaotic planetary orbits in a test considered by Wisdom. We find that increasing smoothness, at negligible extra computational cost in particular tests, improves the Jacobi constant error of the orbits by about 5 orders of magnitude in long-term simulations. The results from this work suggest that smoothness of the N-body Hamiltonian is a property worth preserving in simulations.

scholarly journals An efficient algorithm for simulating ensembles of parameterized flow problems

2018 ◽  
Vol 39 (3) ◽  
pp. 1180-1205 ◽  
Author(s):  
Max Gunzburger ◽  
Nan Jiang ◽  
Zhu Wang
Keyword(s):  
Initial Conditions ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Flow Problems ◽  
Viscosity Coefficients ◽  
Deviation Ratio ◽  
Theoretical Results ◽  
Parameter Deviation

Abstract Many applications of computational fluid dynamics require multiple simulations of a flow under different input conditions. In this paper, a numerical algorithm is developed to efficiently determine a set of such simulations in which the individually independent members of the set are subject to different viscosity coefficients, initial conditions and/or body forces. The proposed scheme, when applied to the flow ensemble, needs to solve a single linear system with multiple right-hand sides, and thus is computationally more efficient than solving for all the simulations separately. We show that the scheme is nonlinearly and long-term stable under certain conditions on the time-step size and a parameter deviation ratio. A rigorous numerical error estimate shows the scheme is of first-order accuracy in time and optimally accurate in space. Several numerical experiments are presented to illustrate the theoretical results.

scholarly journals Effect of a High-Order Filter on a Cubed-Sphere Spectral Element Dynamical Core

2018 ◽  
Vol 146 (7) ◽  
pp. 2047-2064 ◽  
Author(s):  
Hyun-Gyu Kang ◽  
Hyeong-Bin Cheong
Keyword(s):  
Baroclinic Instability ◽  
Spectral Element ◽  
High Order ◽  
Local Domain ◽  
Step Size ◽  
Time Step ◽  
Dynamical Core ◽  
Time Step Size ◽  
Order Filter ◽  
Cubed Sphere

Abstract A high-order filter for a cubed-sphere spectral element model was implemented in a three-dimensional spectral element dry hydrostatic dynamical core. The dynamical core incorporated hybrid sigma–pressure vertical coordinates and a third-order Runge–Kutta time-differencing method. The global high-order filter and the local-domain high-order filter, requiring numerical operation with a huge sparse global matrix and a locally assembled matrix, respectively, were applied to the prognostic variables, except for surface pressure, at every time step. Performance of the high-order filter was evaluated using the baroclinic instability test and quiescent atmosphere with underlying topography test presented by the Dynamical Core Model Intercomparison Project. It was revealed that both the global and local-domain high-order filters could better control the numerical noise in the noisy circumstances than the explicit diffusion, which is widely used for the spectral element dynamical core. Furthermore, by adopting the high-order filter, the effective resolution of the dynamical core could be increased, without weakening the stability of the dynamical core. Computational efficiency of the high-order filter was demonstrated in terms of both the time step size and the wall-clock time. Because of the nature of an implicit diffusion, the dynamical core employing this filter can take a larger time step size, compared to that using the explicit diffusion. The local-domain high-order filter was computationally more efficient than the global high-order filter, but less efficient than the explicit diffusion.

scholarly journals Variable Step Size Adams Methods for BSDEs

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Qiang Han
Keyword(s):  
High Order ◽  
Variable Step Size ◽  
Necessary And Sufficient Condition ◽  
Step Size ◽  
Truncation Errors ◽  
Adams Methods ◽  
Variable Step ◽  
High Order Convergence ◽  
The Stability ◽  
Necessary And Sufficient

For backward stochastic differential equations (BSDEs), we construct variable step size Adams methods by means of Itô–Taylor expansion, and these schemes are nonlinear multistep schemes. It is deduced that the conditions of local truncation errors with respect to Y and Z reach high order. The coefficients in the numerical methods are inferred and bounded under appropriate conditions. A necessary and sufficient condition is given to judge the stability of our numerical schemes. Moreover, the high-order convergence of the schemes is rigorously proved. The numerical illustrations are provided.

Adaptive Semi-Implicit Integrator for Articulated Rigid-Body Systems

2016 ◽  
Author(s):  
Joe Hewlett ◽  
Laszlo Kovacs ◽  
Alfonso Callejo ◽  
Paul G. Kry ◽  
József Kövecses ◽  
...  
Keyword(s):  
Rigid Body ◽  
Adaptive Method ◽  
Point Of View ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Geometric Stiffness ◽  
Starting Point ◽  
The Stability ◽  
Mechanical Dissipation

This paper concerns the dynamic simulation of constrained rigid-body systems in the context of real-time applications and stable integrators. The goal is to adaptively find a balance between the stability of an over-damped implicit scheme and the energetic consistency of the symplectic, semi-implicit Euler scheme. As a starting point, we investigate in detail the properties of a new time stepping scheme proposed by Tournier et al., “Stable constrained dynamics”, ACM transactions on Graphics, 2015, which approximates a full non-linear implicit solution with a single linear system without compromising stability. This introduces a geometric stiffness term that improves numerical stability up to a certain time step size, at the cost of large mechanical dissipation compared to the traditional formulation. Dissipation is sometimes undesirable from a mechanical point of view, especially if the dissipation is not quantified. In this paper, we propose to use an additional control parameter to regulate how “implicit” the Jacobian matrix is, and change the degree to which the geometric stiffness term contributes. For the selection of this parameter, adaptive schemes will be proposed based on the monitoring of energy drift. The proposed adaptive method is verified through the simulation of chain-like systems.

scholarly journals High-order symplectic integrators for planetary dynamics and their implementation in rebound

2019 ◽  
Vol 489 (4) ◽  
pp. 4632-4640 ◽  
Author(s):  
Hanno Rein ◽  
Daniel Tamayo ◽  
Garett Brown
Keyword(s):  
Large Time ◽  
Good Accuracy ◽  
Long Term Evolution ◽  
High Order ◽  
Symplectic Integrators ◽  
Symplectic Methods ◽  
Available N ◽  
Term Evolution ◽  
Planetary Dynamics

ABSTRACT 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 time-steps 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., Laskar & Robutel, and Blanes et al. Since some of these methods are harder to implement and not as readily available to astronomers compared to the standard WH method, they are not used as often. This is somewhat unfortunate given that in typical simulations it is possible to improve the accuracy by up to six orders of magnitude (!) compared to the standard WH method without the need for any additional force evaluations. To change this, we implement a variety of high-order symplectic methods in the freely available N-body integrator rebound. In this paper, we catalogue these methods, discuss their differences, describe their error scalings, and benchmark their speed using our implementations.

High-Order Symplectic Schemes for Stochastic Hamiltonian Systems

2014 ◽  
Vol 16 (1) ◽  
pp. 169-200 ◽  
Author(s):  
Jian Deng ◽  
Cristina Anton ◽  
Yau Shu Wong
Keyword(s):  
Hamiltonian Systems ◽  
High Order ◽  
Symplectic Integrators ◽  
Symplectic Methods ◽  
Computational Tools ◽  
Fully Implicit ◽  
Stochastic Hamiltonian Systems ◽  
Hamiltonian Functions ◽  
Symplectic Schemes

AbstractThe construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general the proposed symplectic schemes are fully implicit, and they become computationally expensive for mean square orders greater than two. However, for stochastic Hamiltonian systems preserving Hamiltonian functions, the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes. A theoretical analysis of the convergence and numerical simulations are reported for several symplectic integrators. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

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