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A MULTIRATE VARIABLE-TIMESTEP ALGORITHM for N-BODY SOLAR SYSTEM SIMULATIONS with COLLISIONS
Abstract
We present and analyze the performance of a new algorithm for performing accurate simulations of the solar system when collisions between massive bodies and test particles are permitted. The orbital motion of all bodies at all times is integrated using a high-order variable-timestep explicit Runge–Kutta Nyström (ERKN) method. The variation in the timestep ensures that the orbital motion of test particles on eccentric orbits or close to the Sun is calculated accurately. The test particles are divided into groups and each group is integrated using a different sequence of timesteps, giving a multirate algorithm. The ERKN method uses a high-order continuous approximation to the position and velocity when checking for collisions across a step. We give a summary of the extensive testing of our algorithm. In our largest simulation—that of the Sun, the planets Earth to Neptune and 100,000 test particles over 100 million years—the relative error in the energy after 100 million years was of the order of 10−11.
... Other possible symplectic methods include those presented in [29][30][31] and [28]. Sharp and Newman [25] presented an algorithm that uses a variable stepsize for all bodies at all times. The solution is advanced using the 12-10 explicit Runge-Kutta Nyström (RKN) pair of [5]. ...
... The order 12 interpolant in [1] is used when checking for a collision between a small body and a massive body. The algorithm of [25] is intended for simulations near limited precision. Sharp implemented the algorithm in the integrator SSS (S olar S ystem S imulator). ...
... We found the ratio of the maximum and minimum CPU times across the 500 processors was 6.53. We were surprised by this ratio because Sharp and Newman [25] had performed the same simulation except with 100,000 test particles and the ratio was 1.33. This unexpected result was caused by a few test particles undergoing many close encounters with a planet. ...
The integrator SSS performs accurate N-body simulations of the Solar System when there is a mix of massive bodies and test particles. The orbital motion of all bodies at all times is integrated using a 12-10 explicit Runge-Kutta Nyström (RKN) pair. The test particles are divided into sets and each set integrated on a different processor. The explicit RKN pair uses an order 12 interpolant for the position and velocity when checking for collisions. We report on two significant improvements to SSS. The first improvement reduced the local round-off error in interpolated values by approximately four orders of magnitude, permitting more accurate modelling of collisions. The technique used to reduce the round-off error can be applied to other high-order interpolants. The second improvement is hand optimization of the implementation of SSS. This optimization increased the speed of SSS by approximately 60%, permitting more accurate modelling through the use of more test particles. We also present a summary of the numerical performance of SSS on a simulation of the Sun, the planets Earth to Neptune, and 500,000 test particles over 100 million years.
It has long been assumed that the planet Jupiter acts as a giant shield, significantly lowering the impact rate of minor bodies upon the Earth, and thus enabling the development and evolution of life in a collisional environment which is not overly hostile. In other words, it is thought that, thanks to Jupiter, mass extinctions have been sufficiently infrequent that the biosphere has been able to diversify and prosper. However, in the past, little work has been carried out to examine the validity of this idea. In the second of a series of papers, we examine the degree to which the impact risk resulting from objects on Centaur-like orbits is affected by the presence of a giant planet, in an attempt to fully understand the impact regime under which life on Earth has developed. The Centaurs are a population of ice-rich bodies which move on dynamically unstable orbits in the outer Solar system. The largest Centaurs known are several hundred kilometres in diameter, and it is certain that a great number of kilometre or sub-kilometre sized Centaurs still await discovery. These objects move on orbits which bring them closer to the Sun than Neptune, although they remain beyond the orbit of Jupiter at all times, and have their origins in the vast reservoir of debris known as the Edgeworth–Kuiper belt that extends beyond Neptune. Over time, the giant planets perturb the Centaurs, sending a significant fraction into the inner Solar System where they become visible as short-period comets. In this work, we obtain results which show that the presence of a giant planet can act to significantly change the impact rate of short-period comets on the Earth, and that such planets often actually increase the impact flux greatly over that which would be expected were a giant planet not present.
Explicit Runge–Kutta Nyström pairs provide an efficient way to find numerical solutions to second-order initial value problems when the derivative is cheap to evaluate. We present new optimal pairs of orders ten and twelve from existing families of pairs that are intended for accurate integrations in double precision arithmetic. We also present a summary of numerical comparisons between the new pairs on a set of eight problems which includes realistic models of the Solar System. Our searching for new order twelve pairs shows that there is often not quantitative agreement between the size of the principal error coefficients and the efficiency of the pairs for the tolerances we are interested in. Our numerical comparisons, as well as establishing the efficiency of the new pairs, show that the order ten pairs are more efficient than the order twelve pairs on some problems, even at limiting precision in double precision.
We revisit the issue of the cause of the dynamical instability during the so-called Nice model, which describes the early dynamical evolution of the giant planets. In particular, we address the problem of the interaction of planets with a distant planetesimal disk in the time interval between the dispersal of the proto-solar nebula and the instability. In contrast to previous works, we assume that the inner edge of the planetesimal disk is several AUs beyond the orbit of the outermost planet, so that no close encounters between planets and planetesimals occur. Moreover, we model the disk's viscous stirring, induced by the presence of embedded Pluto-sized objects. The four outer planets are assumed to be initially locked in a multi-resonant state that most likely resulted from a preceding phase of gas-driven migration. We show that viscous stirring leads to an irreversible exchange of energy between a planet and a planetesimal disk even in the absence of close encounters between the planet and disk particles. The process is mainly driven by the most eccentric planet, which is the inner ice giant in the case studied here. In isolation, this would cause this ice giant to migrate inward. However, because it is locked in resonance with Saturn, its eccentricity increases due to adiabatic invariance. During this process, the system crosses many weak secular resonances—many of which can disrupt the mean motion resonance and make the planetary system unstable. We argue that this basic dynamical process would work in many generic multi-resonant systems—forcing a good fraction of them to become unstable. Because the energy exchange proceeds at a very slow pace, the instability manifests itself late, on a timescale consistent with the epoch of the late heavy bombardment (~700 Myr). In the migration mechanism presented here, the instability time is much less sensitive to the properties of the planetesimal disk (particularly the location of its inner edge) than in the classic Nice model mechanism.
The asteroids are the major source of potential impactors on the Earth today. It has long been assumed that the giant planet Jupiter acts as a shield, significantly lowering the impact rate on the Earth from both cometary and asteroidal bodies. Such shielding, it is claimed, enabled the development and evolution of life in a collisional environment which is not overly hostile. The reduced frequency of impacts, and of related mass extinctions, would have allowed life the time to thrive, where it would otherwise have been suppressed. However, in the past, little work has been carried out to examine the validity of this idea. In the first of several papers, we examine the degree to which the impact risk resulting from a population representative of the asteroids is enhanced or lessened by the presence of a giant planet, in an attempt to fully understand the impact regime under which life on Earth has developed. Our results show that the situation is far less clear cut that has previously been assumed - for example, the presence of a giant planet can act to enhance the impact rate of asteroids on the Earth significantly. Comment: 34 pages, 15 Figures
We describe a GPU implementation of a hybrid symplectic N-body integrator,
GENGA (Gravitational ENcounters with Gpu Acceleration), designed to integrate
planet and planetesimal dynamics in the late stage of planet formation and
stability analysis of planetary systems. GENGA is based on the integration
scheme of the Mercury code (Chambers 1999), which handles close encounters with
very good energy conservation. It uses mixed variable integration (Wisdom &
Holman 1991) when the motion is a perturbed Kepler orbit and combines this with
a direct N-body Bulirsch-Stoer method during close encounters. The GENGA code
supports three simulation modes: Integration of up to 2048 massive bodies,
integration with up to a million test particles, or parallel integration of a
large number of individual planetary systems. GENGA is written in CUDA C and
runs on all Nvidia GPUs with compute capability of at least 2.0. All operations
are performed in parallel, including the close encounter detection and the
grouping of independent close encounter pairs. Compared to Mercury, GENGA runs
up to 30 times faster. GENGA is available as open source code from
https://bitbucket.org/sigrimm/genga.
We quantify the relative contribution of volatiles supplied from outer Solar System planetesimal reservoirs to large wet asteroids during the first few My after the beginning of the Solar System. To that end, we simulate the fate of planetesimals originating within different regions of the Solar System – and thus characterized by different chemical inventories – using a highly accurate integrator tuned to handle close planet/planetesimal encounters. The fraction of icy planetesimals crossing the Asteroid Belt was relatively significant, and our simulations show that planetesimals originating from the Jupiter/Saturn region were orders of magnitude more abundant than those stemming from the Uranus and Neptune regions when the planets were just embryos. As the planets reached their full masses the Jupiter/Saturn and Saturn/Uranus regions contributed similar fractions of planetesimals for any material remaining in these reservoirs late in the stage of planetary formation, This implies that large asteroids like Ceres accreted very little material enriched in low-eutectic volatiles (e.g., methanol, nitrogen and methane ices, etc.) and clathrate hydrates expected to condense at the very low temperatures predicted for beyond Saturn’s orbit in current early solar nebula models. Further, a large fraction of the content in organics of Ceres and neighboring ice-rich objects originates from the outer Solar System.
The masses of the largest bodies that fell onto the planets during their formation may be estimated from the present inclinations of the planetary axes of rotation. Earlier it was found that masses of the largest bodies in the Earth's zone had to be of the order of 10−3−10−2 of the Earth's mass. On the contrary Marcus has found that the masses of the largest bodies were comparable with the mass m0 of the planet (up to ). In the present paper it is shown that this result is a direct consequence of an inverse power law assumed by Marcus for the mass distribution of all bodies including the largest one. A very simple method of estimation of the ratios found by Marcus with the aid of limit theorems of the theory of probability is given. The suggestion by Marcus that the collision speeds of bodies were much lower than the velocity of escape at their surfaces, made to avoid contradiction with the data on the planetary rotation, is untenable.
We present a new symplectic algorithm that has the desirable property of being able to integrate close perihelion passages with the Sun. We have combined this algorithm with our fast and efficient algorithm for studying planetesimal dynamics and planet formation, known as SyMBA. The resulting code can symplectically integrate close encounters between all objects during a simulation, thus removing the main limitation of SyMBA (and related schemes). We believe that this algorithm will be a valuable tool for the study of the violent stages of planet formation.
We present a new symplectic algorithm that has the desirable properties of the sophisticated but highly efficient numerical algorithms known as mixed variable symplectic (MVS) methods and that, in addition, can handle close encounters between objects. This technique is based on a variant of the standard MVS methods, but it handles close encounters by employing a multiple time step technique. When the bodies are well separated, the algorithm has the speed of MVS methods, and whenever two bodies suffer a mutual encounter, the time step for the relevant bodies is recursively subdivided to whatever level is required. We demonstrate the power of this method using several tests of the technique. We believe that this algorithm will be a valuable tool for the study of planetesimal dynamics and planet formation.
From the Publisher:
What is the most accurate way to sum floating point numbers? What are the advantages of IEEE arithmetic? How accurate is Gaussian elimination and what were the key breakthroughs in the development of error analysis for the method? The answers to these and many related questions are included here.
This book gives a thorough, up-to-date treatment of the behavior of numerical algorithms in finite precision arithmetic. It combines algorithmic derivations, perturbation theory, and rounding error analysis. Software practicalities are emphasized throughout, with particular reference to LAPACK and MATLAB. The best available error bounds, some of them new, are presented in a unified format with a minimum of jargon. Because of its central role in revealing problem sensitivity and providing error bounds, perturbation theory is treated in detail.
Historical perspective and insight are given, with particular reference to the fundamental work of Wilkinson and Turing, and the many quotations provide further information in an accessible format.
The book is unique in that algorithmic developments and motivations are given succinctly and implementation details minimized, so that attention can be concentrated on accuracy and stability results. Here, in one place and in a unified notation, is error analysis for most of the standard algorithms in matrix computations. Not since Wilkinson's Rounding Errors in Algebraic Processes (1963) and The Algebraic Eigenvalue Problem (1965) has any volume treated this subject in such depth. A number of topics are treated that are not usually covered in numerical analysis textbooks, including floating point summation, block LU factorization, condition number estimation, the Sylvester equation, powers of matrices, finite precision behavior of stationary iterative methods, Vandermonde systems, and fast matrix multiplication.
Although not designed specifically as a textbook, this volume is a suitable reference for an advanced course, and could be used by instructors at all levels as a supplementary text from which to draw examples, historical perspective, statements of results, and exercises (many of which have never before appeared in textbooks). The book is designed to be a comprehensive reference and its bibliography contains more than 1100 references from the research literature.
Audience
Specialists in numerical analysis as well as computational scientists and engineers concerned about the accuracy of their results will benefit from this book. Much of the book can be understood with only a basic grounding in numerical analysis and linear algebra.
About the Author
Nicholas J. Higham is a Professor of Applied Mathematics at the University of Manchester, England. He is the author of more than 40 publications and is a member of the editorial boards of the SIAM Journal on Matrix Analysis and Applications and the IMA Journal of Numerical Analysis. His book Handbook of Writing for the Mathematical Sciences was published by SIAM in 1993.
Two efficient algorithms for enclosing a zero of a continuous function are presented. They are similar to the recent methods, but together with quadratic interpolation they make essential use of inverse cubic interpolation as well. Since asymptotically the inverse cubic interpolation is always chosen by the algorithms, they achieve higher-efficiency indices: 1.6529… for the first algorithm, and 1.6686… for the second one. It is proved that the second algorithm is optimal in a certain family. Numerical experiments show that the two new methods compare well with recent methods, as well as with the efficient solvers of Dekker, Brent, Bus and Dekker, and Le. The second method from the present article has the best behavior of all 12 methods especially when the termination tolerance is small.
We describe the Harwell-Boeing sparse matrix collection, a set of standard test matrices for sparse matrix problems. Our test set comprises problems in linear systems, least squares, and eigenvalue calculations from a wide variety of scientific and engineering ...
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.
Mixed-variable symplectic integrators exhibit no long-term accumulation of energy error, beyond that owing to round-off, and
they are substantially faster than conventional N-body algorithms. This makes them the integrator of choice for many problems
in Solar system astronomy. However, in their original formulation, they become inaccurate whenever two bodies approach one
another closely. This occurs because the potential energy term for the pair undergoing the encounter becomes comparable to
the terms representing the unperturbed motion in the Hamiltonian. The problem can be overcome using a hybrid method, in which
the close encounter term is integrated using a conventional integrator, whilst the remaining terms are solved symplectically.
In addition, using a simple separable potential technique, the hybrid scheme can be made symplectic even though it incorporates
a non-symplectic component.
Proc. of 12th Computational Techniques and Applications Conf. 46
- Grazier