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.

The Fortran output routine messy enables debugging and error message processing strategies in the design of numerical and mathematical software. It supports separate output from different processes in a parallel computing environment.

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 Nystrom (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).

Collisionless N-body simulations over tens of millions of years are an important tool in understanding the early evolution of planetary systems. We first present a CUDA kernel for evaluating the gravitational acceleration of N bodies that is intended primarily for when N is less than several thousand. We then use the kernel with a variable-order, variable-stepsize Adams method to perform long, collisionless simulations of the Solar System near limiting precision. The varying stepsize means no special scheme is required to integrate close encounters, and the motion of bodies on eccentric orbits or close to the Sun is calculated accurately. Our method is significantly more accurate than symplectic methods and sufficiently fast.

We present, analyze, and test a multirate Störmer-based algorithm for integrating close encounters when performing N-body simulations of the Sun, planets, and a large number of test particles. The algorithm is intended primarily for accurate simulations of the outer solar system. The algorithm uses stepsizes H and hi, i = 1, …, Np, where hi ≪ H and Np is the number of planets. The stepsize H is used for the integration of the orbital motion of the Sun and planets at all times. H is also used as the stepsize for the integration of the orbital motion of test particles when they are not undergoing a close encounter. The stepsize hi is used to integrate the orbital motion of test particles during a close encounter with the ith planet. The position of the Sun and planets during a close encounter is calculated using Hermite interpolation. We tested the algorithm on two contrasting problems, and compared its performance with the existing method which uses the same stepsize for all bodies (this stepsize must be significantly smaller than H to ensure the close encounters are integrated accurately). Our tests show that the integration error for the new and existing methods are comparable when the stepsizes are chosen to minimize the error, and that for this choice of stepsizes the new method requires considerably less CPU time than the existing method.

A new optimal, explicit, Hermite–Obrechkoff method of order 13, denoted by HO(13), that is contractivity-preserving (CP) and has nonnegative coefficients is constructed for solving nonstiff first-order initial value problems. Based on the CP conditions, the new 9-derivative HO(13) has maximum order 13. The new method usually requires significantly fewer function evaluations and significantly less CPU time than the Taylor method of order 13 and the Runge–Kutta method DP(8,7)13M to achieve the same global error when solving standard \(N\)-body problems.

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. (C) 2014 Published by Elsevier Inc.

Numerically integrated ephemerides of the Solar System and the Moon require very accurate integrations of systems of second order ordinary differential equations. We present a new family of 8-9 explicit Runge-Kutta pairs and assess the performance of two new 8-9 pairs on the equations used to create the ephemeris DE102. Part of this work is the introduction of these equations as a test problem for integrators of initial value ordinary differential equations.

Explicit Runge–Kutta Nyström methods with enhanced phase-lag order are intended for long integrations of initial value ordinary differential equations describing free oscillations or free oscillations of high frequency and forced oscillations of low frequency. Numerical comparisons by others of RKN4(3), RKN6(4) and RKN8(6) pairs has established that the pairs with enhanced phase-lag order are more efficient on the intended problems than general purpose pairs.We investigate if these gains in efficiency extend to N-body problems used to model the orbital dynamics of the Solar System. The emphasis in our comparisons is on the RKN8(6) pairs because we are interested in long, accurate integrations. We have included the RKN4(3) and RKN6(4) pairs principally to gain insight about how the gains in efficiency depend on the order.Our main finding is that the gains for the RKN8(6) pair extend to the system of major planets except at severe accuracy requirements, and to the system of regular satellites of these planets. In addition, we found for Kepler’s two-body problem that the gains can be sensitive to small changes in eccentricity.

A four-stage Hermite–Birkhoff–Obrechkoff method of order 14 with four quantized variable steps, denoted by HBOQ(14)4, is constructed for solving non-stiff systems of first-order differential equations of the form y′=f(t,y) with initial conditions y(t0)=y0. Its formula uses y′, y″ and y‴ as in Obrechkoff methods. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep- and Runge–Kutta-type order conditions which are reorganized into linear Vandermonde-type systems. To reduce overhead, simple formulae are derived only once to obtain the values of Hermite–Birkhoff interpolation polynomials in terms of Lagrange basis functions for 16 quantized step size ratios. The step size is controlled by a local error estimator. When programmed in C ++, HBOQ(14)4 is superior to the Dormand–Prince Runge–Kutta pair DP(8,7)13M of order 8 in solving several problems often used to test higher order ODE solvers at stringent tolerances. When programmed in Matlab, it is superior to ode113 in solving costly problems, on the basis of the number of steps, CPU time, and maximum global error. The code is available on the URL www.site.uottawa.ca/~remi.

Coe cients of an explicit Runge Kutta method which might be used for initial value ordinary di erential equations delay di erential equations or Volterra integral equations often require many digits for their representation This can make manual checking of the coe cients unreliable We present a dual precision Fortran package for checking the coe cients of an explicit Runge Kutta method consisting of k formulae In some instances when a coe cient is wrong the output from the package can be used to deduce which coe cient is likely to be wrong Subject Classi cations AMS L CR G