Hermite Integrator Research Articles

SILCC VI – Multiphase ISM structure, stellar clustering, and outflows with supernovae, stellar winds, ionizing radiation, and cosmic rays

ABSTRACT We present simulations of the multiphase interstellar medium (ISM) at solar neighbourhood conditions including thermal and non-thermal ISM processes, star cluster formation, and feedback from massive stars: stellar winds, hydrogen ionizing radiation computed with the novel treeray radiative transfer method, supernovae (SN), and the injection of cosmic rays (CR). N-body dynamics is computed with a 4th-order Hermite integrator. We systematically investigate the impact of stellar feedback on the self-gravitating ISM with magnetic fields, CR advection and diffusion, and non-equilibrium chemical evolution. SN-only feedback results in strongly clustered star formation with very high star cluster masses, a bi-modal distribution of the ambient SN densities, and low volume-filling factors (VFF) of warm gas, typically inconsistent with local conditions. Early radiative feedback prevents an initial starburst, reduces star cluster masses and outflow rates. Furthermore, star formation rate surface densities of $\Sigma _{\dot{M}_\star } = 1.4-5.9 \times 10^{-3}$$\mathrm{M}_\odot \, \mathrm{yr}^{-1}\, \mathrm{kpc}^{-2}$, VFFwarm = 60–80 per cent as well as thermal, kinetic, magnetic, and cosmic ray energy densities of the model including all feedback mechanisms agree well with observational constraints. On the short, 100 Myr, time-scales investigated here, CRs only have a moderate impact on star formation and the multiphase gas structure and result in cooler outflows, if present. Our models indicate that at low gas surface densities SN-only feedback only captures some characteristics of the star-forming ISM and outflows/inflows relevant for regulating star formation. Instead, star formation is regulated on star cluster scales by radiation and winds from massive stars in clusters, whose peak masses agree with solar neighbourhood estimates.

Petar: a high-performance N-body code for modelling massive collisional stellar systems

ABSTRACTThe numerical simulations of massive collisional stellar systems, such as globular clusters (GCs), are very time consuming. Until now, only a few realistic million-body simulations of GCs with a small fraction of binaries ($5{{\ \rm per\ cent}}$) have been performed by using the nbody6++gpu code. Such models took half a year computational time on a Graphic Processing Unit (GPU)-based supercomputer. In this work, we develop a new N-body code, petar, by combining the methods of Barnes–Hut tree, Hermite integrator and slow-down algorithmic regularization. The code can accurately handle an arbitrary fraction of multiple systems (e.g. binaries and triples) while keeping a high performance by using the hybrid parallelization methods with mpi, openmp, simd instructions and GPU. A few benchmarks indicate that petar and nbody6++gpu have a very good agreement on the long-term evolution of the global structure, binary orbits and escapers. On a highly configured GPU desktop computer, the performance of a million-body simulation with all stars in binaries by using petar is 11 times faster than that of nbody6++gpu. Moreover, on the Cray XC50 supercomputer, petar well scales when number of cores increase. The 10 million-body problem, which covers the region of ultracompact dwarfs and nuclear star clusters, becomes possible to be solved.

Modified Hermite integrators of arbitrary order

ABSTRACT We present a family of modified Hermite integrators of arbitrary order possessing superior behaviour for the integration of Keplerian and near-Keplerian orbits. After recounting the derivation of Hermite N-body integrators of arbitrary order, we derive a corrector expression that minimizes integrated errors in the argument of periapsis for any such integrator. In addition to providing an alternate derivation of the modified corrector for the fourth-order Hermite integrator, we focus on improved correctors for the sixth- and eighth-order integrators previously featured in the literature. We present a set of numerical examples and find that the higher order schemes improve performance, even when considering their slightly higher cost in floating point operations. The algorithms presented herein hold promise for systems dominated by central potentials, such as planetary systems and the centres of galaxies. Existing Hermite integrators of any order can be modified to use the expressions presented here with minimal effort. Accordingly, the schemes presented herein can be easily implemented on massively parallel architectures.

Hermite integrator for high-order mesh-free schemes

Abstract In most mesh-free methods, the calculation of interactions between sample points or “particles” is the most time-consuming. When we use mesh-free methods with high spatial orders, the order of the time integration should also be high. If we use usual Runge–Kutta schemes, we need to perform the interaction calculation multiple times per time step. One way to reduce the number of interaction calculations is to use Hermite schemes, which use the time derivatives of the right-hand side of differential equations, since Hermite schemes require a smaller number of interaction calculations than Runge–Kutta schemes do to achieve the same order. In this paper, we construct a Hermite scheme for a mesh-free method with high spatial orders. We performed several numerical tests with fourth-order Hermite schemes and Runge–Kutta schemes. We found that, for both Hermite and Runge–Kutta schemes, the overall error is determined by the error of spatial derivatives, for time steps smaller than the stability limit. The calculation cost at the time-step size of the stability limit is smaller for Hermite schemes. Therefore, we conclude that Hermite schemes are more efficient than Runge–Kutta schemes and thus useful for high-order mesh-free methods for Lagrangian hydrodynamics.

A force evaluation free method to N-body problems: Binary interaction approximation

We recently proposed the binary interaction approximation (BIA) to N-body problems, which, in principle, excludes the interparticle force evaluation if the exact solutions are known for the corresponding two-body problems such as the Coulombic and gravitational interactions. In this article, a detailed introduction to the BIA is given, including the error analysis to give the expressions for the approximation error in the total angular momentum and the total energy of the entire system. It is shown that, although the energy conservation of the BIA scheme is worse than the 4th order Hermite integrator (HMT4) for similar elapsed, or the wall-clock times, the individual errors in position and in velocity are much better than HMT4. The energy error correction scheme to the BIA is also introduced that does not deteriorate the individual errors in position and in velocity. It is suggested that the BIA scheme is applicable to the tree method, the particle–mesh (PM), and the particle–particle-particle–mesh (PPPM) schemes simply by replacing the force evaluation and the conventional time integrator with the BIA scheme.

On the Semianalytical Two-Body Regularization in N-Body Simulations

A two-body regularization for N-body problem based on perturbation theory for Keplerian problem is discussed. We provide analytical estimations of accuracy and conduct N-body experiments in order to compare it with state-of-the-art Hermite integrator. It is shown that this regularization keeps some features that allow overcoming KS-regularization in some particular cases.

GraviDy: a modular, GPU-based, direct-summation N-body code

AbstractThe direct-summation of N gravitational forces is a complex problem for which there is no analytical solution. Dense stellar systems such as galactic nuclei and stellar clusters are the loci of different interesting problems. In this work we present a new GPU, direct-summation N-body integrator written from scratch and based on the Hermite scheme. The first release of the code consists of the Hermite integrator for a system of N bodies with softening. We find an acceleration factor of about ≈ 90 of the GPU version in a single node as compared to the Serial-Single-CPU one. We additionally investigate the impact of using softening in the dynamics of a dense cluster. We study how it affects the two body relaxation, as compared with another code, NBODY6, which uses KS regularization, so as to understand the role of softening in the evolution of the system. This initial release is the first step towards more and more realistic scenarios, starting for a proper treatment for binary evolution, close encounters and the role of a massive black hole.

SEREN – a new SPH code for star and planet formation simulations

We present SEREN, a new hybrid Smoothed Particle Hydrodynamics and N-body code designed to simulate astrophysical processes such as star and planet formation. It is written in Fortran 95/2003 and has been parallelised using OpenMP. SEREN is designed in a flexible, modular style, thereby allowing a large number of options to be selected or disabled easily and without compromising performance. SEREN uses the conservative `grad-h' formulation of SPH, but can easily be configured to use traditional SPH or Godunov SPH. Thermal physics is treated either with a barotropic equation of state, or by solving the energy equation and modelling the transport of cooling radiation. A Barnes-Hut tree is used to obtain neighbour lists and compute gravitational accelerations efficiently, and an hierarchical time-stepping scheme is used to reduce the number of computations per timestep. Dense gravitationally bound objects are replaced by sink particles, to allow the simulation to be evolved longer, and to facilitate the identification of protostars and the compilation of stellar and binary properties. At the termination of a hydrodynamical simulation, SEREN has the option of switching to a pure N-body simulation, using a 4th-order Hermite integrator, and following the ballistic evolution of the sink particles (e.g. to determine the final binary statistics once a star cluster has relaxed). We describe in detail all the algorithms implemented in SEREN and we present the results of a suite of tests designed to demonstrate the fidelity of SEREN and its performance and scalability. Further information and additional tests of SEREN can be found at the web-page http://www.astro.group.shef.ac.uk/seren.

Sixth- and eighth-order Hermite integrator for N-body simulations

We present sixth- and eighth-order Hermite integrators for astrophysical N-body simulations, which use the derivatives of accelerations up to second-order ( snap) and third-order ( crackle). These schemes do not require previous values for the corrector, and require only one previous value to construct the predictor. Thus, they are fairly easy to implement. The additional cost of the calculation of the higher-order derivatives is not very high. Even for the eighth-order scheme, the number of floating-point operations for force calculation is only about two times larger than that for traditional fourth-order Hermite scheme. The sixth-order scheme is better than the traditional fourth-order scheme for most cases. When the required accuracy is very high, the eighth-order one is the best. These high-order schemes have several practical advantages. For example, they allow a larger number of particles to be integrated in parallel than the fourth-order scheme does, resulting in higher execution efficiency in both general-purpose parallel computers and GRAPE systems.

6th and 8th Order Hermite Integrator Using Snap and Crackle

AbstractWe present sixth- and eighth-order Hermite integrators for astrophysical N-body simulations, which use the derivatives of accelerations up to second order (snap) and third order (crackle). These schemes do not require previous values for the corrector, and require only one previous value to construct the predictor. Thus, they are fairly easy to be implemented. The additional cost of the calculation of the higher order derivatives is not very high. Even for the eighth-order scheme, the number of floating-point operations for force calculation is only about two times larger than that for traditional fourth-order Hermite scheme. The sixth order scheme is better than the traditional fourth order scheme for most cases. When the required accuracy is very high, the eighth-order one is the best.

High performance direct gravitational N-body simulations on graphics processing units II: An implementation in CUDA

We present the results of gravitational direct N-body simulations using the graphics processing unit (GPU) on a commercial NVIDIA GeForce 8800GTX designed for gaming computers. The force evaluation of the N-body problem is implemented in “Compute Unified Device Architecture” (CUDA) using the GPU to speedup the calculations. We tested the implementation on three different N-body codes: two direct N-body integration codes, using the 4th order predictor–corrector Hermite integrator with block time-steps, and one Barnes-Hut treecode, which uses a 2nd order leapfrog integration scheme. The integration of the equations of motions for all codes is performed on the host CPU. We find that for N > 512 particles the GPU outperforms the GRAPE-6Af, if some softening in the force calculation is accepted. Without softening and for very small integration time-steps the GRAPE still outperforms the GPU. We conclude that modern GPUs offer an attractive alternative to GRAPE-6Af special purpose hardware. Using the same time-step criterion, the total energy of the N-body system was conserved better than to one in 10 6 on the GPU, only about an order of magnitude worse than obtained with GRAPE-6Af. For N ≳ 10 5 the 8800GTX outperforms the host CPU by a factor of about 100 and runs at about the same speed as the GRAPE-6Af.

High-performance direct gravitational N-body simulations on graphics processing units

We present the results of gravitational direct N-body simulations using the commercial graphics processing units (GPU) NVIDIA Quadro FX1400 and GeForce 8800GTX, and compare the results with GRAPE-6Af special purpose hardware. The force evaluation of the N-body problem was implemented in Cg using the GPU directly to speed-up the calculations. The integration of the equations of motions were, running on the host computer, implemented in C using the 4th order predictor–corrector Hermite integrator with block time steps. We find that for a large number of particles ( N ≳ 10 4) modern graphics processing units offer an attractive low cost alternative to GRAPE special purpose hardware. A modern GPU continues to give a relatively flat scaling with the number of particles, comparable to that of the GRAPE. The GRAPE is designed to reach double precision, whereas the GPU is intrinsically single-precision. For relatively large time steps, the total energy of the N-body system was conserved better than to one in 10 6 on the GPU, which is impressive given the single-precision nature of the GPU. For the same time steps, the GRAPE gave somewhat more accurate results, by about an order of magnitude. However, smaller time steps allowed more energy accuracy on the grape, around 10 −11, whereas for the GPU machine precision saturates around 10 −6 For N ≳ 10 6 the GeForce 8800GTX was about 20 times faster than the host computer. Though still about a factor of a few slower than GRAPE, modern GPUs outperform GRAPE in their low cost, long mean time between failure and the much larger onboard memory; the GRAPE-6Af holds at most 256k particles whereas the GeForce 8800GTX can hold 9 million particles in memory.

A Modified Hermite Integrator for Planetary Dynamics

A Modified Hermite Integrator for Planetary Dynamics

Stellar Orbits at the Center of the Milky Way

During the past ten years, measurements of stellar proper motion, and radial velocities (Eckart et al. 2002, Genzel et al. 2000, Ghez et al. 2000) as well as variable X-ray emission (Baganoff et al. 2001) near the center of theMilkyWay have convincingly proven the presence of a super-massive 3 million solar masses black hole in the center of our Galaxy. We discuss the possible amount of the unresolved mass present at the inner cusp, and its translation into Newtonian periastron-shifts for stellar orbits in the central cluster. For this purpose we use a 4th-order Hermite integrator. Further calculations provide valuable additional information on the three dimensional distribution and dynamics of the He-Stars. We also discuss how future observations with infrared interferometers (LBT, VLTI, Keck) will help to improve our understanding of the dynamics and distribution of the stars in this region.

On a time-symmetric Hermite integrator for planetary N-body simulation

We describe a P(EC)n Hermite scheme for planetary N-body simulation. The fourth-order implicit Hermite scheme is a time-symmetric integrator that has no secular energy error for the integration of periodic orbits with time-symmetric time-steps. In general N-body problems, however, this advantage is of little practical significance, since it is difficult to achieve time-symmetry with individual variable time-steps. However, we can easily enjoy the benefit of the time-symmetric Hermite integrator in planetary N-body systems, where all bodies spend most of the time on nearly circular orbits. These orbits are integrated with almost constant time-steps even if we adopt the individual time-step scheme. The P(EC)n Hermite scheme and almost constant time-steps reduce the integration error greatly. For example, the energy error of the P(EC)2 Hermite scheme is two orders of magnitude smaller than that of the standard PEC Hermite scheme in the case of an N = 100, m = 1025 g planetesimal system with the rms eccentricity 〈e2〉1/2 ≲0.03.