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We present sixth- and eighth-order Hermite integrators for astrophysical $N$-body simulations, which use the derivatives of accelerations up to second order ({it snap}) and third order ({it 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 implemente. 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.
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In most of 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 one 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 smaller number of interaction calculations than RK 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 of Hermite and Runge-Kutta schemes, the overall error is determined by the error of spatial derivatives, for timesteps smaller than the stability limit. The calculation cost at the timestep 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.
The N-body problem has become one of the hottest topics in the fields of computational dynamics and cosmology. The large dynamical range in some astrophysical problems led to the use of adaptive time steps to integrate particle trajectories, however, the search of optimal strategies is still challenging. We quantify the performance of the hierarchical time step integrator Hamiltonian Splitting (HamSp) for collisionless multistep simulations. We compare with the constant step Leap-Frog (LeapF) integrator and the adaptive one (AKDK). Additionally, we explore the impact of different time step assigning functions. There is a computational overhead in HamSp however there are two interesting advantages: choosing a convenient time-step function may compensate and even turn around the efficiency compared with AKDK. We test both reversibility and time symmetry. The symmetrized nature of the HamSp integration is able to provide time-reversible integration for medium time scales and overall deliver better energy conservation for long integration times, and the linear and angular momentum are preserved at machine precision. We address the impact of using different integrators in astrophysical systems. We found that in most situations both AKDK and HamSp are able to correctly simulate the problems. We conclude that HamSp is an attractive and competitive alternative to AKDK, with, in some cases, faster and with better energy and momentum conservation. The use of recently discussed Bridge splitting techniques with HamSp may allow to reach considerably high efficiency.
We use gauge-invariant cosmological perturbation theory to calculate the displacement field that sets the initial conditions for $N$-body simulations. Using first and second-order fully relativistic perturbation theory in the synchronous-comoving gauge, allows us to go beyond the Newtonian predictions and to calculate relativistic corrections to it. We use an Einstein--de Sitter model, including both growing and decaying modes in our solutions. The impact of our results should be assessed through the implementation of the featured displacement in cosmological $N$-body simulations.
Gravitational N-body simulations, that is numerical solutions of the equations of motions for N particles interacting gravitationally, are widely used tools in astrophysics, with applications from few body or solar system like systems all the way up to galactic and cosmological scales. In this article we present a summary review of the field highlighting the main methods for N-body simulations and the astrophysical context in which they are usually applied.
Initial conditions for (Newtonian) cosmological N-body simulations are usually set by re-scaling the present-day power spectrum obtained from linear (relativistic) Boltzmann codes to the desired initial redshift of the simulation. This back-scaling method can account for the effect of inhomogeneous residual thermal radiation at early times, which is absent in the Newtonian simulations. We analyse this procedure from a fully relativistic perspective, employing the recently-proposed Newtonian motion gauge framework. We find that N-body simulations for LambdaCDM cosmology starting from back-scaled initial conditions can be self-consistently embedded in a relativistic space-time with first-order metric potentials calculated using a linear Boltzmann code. This space-time coincides with a simple N-body gauge for z<50 for all observable modes. Care must be taken, however, when simulating non-standard cosmologies. As an example, we analyse the back-scaling method in a cosmology with decaying dark matter, and show that metric perturbations become large at early times in the back-scaling approach, indicating a breakdown of the perturbative description. We suggest a suitable forwards approach for such cases.
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