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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.

The N-body problem appears to be one of the most famous unsolved problems of mathematical physics. The area of its applications is tremendously wide, from simulations of DNA and ferrofluids [

In order to simulate many-body systems with a required level of precision, the corresponding algorithm should be selected. For so-called collisional [

this number seems to be of little avail for

sional code [

One of the classical difficulties of the collisional N-body problem is the regularization of equations of motion. It is clear that the accelerations of two particles approach infinity when

The Hamiltonian of the gravitational N-body problem is written in the form of:

This Hamiltonian leads to the equations of motion:

where

The approach based on the perturbation theory for the Keplerian problem previously was discussed in [

From the view point of analytical mechanics, the solution of the Keplerian problem looks simplest in action variables. Let us now assume that the pair formed by the kth and pth bodies is sufficiently close (the tidal forces from the interaction with other bodies are much weaker than the interaction force between the two chosen bodies). The exact trajectories of these two bodies can then be considered as an unperturbed motion, while we will take into account the influence of the surrounding bodies as a small correction. In this case, the integration step will be determined by the perturbation

As we consider

In order to resolve the trajectory with the fixed level of accuracy, we have to choose timestep, for example, from the formula

where

below allows to set up the integration timestep up to infinity [

which means that if the normal timestep

values and substitute them in criteria (12), we get the condition

two others body, and if we assume their velocities to be of the same order of magnitude, we could get that

In development of presented method some improvements were made. First of all, calculation of the jerk and snap (first and second time derivatives of acceleration) of each body are corrected to the time in which we integrate bodies in the current iteration. Besides, prediction of the position, velocity and acceleration of the bodies which are integrated by straight scheme are performed by straight scheme, and the prediction of these parameters for regularized scheme are done with itself. So, the accuracy of method was increased using these predictor- like operations without any additional complexity.

Some further corrections were made with integration timestep calculation. Using only straight scheme of integration without regularization, it is simple to estimate the third derivative of acceleration by dividing snap difference for the end and the beginning of the step by its duration. But the regularized procedure doesn’t allow estimating the third derivative for acceleration in such a simple way, because time step is much larger. So the Aarseth semiempiric formula for time step is difficult to use straightly, as it uses the third acceleration derivative. To simplify our formula and to avoid zeroes in denominator, the following procedure was suggested. Let us

suppose that orders of values of derivatives decrease proportionally to their order, i.e.

approximation doesn’t use the third acceleration derivative and the denominator consists of 2 composes not proportional to each other as in Aarseth formula. The numeric testing has shown the effectiveness of such time step approximation.

As it was mentioned above the suggested method uses individual integration timestep for each body, but it was not discussed in detail in [_{ }its current integration time step. So let’s consider the concrete step for body with number

In order to check the level of precision that can be achieved by the code described above, we conduct series of N-body experiments and compare the state-of-the-art Hermite code with our version. The algorithm of the verification is the following―we generate an initial snapshot with coordinates and velocities taken randomly in some range. All particles have masses equal to one solar mass. Using these initial conditions, we start a simulation within framework of regularized code and integrate only one timestep. The output of this calculation in which we are interested in are the new acceleration values of one or two bodies moved by one step of the algorithm. The same input is used for Hermite code, so we start the simulation and carry it up to the time that we have integrated by the regularized code. After these actions, we can compare calculated accelerations, considering the Hermite code output as the true solution of the N-body problem. There is a number of ways to compare two codes that have been discussed in [

The average error is on a level of

Another useful experiment is carried out to distinguish the reduction of the timestep for the ordinary algorithm from the precision level that is allowed by two-body regularization. As is said above, we formulate the criteria to choose the integration either by Taylor expansion or with the use of the regularization. If we increase the parameter

We also have conducted a numerical experiment analogous to one described in [

We have discussed the final adjustments of the algorithm presented in [

Sergey Chernyagin,Kirill Lezhnin, (2015) On the Semianalytical Two-Body Regularization in N-Body Simulations. Journal of Applied Mathematics and Physics,03,124-129. doi: 10.4236/jamp.2015.32018