New features of parallel implementation of N-body problems on GPU

This paper focuses on the parallel implementation of a direct $N$-body method~(particle-particle algorithm) and the application of multiple GPUs for galactic dynamics simulations. Application of a hybrid OpenMP-CUDA technology is considered for models with a number of particles $N \sim 10^5 \div 10^7$. By means of $N$-body simulations of gravitationally unstable stellar galactic we have investigated the algorithms parallelization efficiency for various Nvidia Tesla graphics processors~(K20, K40, K80). Particular attention was paid to the parallel performance of simulations and accuracy of the numerical solution by comparing single and double floating-point precisions~(SP and DP). We showed that the double-precision simulations are slower by a factor of~$1.7$ than the single-precision runs performed on Nvidia Tesla K-Series processors. We also claim that application of the single-precision operations leads to incorrect result in the evolution of the non-axisymmetric gravitating $N$-body systems. In particular, it leads to significant quantitative and even qualitative distortions in the galactic disk evolution. For instance, after $10^4$ integration time steps for the single-precision numbers the total energy, momentum, and angular momentum of a system with $N = 2^{20}$ conserve with accuracy of $10^{-3}$, $10^{-2}$ and $10^{-3}$ respectively, in comparison to the double-precision simulations these values are $10^{-5}$, $10^{-15}$ and $10^{-13}$, respectively. Our estimations evidence in favour of usage of the second-order accuracy schemes with double-precision numbers since it is more efficient than in the fourth-order schemes with single-precision numbers.