Implementation features of the Treecode algorithm for solving $N$-body problems on gpus

Hierarchical methods for calculating gravitational forces in a $N$-body system significantly increase the quality of numerical simulations when solving various astrophysical problems by increasing the number of $N$ elements, since we have the computational complexity $N\log(N)$ for the TreeCode approximate method instead of $\sim O(N^2)$ for the direct method, which allows to greatly increase the number of particles in the models. We developed new software for solving a dynamic problem with a large number of particles for modeling the collisionless components of the galaxies, in particular, stellar subsystem and dark matter. The paper presents the test results for the parallel implementation of the Treecode algorithm for the NVidia Tesla GPUs. To construct a hierarchical grid structure, we implemented a fast parallel octree-construction algorithm based on Morton's space-filling curve. To assess the quality of the constructed numerical model, we use the simulation results based on the direct calculation of the interaction forces between all $N$ particles of the system. We have compared the performance of the different implementations of algorithms for solving the $N$-body problem and an analisis of the fulfillment of the integral physical conservation laws of a self-gravitational system. The analysis of the fulfillment of the conservation laws of total energy and angular momentum is carried out for a rotating self-gravitating disk. Models with different criteria for a particle remoteness and value of the opening angle $\theta$ are considered.