GOTHIC: Gravitational oct-tree code accelerated by hierarchical time step controlling

The tree method is a widely implemented algorithm for collisionless $N$-body simulations in astrophysics well suited for GPU(s). Adopting hierarchical time stepping can accelerate $N$-body simulations; however, it is infrequently implemented and its potential remains untested in GPU implementations. We have developed a Gravitational Oct-Tree code accelerated by HIerarchical time step Controlling named \texttt{GOTHIC}, which adopts both the tree method and the hierarchical time step. The code adopts some adaptive optimizations by monitoring the execution time of each function on-the-fly and minimizes the time-to-solution by balancing the measured time of multiple functions. Results of performance measurements with realistic particle distribution performed on NVIDIA Tesla M2090, K20X, and GeForce GTX TITAN X, which are representative GPUs of the Fermi, Kepler, and Maxwell generation of GPUs, show that the hierarchical time step achieves a speedup by a factor of around 3--5 times compared to the shared time step. The measured elapsed time per step of \texttt{GOTHIC} is 0.30~s or 0.44~s on GTX TITAN X when the particle distribution represents the Andromeda galaxy or the NFW sphere, respectively, with $2^{24} =$~16,777,216 particles. The averaged performance of the code corresponds to 10--30\% of the theoretical single precision peak performance of the GPU.