Particle mesh multipole method: An efficient solver for gravitational/electrostatic forces based on multipole method and fast convolution over a uniform mesh

We propose an efficient algorithm for the evaluation of the potential and its gradient of gravitational/electrostatic $N$-body systems, which we call particle mesh multipole method (PMMM or PM$^3$). PMMM can be understood both as an extension of the particle mesh (PM) method and as an optimization of the fast multipole method (FMM).In the former viewpoint, the scalar density and potential held by a grid point are extended to multipole moments and local expansions in $(p+1)^2$ real numbers, where $p$ is the order of expansion. In the latter viewpoint, a hierarchical octree structure which brings its $\mathcal O(N)$ nature, is replaced with a uniform mesh structure, and we exploit the convolution theorem with fast Fourier transform (FFT) to speed up the calculations. Hence, independent $(p+1)^2$ FFTs with the size equal to the number of grid points are performed. The fundamental idea is common to PPPM/MPE by Shimada et al. (1993) and FFTM by Ong et al. (2003). PMMM differs from them in supporting both the open and periodic boundary conditions, and employing an irreducible form where both the multipole moments and local expansions are expressed in $(p+1)^2$ real numbers and the transformation matrices in $(2p+1)^2$ real numbers. The computational complexity is the larger of $\mathcal O(p^2 N)$ and $\mathcal O(N \log (N/p^2))$, and the memory demand is $\mathcal O(N)$ when the number of grid points is $\propto N/p^2$.