Comparison of scalable fast methods for long-range interactions

Based on a parallel scalable library for Coulomb interactions in particle systems, a comparison between the fast multipole method (FMM), multigrid-based methods, fast Fourier transform (FFT)-based methods, and a Maxwell solver is provided for the case of three-dimensional periodic boundary conditions. These methods are directly compared with respect to complexity, scalability, performance, and accuracy. To ensure comparable conditions for all methods and to cover typical applications, we tested all methods on the same set of computers using identical benchmark systems. Our findings suggest that, depending on system size and desired accuracy, the FMM- and FFT-based methods are most efficient in performance and stability.

Figure 1

The charge distribution consisting of point charges (black bars) is split into a smooth part only (dashed blue lines) and the rest; compare with (3), taken from [44].

Figure 2

Depiction of typical V cycle.

Figure 3

Schematic of the MEMD lattice interpolation. The electric fields $\mathbf{D}$ are placed on lattice sites and the magnetic fields $\mathbf{B}$ in rotated dual space on the plaquettes. The current $\mathbf{j}$ is interpolated from the moving charges.

Figure 4

The cloud-wall system (300 charges): two oppositely charged walls in the center of the box and a surrounding diffuse cloud. The system was artificially created to contain a strong long-range field component.

Figure 5

Silica melt ($12960$ charges): a system that is sufficiently homogeneous while retaining a significant long-range contribution.

Figure 6

Kinetic energy per particle of the center-of-mass motion of the system according to different methods and accuracies: ${\varepsilon }_{\mathrm{pot}}={10}^{-3}$ (top); ${\varepsilon }_{\mathrm{pot}}={10}^{-5}$ (bottom). Note that the P${}^3$M method used here uses an $ik$ differentiation for the forces instead of analytical differentiation and does not apply interlacing. With MEMD, only an accuracy of ${\varepsilon }_{\mathrm{pot}}={10}^{-3}$ could be reached.

Figure 7

Total energy per particle according to different methods at accuracies ${\varepsilon }_{\mathrm{pot}}={10}^{-3}$ (top) and ${\varepsilon }_{\mathrm{pot}}={10}^{-5}$ (bottom). The initial energy has been subtracted in order to make the fluctuations and the drift better visible. Note the different scales of the two figures (units of ${10}^{-3}$ and ${10}^{-5}$ eV). Note that the P${}^3$M method used here uses an $ik$ differentiation for the forces instead of analytical differentiation and does not apply interlacing. With MEMD, only an accuracy of ${\varepsilon }_{\mathrm{pot}}={10}^{-3}$ could be reached.

Figure 8

Required wall clock time per particle for $102900$ charges versus the relative rms potential error ${\varepsilon }_{\mathrm{pot}}$ on one core of JUROPA.

Figure 9

Required wall clock time per particle at relative rms potential error ${\varepsilon }_{\mathrm{pot}}<{10}^{-3}$ on one core of JUROPA.

Figure 10

Required wall clock time of the cloud-wall test case with $1012500$ charges versus the number of cores on JUROPA at relative rms potential error ${\varepsilon }_{\mathrm{pot}}<{10}^{-3}$.

Figure 11

Relative efficiencies $e\left(P\right)$ [see (15)] of the cloud-wall test case at different sizes versus the number of cores on JUROPA at relative rms potential error ${\varepsilon }_{\mathrm{pot}}<{10}^{-3}$. (a) Cloud-wall test case with 8100 charges. $t_{\text{best}}=6.35\times {10}^{-2}\mathrm{s}$. (b) Cloud-wall test case with $102900$ charges. $t_{\text{best}}=6.60\times {10}^{-1}\mathrm{s}$. (c) Cloud-wall test case with $1012500$ charges. $t_{\text{best}}=7.25\mathrm{s}$.

Figure 12

Relative efficiency [see (15)] of the cloud-wall test case with relative rms potential error ${\varepsilon }_{\mathrm{pot}}<{10}^{-3}$ on the JUGENE architecture. Note that the P${}^3$M method used here uses an $ik$ differentiation for the forces instead of analytical differentiation and does not apply interlacing. (a) Cloud wall test case with $1012500$ charges. $t_{\text{best}}=106.7\mathrm{s}$. (b) Cloud wall test case with $9830400$ charges. $t_{\text{best}}=454.8\mathrm{s};P_{\text{min}}=2$. (c) Cloud wall test case with $102900000$ charges. $t_{\text{best}}=753.1\mathrm{s};P_{\text{min}}=16.$