Gauss–Legendre-spherical- <i>t</i> (GLST) cubature-based factorization of long-range electrostatics in simulations

Abstract

We develop a highly parallelizable algorithm to calculate long-range electrostatic interactions named the Gauss–Legendre-Spherical-t (GLST) cubature method. Motivated by our recent spherical grid and treecode method, we utilize the Gauss–Legendre quadrature for integration over a finite range and spherical t-design for integration over a unit sphere. The resulting GLST cubature breaks the long-range interaction term into a sum of terms that can be calculated in parallel with minimal inter-processor communication. The simulation box is divided into cells that are grouped with a separate GLST cubature applied to each group, based on their distance from the atom or cell for which the long-range interaction is calculated. Periodic boundary conditions are handled at two levels: first by “wrapping-around” other cells about the cell under consideration, then by repeating the wrapped-around box over a pre-computed number of times to make the relative error of the calculated force meet the target accuracy. With its high granularity, tunable accuracy, and adaptability to different box geometries, the GLST method is suitable for the simulation of large systems on computer hardware where many cores or threads are available.