Accurate and efficient computation of nonlocal potentials based on Gaussian-sum approximation

Abstract

We introduce an accurate and efficient method for the numerical evaluation of nonlocal potentials, including the 3D/2D Coulomb, 2D Poisson and 3D dipole–dipole potentials. Our method is based on a Gaussian-sum approximation of the singular convolution kernel combined with a Taylor expansion of the density. Starting from the convolution formulation of the nonlocal potential, for smooth and fast decaying densities, we make a full use of the Fourier pseudospectral (plane wave) approximation of the density and a separable Gaussian-sum approximation of the kernel in an interval where the singularity (the origin) is excluded. The potential is separated into a regular integral and a near-field singular correction integral. The first is computed with the Fourier pseudospectral method, while the latter is well resolved utilizing a low-order Taylor expansion of the density. Both parts are accelerated by fast Fourier transforms (FFT). The method is accurate (14–16 digits), efficient (O(Nlog⁡N) complexity), low in storage, easily adaptable to other different kernels, applicable for anisotropic densities and highly parallelizable.