Young Researchers Workshop: Stochastic and deterministic methods in kinetic theory

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

Yong Zhang

New York University, Courant Institute


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(NlogN) complexity), low in storage, easily adaptable to other different kernels, applicable for anisotropic densities and highly parallelizable.