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

• Published in: Journal of computational physics 2016-12, Vol.327, p.629-642
• Main Authors: Exl, Lukas, Mauser, Norbert J., Zhang, Yong
• Format: Article
• Language: English
• Subjects: Accuracy; Approximation; Computational physics; Convolution; Convolution integral; Coulomb/Poisson/dipole–dipole potential; Decay rate; Density; Digits; Dipoles; Efficiency; Fast Fourier transformations; Integrals; Mathematical analysis; Nonlocal potential solver; Normal distribution; Parallel processing; Separable Gaussian-sum approximation; Singular correction integral; Spectral methods; Studies
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2016.09.045

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.