Fast convolution solvers using moment-matching

Abstract

We propose two easy-to-implement fast algorithms based on moment-matching to compute the nonlocal potential $φ(\textbf{x})=(U\ast ρ)(\textbf{x})$ on bounded domain, where the kernel $U$ is singular at the origin and the density $ρ$ is a fast-decaying smooth function. Each method requires merely minor modifications to commonly-used existing methods, i.e., the sine spectral/Fourier quadrature method, and achieves a much better convergence rate. The key lies in the introduction of a smooth auxiliary function $ρ_1$ whose moments match those of the density up to an integer order $m$. Specifically, $ρ_1$ is constructed using Gaussian function in an explicit way and the associated potential can be calculated analytically. The moments of residual density vanish up to order $m$, and the corresponding residual potential $U \ast (ρ-ρ_1)$ decays much faster than the original potential $φ$ at the far field. As for the residual potential evaluation, for classical kernels (e.g., the Coulomb kernel), we solve a differential/pseudo-differential equation on a rectangular domain with homogeneous Dirichlet boundary conditions via sine pseudospectral method, and achieve an arbitrary high convergence rate. While, for general kernels, the regularity of Fourier integrand increase by $m$ thanks to the moments-vanishing property, therefore, the standard trapezoidal rule/midpoint quadrature also converges much faster. To gain a better numerical performance, we utilize the domain expansion technique to obtain better accuracy, and improve the efficiency by simplifying the quadrature into one discrete convolution and applying Fast Fourier Transform (FFT) to a double-sized vector. Rigorous error estimates and extensive numerical investigations showcase the accuracy and efficiency for different nonlocal potentials.