Harmonic surface mapping algorithm for fast electrostatic sums

Abstract

We propose a harmonic surface mapping algorithm (HSMA) for electrostatic pairwise sums of an infinite number of image charges. The images are induced by point sources within a box due to a specific boundary condition which can be non-periodic. The HSMA first introduces an auxiliary surface such that the contribution of images outside the surface can be approximated by the least-squares method using spherical harmonics as basis functions. The so-called harmonic surface mapping is the procedure to transform the approximate solution into a surface charge and a surface dipole over the auxiliary surface, which becomes point images by using numerical integration. The mapping procedure is independent of the number of the sources and is considered to have a low complexity. The electrostatic interactions are then among those charges within the surface and at the integration points, which are all the forms of Coulomb potential and can be accelerated straightforwardly by the fast multipole method to achieve linear scaling. Numerical calculations of the Madelung constant of a crystalline lattice, electrostatic energy of ions in a metallic cavity, and the time performance for large-scale systems show that the HSMA is accurate and fast, and thus is attractive for many applications.

A High-Accurate Fast Poisson Solver Based on Harmonic Surface Mapping Algorithm

“…For convenience of description, one supposes that the source term f (r) is piecewise smooth in Ω. The case of f (r) being the sum of delta functions corresponds to a point-charge distribution, for which the solution can also be accurately calculated as described in our prior work [27]. The boundary condition on each face of ∂Ω can take either Dirichlet type u =0, Neumann type ∂u/∂n = 0, periodic, dielectric or free-space boundary condition.…”

“…In the literature, there are extensive efforts [1,21,[32][33][34][35] in developing fast algorithms for periodic sums based on the periodic structure of the source term. Here we adapt the HSMA algorithm [27] to construct an efficient and accurate method for general boundary conditions.…”

“…where V(r) and W(r) represent the integrals inside and outside the ball, respectively. Since W(r) is a harmonic function in B, it can be approximated by the spherical harmonic expansion in spherical coordinate r = (r,θ,ψ) [21,27], truncated at n = P,…”

“…Recently, a harmonic surface mapping algorithm (HSMA), which combines the method of image charges and the boundary integral method to calculate the Green's function, has been proposed and applied to particle systems [27,28]. Its basic idea is that an auxiliary surface is introduced such that the contribution of image charges outside the surface is represented by image charges distributed over the surface.…”

“…The asymptotic error can be controlled by a parameter such that the error can be smaller than a given tolerance. In addition, we improve the surface mapping result of Zhao et al [27] where the surface dipole derived by the interior boundary integral expression is replaced by a surface charge by the exterior boundary integral, which reduces the complexity in approximating the boundary integral. The computational complexity of the algorithm is linear to the quadrature points through the FMM acceleration.…”