Global field interpolation for particle methods

A common problem in particle simulations is effective field interpolation. Field interpolation is any method for creating an accurate representation of a given continuous field by a linear combination of overlapping basis functions. This paper presents two techniques for field interpolation, based on a radial basis function (RBF) formulation using Gaussians. The application in mind is vortex methods, where one needs to determine the circulation (or strength) of individual vortex particles with known position and scale to represent a given vorticity field. This process is required both to initially discretize a given vorticity field, and to replace a configuration of particles with another for the purposes of maintaining spatial accuracy. The first technique presented is formulated as an RBF collocation problem, and obtains a solution accurately and with excellent algorithmic efficiency by means of a preconditioned iterative method. The preconditioner is a sparse approximation, based on localization, to the dense coefficient matrix of the RBF system. The second technique uses approximate solutions to the reverse heat equation, recognizing that the standard regularization used in vortex methods (estimating particle strengths using the local value of vorticity multiplied by particle area/volume) corresponds to a Gaussian blurring of the original field. A single time step is used, thus avoiding amplification of high frequencies, and accurate solutions are produced using explicit finite difference methods. Computational experiments were performed in two dimensions, demonstrating the accuracy and convergence of the proposed techniques. Application in three dimensions is straightforward, as radial basis function interpolation is neutral to dimension, but will require more computational effort.