An Efficient Implementation of Particle Methods for the Incompressible Euler Equations

Abstract

A fast algorithm is presented that reduces the amount of work necessary for computing pairwise interactions in a system of n particles from $\mathcal{O}(n^2 )$ to $\mathcal{O}(n\log ^{d + 1} n)$, where d is the dimension of the computational domain. Theoretical error and work estimates are given. The main advantages of the new method over other fast algorithms of this kind are its applicability to a class of kernels not necessarily associated with the Laplace equation and the straightforward extension to three-dimensional problems. The algorithm is used to obtain a fast summation of the velocity field induced by (a) point vortices and (b) vortex blobs in the approximation of the incompressible Euler equations. It is shown that, with a proper choice of parameters, this technique preserves the accuracy of these methods. Numerical results are presented for the two-dimensional case.