A novel nonlocal potential solver based on nonuniform FFT for efficient simulation of the Davey−Stewartson equations

¹ Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
norbert.mauser@univie.ac.at
² ATI, Vienna University of Technology, Stadionallee 2, 1020 Vienna, Austria.
sunny5zhang@gmail.com
³ Wolfgang Pauli Institute, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
⁴ Université de Rennes 1, IRMAR, Campus de Beaulieu, 35042 Rennes Cedex, France.
hans.peter.stimming@univie.ac.at

Received: 3 October 2014
Revised: 16 March 2016
Accepted: 9 November 2016

Abstract

We propose an efficient and accurate solver for the nonlocal potential in the Davey−Stewartson equations using nonuniform FFT (NUFFT). A discontinuity in the Fourier transform of the nonlocal potential causes “accuracy locking” if the potential is solved by standard FFT with periodic boundary conditions on a truncated domain. Using the fact that the discontinuity disappears in polar coordinates, we reformulate the potential integral and split it into high and low frequency parts. The high frequency part can be approximated by the standard FFT method, while the low frequency part is evaluated with a high order Gauss quadrature accelerated by nonuniform FFT. The NUFFT solver has O(Nlog N) complexity, where N is the total number of discretization points, and achieves higher accuracy than standard FFT solver, which makes its use in simulations very attractive. Extensive numerical results show the efficiency and accuracy of the proposed new method.