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

Abstract: We propose an efficient and accurate solver for the nonlocal potential in the Davey-Stewartson equation 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 $O(N\log N)$ complexity, where $N$ is the total number of discretization points, and achieves higher accuracy than standard FFT solver, which makes it a good alternative in simulation. Extensive numerical results show the efficiency and accuracy of the proposed method.