A Fast Algorithm for Chebyshev, Fourier & Sinc Interpolation Onto an Irregular Grid

A Chebyshev or Fourier series may be evaluated on the standard collocation grid by the Fast Fourier Transform (FFT). Unfortunately, the FFT does not apply when one needs to sum a spectral series at N points which are spaced irregularly. The cost becomes O(N2) operations instead of the FFT's O(N log N). This sort of "off-grid" interpolation is needed by codes which dynamically readjust the grid every few time steps to resolve a shock wave or other narrow features. It is even more crucial to semi-Lagrangian spectral algorithms for solving convection-diffusion and NavierStokes problems because off-grid interpolation must be performed several times per time step. In this work, we describe an alternative algorithm. The first step is to pad the set of spectral coefficients {an} with zeros and then take an FFT of length 3N to interpolate the Chebyshev series to a very fine grid. The second step is to apply either the M-th order Euler sum acceleration or (2M+1)-point Lagrangian interpolation to approximate the sum of the series on the irregular grid. We show that both methods yield full precision with M << N, allowing an order of magnitude reduction in cost with no loss of accuracy . A Fast Algorithm for Chebyshev "Off-Grid" Interpolation J. P. Boyd 5