The EPS method: A new method for constructing pseudospectral derivative operators

We present a stable algorithm for constructing derivative matrices with small norm for pseudospectral methods. In particular, we construct second derivative matrices that incorporate Dirichlet or Neumann boundary conditions on an interval and on the disk. By construction, the norm of these matrices grows at the optimal rate O(N) for N -by-N matrices, in contrast to standard pseudospectral constructions that result in O(N) growth of the norm. The smaller norm has a big advantage when using the derivative matrix for solving time dependent problems such as wave propagation. The construction can be used with any quadrature, but we have found that by using quadratures based on Prolate Spheroidal Wave Functions, we can achieve a near optimal sampling rate close to 2 points per wavelength, even for nonperiodic problems. We provide numerical results for the new construction and demonstrate that the construction achieves similar or better accuracy than traditional pseudospectral derivative matrices, while resulting in a norm that is orders of magnitude smaller than the standard construction. To demonstrate the advantage of the new construction, we also apply the method for solving the wave equation in constant and discontinuous media and for solving PDEs on the unit disk. We also present two compression algorithms for applying the derivative matrices in O(N logN) operations. Email addresses: [email protected] (Kristian Sandberg), [email protected] (Keith J. Wojciechowski) Preprint submitted to Journal of Computational Physics March 18, 2011