The Chebyshev fast Gauss and nonuniform fast Fourier transforms and their application to the evaluation of distributed heat potentials

  • Authors:

  • Shravan K. Veerapaneni;George Biros

  • Affiliations:

  • Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA;Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA and Department of Computer and Information Science, University of Pennsylvania, ...

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2008

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Abstract

We present a method for the fast and accurate computation of distributed heat potentials in two dimensions. The distributed source is assumed to be given in terms of piecewise space-time Chebyshev polynomials. We discretize uniformly in time, whereas in space the polynomials are defined on the leaf nodes of a quadtree. The quadtree can vary at each time step. We combine a product integration rule with fast algorithms (fast heat potentials, nonuniform FFT, fast Gauss transform) to obtain a high-order accurate method with optimal complexity. If N is the number of time steps, M is the maximum number of leaf nodes over all the time steps and the input contains a qth-order polynomial representation of f, then, our method requires O(q^3MNlogM) work to evaluate the heat potential at arbitrary MN space-time target locations. The overall convergence rate of the method is of order q. We present numerical experiments for q=4, 8, and 16, and we verify the theoretical convergence rate of the method. When the solution is sufficiently smooth, the 16th-order variant results in significant computational savings, even in the case in which we require only a few digits of accuracy.