Departmental Papers (MEAM)
Document Type
Journal Article
Date of this Version
November 2007
Abstract
We describe a fast high-order accurate method for the solution of the heat equation in domains with moving Dirichlet or Neumann boundaries and distributed forces. We assume that the motion of the boundary is prescribed. Our method extends the work of Greengard and Strain [Comm. Pure Appl. Math., XLIII (1990), pp. 949–963]. Our scheme is based on a time-space Chebyshev pseudo-spectral collocation discretization, which is combined with a recursive product quadrature rule to accurately and efficiently approximate convolutions with Green's function for the heat equation. We present numerical results that exhibit up to eighth-order convergence rates. Assuming N time steps and M spatial discretization points, the evaluation of the solution of the heat equation at the same number of points in space-time requires O(NM log M) work. Thus, our scheme can be characterized as "fast"; that is, it is work-optimal up to a logarithmic factor.
Keywords
integral equations, spectral methods, Chebyshev polynomials, moving boundaries, heat equation, quadratures, Nyström's method, collocation methods, potential theory
Date Posted: 15 January 2008
This document has been peer reviewed.
Comments
© SIAM 2007. Reprinted from SIAM Journal on Scientific Computing, Volume 29, Issue 6, November 2007, pages 2581-2606.
Publisher URL: http://dx.doi.org/10.1137/060677896