Departmental Papers (MEAM)

Document Type

Journal Article

Date of this Version

10-24-2007

Comments

Suggested Citation:
S.K. Veerapaneni and G. Biros. (2007). A High-Order Solver for the Heat Equation in 1D Domains with Moving Boundaries. SIAM Journal on Scientific Computing. Vol. 29, No. 6, pp. 2581-2606.

© 2007 Society for Industrial and Applied Mathematics

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(N M 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

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Date Posted: 11 May 2011

This document has been peer reviewed.