On the numerical solution of the heat equation I: Fast solvers in free space

Jing Rebecca Li, Leslie Greengard

Research output: Contribution to journalArticlepeer-review


We describe a fast solver for the inhomogeneous heat equation in free space, following the time evolution of the solution in the Fourier domain. It relies on a recently developed spectral approximation of the free-space heat kernel coupled with the non-uniform fast Fourier transform. Unlike finite difference and finite element techniques, there is no need for artificial boundary conditions on a finite computational domain. The method is explicit, unconditionally stable, and requires an amount of work of the order O (NM log N), where N is the number of discretization points in physical space and M is the number of time steps. We refer to the approach as the fast recursive marching (FRM) method.

Original languageEnglish (US)
Pages (from-to)1891-1901
Number of pages11
JournalJournal of Computational Physics
Issue number2
StatePublished - Oct 1 2007


  • Free space
  • Heat equation
  • Integral representation
  • Spectral approximation
  • Unbounded domain

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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