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

Jing Rebecca Li, Leslie Greengard

Research output: Contribution to journalArticle

Abstract

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
Volume226
Issue number2
DOIs
StatePublished - Oct 1 2007

Fingerprint

thermodynamics
Fast Fourier transforms
Boundary conditions
boundary conditions
heat
approximation
Hot Temperature

Keywords

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

ASJC Scopus subject areas

  • Computer Science Applications
  • Physics and Astronomy(all)

Cite this

On the numerical solution of the heat equation I : Fast solvers in free space. / Li, Jing Rebecca; Greengard, Leslie.

In: Journal of Computational Physics, Vol. 226, No. 2, 01.10.2007, p. 1891-1901.

Research output: Contribution to journalArticle

@article{0201dbb11ff64aa984eaba70a8a0cd09,
title = "On the numerical solution of the heat equation I: Fast solvers in free space",
abstract = "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.",
keywords = "Free space, Heat equation, Integral representation, Spectral approximation, Unbounded domain",
author = "Li, {Jing Rebecca} and Leslie Greengard",
year = "2007",
month = "10",
day = "1",
doi = "10.1016/j.jcp.2007.06.021",
language = "English (US)",
volume = "226",
pages = "1891--1901",
journal = "Journal of Computational Physics",
issn = "0021-9991",
publisher = "Academic Press Inc.",
number = "2",

}

TY - JOUR

T1 - On the numerical solution of the heat equation I

T2 - Fast solvers in free space

AU - Li, Jing Rebecca

AU - Greengard, Leslie

PY - 2007/10/1

Y1 - 2007/10/1

N2 - 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.

AB - 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.

KW - Free space

KW - Heat equation

KW - Integral representation

KW - Spectral approximation

KW - Unbounded domain

UR - http://www.scopus.com/inward/record.url?scp=34548710338&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34548710338&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2007.06.021

DO - 10.1016/j.jcp.2007.06.021

M3 - Article

AN - SCOPUS:34548710338

VL - 226

SP - 1891

EP - 1901

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 2

ER -