### 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 language | English (US) |
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Pages (from-to) | 1891-1901 |

Number of pages | 11 |

Journal | Journal of Computational Physics |

Volume | 226 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1 2007 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 226, no. 2, pp. 1891-1901. https://doi.org/10.1016/j.jcp.2007.06.021

}

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 -