The effect of the 2-D Laplacian operator approximation on the performance of finite-difference time-domain schemes for Maxwell's equations

Anastasios H. Panaretos, James Aberle, Rodolfo Diaz

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

The behavior of the finite-difference time-domain method (FDTD) is investigated with respect to the approximation of the two-dimensional Laplacian, associated with the curl-curl operator. Our analysis begins from the observation that in a two-dimensional space the Yee algorithm approximates the Laplacian operator via a strongly anisotropic 5-point approximation. It is demonstrated that with the aid of a transversely extended-curl operator any 9-point Laplacian can be mapped onto FDTD update equations. Our analysis shows that the mapping of an isotropic Laplacian approximation results in an isotropic and less dispersive FDTD scheme. The properties of the extended curl are further explored and it is proved that a unity Courant number can be achieved without the resulting scheme suffering from grid decoupling. Additionally, the case of a 25-point isotropic Laplacian is examined and it is shown that the corresponding scheme is fourth order accurate in space and exhibits isotropy up to sixth order. Representative numerical simulations are performed that validate the theoretically derived results.

Original languageEnglish (US)
Pages (from-to)513-536
Number of pages24
JournalJournal of Computational Physics
Volume227
Issue number1
DOIs
StatePublished - Nov 10 2007

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Finite difference time domain method
Maxwell equations
finite difference time domain method
Maxwell equation
operators
approximation
isotropy
decoupling
Mathematical operators
unity
grids
Computer simulation
simulation

Keywords

  • Curl operator
  • Electromagnetics
  • Finite-difference time-domain method
  • Laplacian operator

ASJC Scopus subject areas

  • Computer Science Applications
  • Physics and Astronomy(all)

Cite this

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title = "The effect of the 2-D Laplacian operator approximation on the performance of finite-difference time-domain schemes for Maxwell's equations",
abstract = "The behavior of the finite-difference time-domain method (FDTD) is investigated with respect to the approximation of the two-dimensional Laplacian, associated with the curl-curl operator. Our analysis begins from the observation that in a two-dimensional space the Yee algorithm approximates the Laplacian operator via a strongly anisotropic 5-point approximation. It is demonstrated that with the aid of a transversely extended-curl operator any 9-point Laplacian can be mapped onto FDTD update equations. Our analysis shows that the mapping of an isotropic Laplacian approximation results in an isotropic and less dispersive FDTD scheme. The properties of the extended curl are further explored and it is proved that a unity Courant number can be achieved without the resulting scheme suffering from grid decoupling. Additionally, the case of a 25-point isotropic Laplacian is examined and it is shown that the corresponding scheme is fourth order accurate in space and exhibits isotropy up to sixth order. Representative numerical simulations are performed that validate the theoretically derived results.",
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AU - Panaretos, Anastasios H.

AU - Aberle, James

AU - Diaz, Rodolfo

PY - 2007/11/10

Y1 - 2007/11/10

N2 - The behavior of the finite-difference time-domain method (FDTD) is investigated with respect to the approximation of the two-dimensional Laplacian, associated with the curl-curl operator. Our analysis begins from the observation that in a two-dimensional space the Yee algorithm approximates the Laplacian operator via a strongly anisotropic 5-point approximation. It is demonstrated that with the aid of a transversely extended-curl operator any 9-point Laplacian can be mapped onto FDTD update equations. Our analysis shows that the mapping of an isotropic Laplacian approximation results in an isotropic and less dispersive FDTD scheme. The properties of the extended curl are further explored and it is proved that a unity Courant number can be achieved without the resulting scheme suffering from grid decoupling. Additionally, the case of a 25-point isotropic Laplacian is examined and it is shown that the corresponding scheme is fourth order accurate in space and exhibits isotropy up to sixth order. Representative numerical simulations are performed that validate the theoretically derived results.

AB - The behavior of the finite-difference time-domain method (FDTD) is investigated with respect to the approximation of the two-dimensional Laplacian, associated with the curl-curl operator. Our analysis begins from the observation that in a two-dimensional space the Yee algorithm approximates the Laplacian operator via a strongly anisotropic 5-point approximation. It is demonstrated that with the aid of a transversely extended-curl operator any 9-point Laplacian can be mapped onto FDTD update equations. Our analysis shows that the mapping of an isotropic Laplacian approximation results in an isotropic and less dispersive FDTD scheme. The properties of the extended curl are further explored and it is proved that a unity Courant number can be achieved without the resulting scheme suffering from grid decoupling. Additionally, the case of a 25-point isotropic Laplacian is examined and it is shown that the corresponding scheme is fourth order accurate in space and exhibits isotropy up to sixth order. Representative numerical simulations are performed that validate the theoretically derived results.

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KW - Electromagnetics

KW - Finite-difference time-domain method

KW - Laplacian operator

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