Examination, clarification, and simplification of stability and dispersion analysis for ADI-FDTD and CNSS-FDTD schemes

S. Ogurtsov, George Pan, Rodolfo Diaz

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

We describe a rigorous analysis of unconditional stability for the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) and Crank-Nicholson split step (CNSS-FDTD) schemes avoiding use of the von Neumann spectral criterion. The proof is performed in the spectral domain, and uses skew-Hermitivity of matrix terms in the ADI and CNSS total amplification matrices. A bound for the total ADI amplification matrix is provided. While the CN and CNSS-FDTD amplification matrices are unitary, the bound on the ADI-FDTD depends on the Courant number. Importantly, we have found that the ADI-FDTD amplification matrix is not normal, i.e., the unit spectral radius alone cannot be used to prove the ADI-FDTD unconditional stability. The paper also shows that the ADI-FDTD and CNSS-FDTD schemes share the same dispersion equation by a similarity of their total amplification matrices, and the two schemes have identical numerical dispersion in the frame of plane waves.

Original languageEnglish (US)
Pages (from-to)3595-3602
Number of pages8
JournalIEEE Transactions on Antennas and Propagation
Volume55
Issue number12
DOIs
StatePublished - Dec 2007

Fingerprint

Amplification

Keywords

  • Amplification
  • Amplification matrix
  • Dispersion equation
  • Finite difference time domain analysis
  • Finite difference time main (FDTD) schemes
  • Hermitian matrices
  • Matrix algebra
  • Normal matrices
  • Skew-Hermitian matrices
  • Stability
  • stability analysis
  • Time domain analysis
  • Unconditional stability
  • Unitary matrices

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Computer Networks and Communications

Cite this

@article{c6f2d8c7f3d44d83844a8e09b9bab7d9,
title = "Examination, clarification, and simplification of stability and dispersion analysis for ADI-FDTD and CNSS-FDTD schemes",
abstract = "We describe a rigorous analysis of unconditional stability for the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) and Crank-Nicholson split step (CNSS-FDTD) schemes avoiding use of the von Neumann spectral criterion. The proof is performed in the spectral domain, and uses skew-Hermitivity of matrix terms in the ADI and CNSS total amplification matrices. A bound for the total ADI amplification matrix is provided. While the CN and CNSS-FDTD amplification matrices are unitary, the bound on the ADI-FDTD depends on the Courant number. Importantly, we have found that the ADI-FDTD amplification matrix is not normal, i.e., the unit spectral radius alone cannot be used to prove the ADI-FDTD unconditional stability. The paper also shows that the ADI-FDTD and CNSS-FDTD schemes share the same dispersion equation by a similarity of their total amplification matrices, and the two schemes have identical numerical dispersion in the frame of plane waves.",
keywords = "Amplification, Amplification matrix, Dispersion equation, Finite difference time domain analysis, Finite difference time main (FDTD) schemes, Hermitian matrices, Matrix algebra, Normal matrices, Skew-Hermitian matrices, Stability, stability analysis, Time domain analysis, Unconditional stability, Unitary matrices",
author = "S. Ogurtsov and George Pan and Rodolfo Diaz",
year = "2007",
month = "12",
doi = "10.1109/TAP.2007.910323",
language = "English (US)",
volume = "55",
pages = "3595--3602",
journal = "IEEE Transactions on Antennas and Propagation",
issn = "0018-926X",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
number = "12",

}

TY - JOUR

T1 - Examination, clarification, and simplification of stability and dispersion analysis for ADI-FDTD and CNSS-FDTD schemes

AU - Ogurtsov, S.

AU - Pan, George

AU - Diaz, Rodolfo

PY - 2007/12

Y1 - 2007/12

N2 - We describe a rigorous analysis of unconditional stability for the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) and Crank-Nicholson split step (CNSS-FDTD) schemes avoiding use of the von Neumann spectral criterion. The proof is performed in the spectral domain, and uses skew-Hermitivity of matrix terms in the ADI and CNSS total amplification matrices. A bound for the total ADI amplification matrix is provided. While the CN and CNSS-FDTD amplification matrices are unitary, the bound on the ADI-FDTD depends on the Courant number. Importantly, we have found that the ADI-FDTD amplification matrix is not normal, i.e., the unit spectral radius alone cannot be used to prove the ADI-FDTD unconditional stability. The paper also shows that the ADI-FDTD and CNSS-FDTD schemes share the same dispersion equation by a similarity of their total amplification matrices, and the two schemes have identical numerical dispersion in the frame of plane waves.

AB - We describe a rigorous analysis of unconditional stability for the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) and Crank-Nicholson split step (CNSS-FDTD) schemes avoiding use of the von Neumann spectral criterion. The proof is performed in the spectral domain, and uses skew-Hermitivity of matrix terms in the ADI and CNSS total amplification matrices. A bound for the total ADI amplification matrix is provided. While the CN and CNSS-FDTD amplification matrices are unitary, the bound on the ADI-FDTD depends on the Courant number. Importantly, we have found that the ADI-FDTD amplification matrix is not normal, i.e., the unit spectral radius alone cannot be used to prove the ADI-FDTD unconditional stability. The paper also shows that the ADI-FDTD and CNSS-FDTD schemes share the same dispersion equation by a similarity of their total amplification matrices, and the two schemes have identical numerical dispersion in the frame of plane waves.

KW - Amplification

KW - Amplification matrix

KW - Dispersion equation

KW - Finite difference time domain analysis

KW - Finite difference time main (FDTD) schemes

KW - Hermitian matrices

KW - Matrix algebra

KW - Normal matrices

KW - Skew-Hermitian matrices

KW - Stability

KW - stability analysis

KW - Time domain analysis

KW - Unconditional stability

KW - Unitary matrices

UR - http://www.scopus.com/inward/record.url?scp=37249011877&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=37249011877&partnerID=8YFLogxK

U2 - 10.1109/TAP.2007.910323

DO - 10.1109/TAP.2007.910323

M3 - Article

VL - 55

SP - 3595

EP - 3602

JO - IEEE Transactions on Antennas and Propagation

JF - IEEE Transactions on Antennas and Propagation

SN - 0018-926X

IS - 12

ER -