Investigation of numerical time-integrations of Maxwell's equations using the staggered grid spatial discretization

I. Faragó, R. Horváth, W. H A Schilders

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

The Yee-method is a simple and elegant way of solving the time-dependent Maxwell's equations. On the other hand, this method has some inherent drawbacks too. The main one is that its stability requires a very strict upper bound for the possible time-steps. This is why, during the last decade, the main goal was to construct such methods that are unconditionally stable. This means that the time-step can be chosen based only on accuracy instead of stability considerations. In this paper we give a uniform treatment of methods that use the same spatial staggered grid approximation as the classical Yee-method. Three other numerical methods are discussed: the Namiki-Zheng-Chen-Zhang alternating direction implicit method (NZCZ), the Kole-Figge-de Raedt method (KFR) and a Krylov-space method. All methods are discussed with non-homogeneous material parameters. We show how the existing finite difference numerical methods are based on the approximation of a matrix exponential. With this formulation we prove the unconditional stability of the NZCZ method without any computer algebraic tool. Moreover, we accelerate the Krylov-space method with a skew-symmetric formulation of the semi-discretized equations. Our main goal is to compare the methods from the point of view of the computational speed. This question is investigated in ID numerical tests.

Original languageEnglish
Pages (from-to)149-169
Number of pages21
JournalInternational Journal of Numerical Modelling: Electronic Networks, Devices and Fields
Volume18
Issue number2
DOIs
Publication statusPublished - Mar 2005

Fingerprint

Staggered Grid
Maxwell equations
Time Integration
Maxwell's equations
Discretization
Numerical methods
Numerical Methods
Alternating Direction Implicit Method
Unconditional Stability
Matrix Exponential
Formulation
Unconditionally Stable
Approximation
Skew
Difference Method
Accelerate
Finite Difference

Keywords

  • FDTD method
  • Maxwell's equations
  • Stability
  • Unconditional stability

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Applied Mathematics

Cite this

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AB - The Yee-method is a simple and elegant way of solving the time-dependent Maxwell's equations. On the other hand, this method has some inherent drawbacks too. The main one is that its stability requires a very strict upper bound for the possible time-steps. This is why, during the last decade, the main goal was to construct such methods that are unconditionally stable. This means that the time-step can be chosen based only on accuracy instead of stability considerations. In this paper we give a uniform treatment of methods that use the same spatial staggered grid approximation as the classical Yee-method. Three other numerical methods are discussed: the Namiki-Zheng-Chen-Zhang alternating direction implicit method (NZCZ), the Kole-Figge-de Raedt method (KFR) and a Krylov-space method. All methods are discussed with non-homogeneous material parameters. We show how the existing finite difference numerical methods are based on the approximation of a matrix exponential. With this formulation we prove the unconditional stability of the NZCZ method without any computer algebraic tool. Moreover, we accelerate the Krylov-space method with a skew-symmetric formulation of the semi-discretized equations. Our main goal is to compare the methods from the point of view of the computational speed. This question is investigated in ID numerical tests.

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