A Crank-Nicholson-based unconditionally stable time-domain algorithm for 2D and 3D problems

Xin Xie, George Pan, Stephen Hall

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


It has been shown that both ADI-FDTD and CN-FDTD are unconditionally stable. While the ADI is a second-order approximation, CN is only in the first order. However, analytical expressions reveal that the CN-FDTD has much smaller truncation errors and is more accurate than the ADI-FDTD. Nonetheless, it is more difficult to implement the CN than the ADI, especially for 3D problems. In this paper, we present an unconditionally stable time-domain method, CNRG-TD, which is based upon the Crank-Nicholson scheme and implemented with the Ritz-Galerkin procedure. We provide a physically meaningful stability proof, without resorting to tedious symbolic derivations. Numerical examples of the new method demonstrate high precision and high efficiency. In a 2D capacitance problem, we have enlarged the time step, A/, 400 times of the CFL limit, yet preserved good accuracy. In the 3D antenna case, we use the time step, Δt, 7.6 times larger that that of the ADI-FDTD i.e., more than 38 times of the CFL limit, with excellent agreement of the benchmark solution.

Original languageEnglish (US)
Pages (from-to)261-265
Number of pages5
JournalMicrowave and Optical Technology Letters
Issue number2
StatePublished - Feb 2007


  • Couranl-friedrich-levy condition
  • FDTD
  • Maxwell's equations
  • Rayleigh-ritz procedure

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Atomic and Molecular Physics, and Optics
  • Condensed Matter Physics
  • Electrical and Electronic Engineering


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