The Accuracy of ADI-FDTD: Recent Insights about Truncation Errors and Source Conditions

1. The Accuracy of ADI-FDTD: Recent Insights about Truncation Errors and Source Conditions Salvador González García, Amelia Rubio Bretones Depto. Electromagnetismo y Física de la Materia University of Granada, SPAIN Susan C. Hagness Department of Electrical and Computer Engineering University of Wisconsin-Madison, USA

2. Motivation for ADI-FDTD • For EM wave interaction problems involving structures with fine-scale geometric details, the Courant stability bound may be too restrictive. • Alternating-direction implicit (ADI)* time-stepping yields unconditional numerical stability for FDTD • Namiki, IEEE T-MTT, vol. 47, pp. 2003-2007, Oct. 1999 • Zheng, Chen and Zhang, IEEE MGWL, vol. 9, pp. 441-443, Nov. 1999 • ADI-FDTD is a computationally efficient approach when the FDTD time step required for stability is much smaller than that required for accuracy. *Peaceman and Rachford, J. Soc. Indust. Appl. Math., vol. 3, pp. 28-41, 1955 Douglas, J. Soc. Indust. Appl. Math., vol. 3, pp. 42-65, 1955

3. Overview of ADI-FDTD E and H are staggered in space, collocated in time. sub-iteration 1 sub-iteration 2 • Update Eximplicitly along y direction for all x, z • Update Eyimplicitly along z direction for all x, y • Update Ezimplicitly along x direction for all y, z • Update Eximplicitly along z direction for all x, y • Update Eyimplicitly along x direction for all z, y • Update Ezimplicitly along y direction for all x, z Update H-fields explicitly for all x, y, z Update H-fields explicitly for all x, y, z Implicit equations represent tridiagonal matrix systems.

4. ADI-FDTD:Perturbation of CN-FDTD • Systematic framework for • Proving unconditional stability • Analyzing accuracy limitations imposed the truncation error • limitations not revealed by numerical dispersion analyses • Deriving the proper formulation for current source implementation • lack of consensus in the literature about source implementation S. G. Garcia, T.-W. Lee, and S. C. Hagness,“On the accuracy of the ADI-FDTD method,” IEEE AWPL, 2002. S. G. Garcia, A. R. Bretones, R. G. Martin, and S. C. Hagness,“Accurate implementation of current sources in the ADI–FDTD scheme,”IEEE AWPL, accepted (in press).

5. Composite field vector Source vector Time-independent operator (Curl, Ohmic losses) Maxwell’s Curl Equations

6. Finite Difference Notation Discrete Continuum

7. FDTD Schemes: Yee and CN Maxwell’s curl equations Yee scheme (explicit)  Conditional stability Crank-Nicolsonscheme (implicit)  Unconditional stability

8. If operators and are chosen so that which decouples into… ADI-FDTD scheme (implicit)  Unconditional stability Alternate Scheme: ADI-FDTD • It is possible to factorize the CN-FDTD scheme by adding a 2nd-order perturbation term:

9. Summary of Schemes Analytical Yee FDTD CN-FDTD ADI-FDTD decoupled:

10. Global 2nd-Order Consistency(source-free case) Analytical Yee FDTD CN-FDTD ADI-FDTD

11. Accuracy of ADI-FDTD • Unique feature of ADI-FDTD*: • the dominant 2nd-order terms of the truncation error depend on • These terms are problematic when inhomogeneities introduce spatial variations in the fields that are on a much smaller scale than that of the wavelength of the source excitation! Yee-FDTD: Finer mesh req’d in regions of significant spatial variations of the field ADI-FDTD: Finer mesh + low Courant number req’d *Garcia, Lee, and Hagness, IEEE AWPL, vol. 1, no. 1, pp. 31-34, 2002.

12. Numerical Example:Comparison of Yee, CN, and ADI Source: 750-kHz raised cosine (0=400 m) x=0.2 m (2000 pts/0) t=0.33 ns (2000 samples over 0.66 s)

13. Simplified Form of ADI-FDTD • Starting with… • The familiar two-step ADI-FDTD scheme is obtained using where

14. Simplified Form of ADI-FDTD(lossless case) sub-iteration 1 • implicit (tridiagonal) updating for E • explicit updating for H sub-iteration 2 • implicit updating for E • explicit updating for H Note: electric current sources must be embedded in the tridiagonal system* *T. W. Lee and S. C. Hagness, IEEE AP-S International Symposium Digest, vol. 4, pp. 142-145, Boston, MA, July 2001.

15. time derivatives and fields: • currents: • time derivatives and fields: • currents: Source Implementation: Method #1 sub-iteration 1 sub-iteration 2 Global consistency, but no apparent intrastep consistency!

16. time derivatives and fields: • currents: • time derivatives and fields: • currents: Source Implementation: Method #2 sub-iteration 1 sub-iteration 2 Apparent intrastep consistency, but what happens globally?

17. Analytical Comparison ofSource Implementations • Converting back to a single-step scheme… Method #1 Method #2 approximates

18. Numerical Example:Comparison of ADI Source Implementations plane wave propagation inside Huygens’ surface

19. Summary and Conclusions • Demonstrated systematic approach to building ADI-FDTD as a perturbation of the CN-FDTD scheme • Proves unconditional stability of ADI-FDTD • Reveals accuracy limitations imposed the truncation error • Second-order terms of the truncation errorgive rise to potentially large errors as the time step increases even though key temporal features may be highly resolved. • Resolves issues about source conditions • Current source waveforms should be evaluated at in both sub-iterations. • Presented numerical examples that support these insights. Garcia, Lee, and Hagness, IEEE AWPL, 2002 Garcia, Bretones, Martin, and Hagness, IEEE AWPL, in press

20. Acknowledgements • This work was supported by • Spanish National Research Project TIC-2001-3236-C02-01 • National Science Foundation Presidential Early Career Award for Scientists and Engineers (PECASE)ECS-9985004