The frequency-domain finite-difference (FD-FD) methods have been successfully used to obtain numerical solutions of two-dimensional (2-D) Helmholtz equation. The standard second-order accurate FD-FD scheme is known to produce unwanted numerical spatial and temporal dispersions when the sampling is inadequate. Recently compact higher-order accurate FD-FD methods have been proposed to reduce the spatial sampling density. We present a semi-analytical solution of 2-D homogeneous Helmholtz equation by connecting overlapping square patches of local fields where each patch is expanded in a set of Fourier-Bessel (FB) series. These local FB coefficients are related to total eight points, four on the sides and four on the corners, on the square patch. The local field expansion (LFE) analysis leads to an improved, compact FD-like, nine-point stencil for the 2-D homogeneous Helmholtz equation. We show that LFE formulation possesses superior numerical properties of being low dispersive and nearly isotropic because this method of connecting local fields merely ties these overlapping EM field patches already satisfy the Helmholtz equation.
"Semi-Analytical Solutions of 2-d
Homogeneous Helmholtz Equation by the Method of Connected Local Fields," Progress In Electromagnetics Research,
Vol. 109, 399-424, 2010. doi:10.2528/PIER10092807
1. Hadley, G. R., "High-accuracy finite-difference equations for dielectric waveguides I: Uniform regions and dielectric interfaces," Journal of Lightwave Technology, Vol. 20, No. 7, 1210-1218, 2002. doi:10.1109/JLT.2002.800361
2. Hadley, G. R., "High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners," Journal of Lightwave Technology, Vol. 20, No. 7, 1219-1231, 2002. doi:10.1109/JLT.2002.800371
3. Li, L.-Y. and J.-F. Mao, "An improved compact 2-D finite-difference frequency-domain method for guided wave structures," IEEE Microwave and Wireless Components Letters, Vol. 13, No. 12, 520-522, 2003. doi:10.1109/LMWC.2003.819956
4. Yu, C.-P. and H. C. Chang, "Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals," Optics Express, Vol. 12, 1397-1408, 2004. doi:10.1364/OPEX.12.001397
5. Chang, H. W. and W. C. Cheng, "Analysis of dielectric waveguide termination with tilted facets by analytic continuity method," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 12, 1653-1662, 2007.
6. Zhao, W., H. W. Deng, and Y. J. Zhao, "Application of 4-component compact 2-D FDFD method in analysis of lossy circular metal waveguide," Journal of Electromagnetic Waves and Applications, Vol. 22, No. 18-18, 2297-2308, 2008. doi:10.1163/156939308787543930
7. Kusiek, A. and J. Mazur, "Analysis of scattering from arbitrary configuration of cylindrical objects using hybrid FD mode-matching method," Progress In Electromagnetics Research, Vol. 97, 105-127, 2009. doi:10.2528/PIER09072804
8. Chang, H.-W., Y.-H. Wu and W.-C. Cheng, "Hybrid FD-FD analysis of crossing waveguides by exploiting both the plus and the cross structural symmetry," Progress In Electromagnetics Research, Vol. 103, 217-240, 2010. doi:10.2528/PIER10030202
9. Chang, H.-W. and Y.-H. Wu, "Analysis of perpendicular crossing dielectric waveguides with various typical index contrasts and intersection profiles," Progress In Electromagnetics Research, Vol. 108, 323-341, 2010. doi:10.2528/PIER10081008
10. Mittra, R. and U. Pekel, "A new look at the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves," IEEE Microwave and Guided Wave Letter, Vol. 5, No. 3, 84-86, 1995. doi:10.1109/75.366461
11. Zheng, S., W.-Y. Tam, D.-B. Ge, and J.-D. Xu, "Uniaxial PML absorbing boundary condition for truncating the boundary of DNG metamaterials," Progress In Electromagnetics Research Letters, Vol. 8, 125-134, 2009. doi:10.2528/PIERL09030901
12. Chang, H.-W., W.-C. Cheng, and S.-M. Lu, "Layer-mode transparent boundary condition for the hybrid FD-FD method," Progress In Electromagnetics Research, Vol. 94, 175-195, 2009. doi:10.2528/PIER09061606
13. Jo, C.-H., C. Shin, and J.-H. Suh, "An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator," Geophysics, Vol. 61, No. 2, 529-537, 1996. doi:10.1190/1.1443979
14. Singer, I. and E. Turkel, "High-order finite difference method for the Helmholtz equation," Computer Methods in Applied Mechanics and Engineering, Vol. 163, 343-358, 1998. doi:10.1016/S0045-7825(98)00023-1
15. Singer, I. and E. Turkel, "Sixth order accurate finite difference schemes for the Helmholtz equation," Journal of Computational Acoustics, Vol. 14, 339-351, 2006. doi:10.1142/S0218396X06003050
16. Nabavi, M., M. H. K. Siddiqui, and J. Dargahi, "A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation," Journal of Sound and Vibration, Vol. 307, 972-982, 2007. doi:10.1016/j.jsv.2007.06.070
17. Sutmann, G., "Compact finite difference schemes of sixth order for the Helmholtz equation," Journal of Computational and Applied Mathematics, Vol. 203, 15-31, 2007. doi:10.1016/j.cam.2006.03.008
18. Smith, G. D., Numerical Solution of Partial Differential Equations, Finite Difference Methods, 3 Ed., ISBN 19 859650 2, Glarendon Press, Oxford, 1985.
19. Hall, C. A. and T. A. Porsching, Numerical Analysis of Partial Differential Equations, 2 Ed., ISBN 9780136265573, Prentice Hall, Englewood Cliffs, NJ 07632, 1990.
20. Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equations, 2 Ed., ISBN 0-89871-567-9, Siam, 2004. doi:10.1137/1.9780898717938
21. Chang, H.-W. and S.-Y. Mu, "Novel nine-point FD coefficients for the two-dimensional Helmholtz equation," Cross Strait Tri-Regional Radio Science and Wireless Technology Conference, Hanan, China, 2010.
22. Nehrbass, J. W., J. O. Jevti'c, and R. Lee, "Reducing the phase error for finite-difference methods without increasing the order," IEEE Trans. on Antennas and Propagation, Vol. 46, No. 8, 1194-1201, 1998. doi:10.1109/8.718575
23. Burden, R. L. and J. D. Faires, Numerical Analysis, Brooks/Cole, Pacific Grove, CA, 2001.
24. Lehmann, T. M., C. Gonner, and K. Spitzer, "Survey: Interpolation methods in medical image processing," IEEE Transactions on Medical Imaging, Vol. 18, No. 11, 1049-1075, 1999. doi:10.1109/42.816070
25. Elsherbeni, A. and V. Demir, The Finite-difference Time-domain Method for Electromagnetics with MATLAB Simulations, ISBN 9789746521048, SciTech Pub., Raleigh, NC 27615, 2009.
26. Ishimaru, A., Electromagnetic Propagation, Radiation, and Scattering, Prentice Hall, Englewood Cliffs, N.J., 1991.
27. Saleh, B. E. A. and M. C. Teich, Fundamental of Photonics, John Wiley & Son, New York, 1991. doi:10.1002/0471213748
28. Chang, H.-W., Y.-H. Wu, S.-M. Lu, W.-C. Cheng, and M.-H. Sheng, "Field analysis of dielectric waveguide devices based on coupled transverse-mode integral equation --- Numerical investigation," Progress In Electromagnetics Research, Vol. 97, 159-176, 2009. doi:10.2528/PIER09091402
29. Chang, H.-W. and M.-H. Sheng, "Field analysis of dielectric waveguide devices based on coupled transverse-mode integral equation --- Mathematical and numerical formulations," Progress In Electromagnetics Research, Vol. 78, 329-347, 2008. doi:10.2528/PIER07091002
30. Reiter, J. M. and F. Arndt, "Rigorous analysis of arbitrarily shaped H- and E-plane discontinuities in rectangular waveguides by a full-wave boundary contour mode-matching method," IEEE Trans. on Microwave Theory and Techniques, Vol. 43, No. 6, 796-801, 1995. doi:10.1109/22.375226
31. Kusiek, A. and J. Mazur, "Analysis of scattering from arbitrary configuration of cylindrical objects using hybrid FD mode-matching method," Progress In Electromagnetics Research, Vol. 97, 105-127, 2009. doi:10.2528/PIER09072804
32. Mohammad, R. Z., K. C. Donepudi, J.-M. Jin, and W. C. Chew, "Efficient time-domain and frequency-domain finite-element solution of Maxwell's equations using spectral Lanczos decomposition method," IEEE Trans. on Microwave Theory and Techniques, Vol. 46, No. 8, 1141-1149, 1998. doi:10.1109/22.704957
33. Fan, Z., D.-Z. Ding, and R.-S. Chen, "The efficient analysis of electromagnetic scattering from composite structures using hybrid CFIE-IEFIE," Progress In Electromagnetics Research B, Vol. 10, 131-143, 2009.
34. Chiou, Y. P., Y. C. Chiang, and H. C. Chang, "Improved three-point formulae considering the interface conditions in the finite-difference analysis of step-index optical devices," Journal of Lightwave Technology, Vol. 18, No. 2, 243-251, 2000. doi:10.1109/50.822799