We present the dispersion and local-error analysis of the twenty-seven point local field expansion (LFE-27) formula for obtaining highly accurate semi-analytical solutions of the Helmholtz equation in a 3D homogeneous medium. Compact finite-difference (FD) stencils are the cornerstones in frequency-domain FD methods. They produce block tri-diagonal matrices which require much less computing resources compared to other non-compact stencils. LFE-27 is a 3D compact FD-like stencil used in the method of connected local fields (CLF) . In this paper, we show that LFE-27 possesses such good numerical quality that it is accurate to the sixth order. Our analyses are based on the relative error studies of numerical phase and group velocities. The classical second-order FD formula requires more than twenty sampling points per wavelength to achieve less than 1% relative error in both phase and group velocities, whereas LFE-27 needs only three points per wavelength to match the same performance.
2. Smith, G. D., Numerical Solution of Partial Differential Equations, 2nd Ed., Oxford University Press, 1978.
3. Hall, C. A. and T. A. Porsching, "Numerical Analysis of Partial Differential Equations," Prentice-Hall, 1990.
4. 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.
5. Nehrbass, J. W., J. O. Jevtic, and R. Lee, "Reducing the phase error for finite-difference methods without increasing the order," IEEE Transactions on Antennas and Propagation, Vol. 46, No. 8, 1194-1201, 1998.
6. 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.
7. Singer, I. and E. Turkel, "Sixth order accurate finite difference schemes for the Helmholtz equation," Journal of Computational Acoustics, Vol. 14, 339-351, 2006.
8. Sutmann, G., "Compact finite difference schemes of sixth order for the Helmholtz equation," Journal of Computational and Applied Mathematics, Vol. 203, 15-31, 2007.
9. Hadley, G. R., "High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces ," Journal of Lightwave Technology, Vol. 20, No. 7, 1210-1218, 2002.
10. Hadley, G. R., "High-accuracy finite-diffference equations for dielectric waveguide analysis II: Dielectric corners," Journal of Lightwave Technology, Vol. 20, No. 7, 1219-1231, 2002.
11. Chang, H.-W. and S.-Y. Mu, "Semi-analytical solutions of the 2-D Homogeneous Helmholtz equation by the method of connected local fields," Progress In Electromagnetics Research, Vol. 109, 399-424, 2010.
12. Mu, S.-Y. and H.-W. Chang, "Theoretical foundation for the method of connected local fields," Progress In Electromagnetics Research, Vol. 114, 67-88, 2011.
13. Tsukerman, I., "Electromagnetic applications of a new finite-difference calculus," IEEE Transaction on Magnetics, Vol. 41, No. 7, 2206-2225, 2005.
14. Fernandes, D. T. and A. F. D. Loula, "Quasi optimal finite difference method for Helmholtz problem on unstructured grids," Int. J. Numer. Meth. Engng., Vol. 82, 1244-1281, 2010.
15. 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.
16. Engquist, B. and A. Majda, "Absorbing boundary conditions for numerical simulation of waves," Applied Mathematical Science, Vol. 74, 1765-1766, 1977.
17. 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.
18. Harari, I. and E. Turkel, "Accurate finite difference methods for time-harmonic wave propagation," Journal of Computational Physics, Vol. 119, No. 2, 252-270, 1995.
19. Trefethen, L. N., "Group velocity in finite difference schemes," SIAM Rev., Vol. 24, No. 2, 113-136, 1982.
20. Anne, L. and Q. H. Tran, "Dispersion and cost analysis of some ¯nite di®erence schemes in one-parameter acoustic wave modeling," Computational Geosciences, Vol. 1, 1-33, 1997.
21. Peterson, A. F., S. L. Ray, and R. Mittra, Computational Method for Electromagnetics, IEEE Press, New York, 1998.
22. Rao, K. R., J. Nehrbass, and R. Lee, "Discretization errors in finite methods: Issues and possible solutions," Comput. Methods Appl. Mech. Engrg., Vol. 169, 219-236, 1999.
23. Taflove, , A. and S. C. Hagness, "Computational Electrodynamics: The Finite-difference Time-domain Method," Artech House, 2005.
24. Spotz, W. F. and G. F. Carey, "A high-order compact formulation for the 3D Poisson equation," Numerical Methods for Partial Differential Equations, Vol. 12, No. 2, 235-243, 1996.
25. Chang, H.-W. and S.-Y. Mu, "3-D LFE-27 formulae for the method of connected local fields," Optics & Photonics Taiwan, International Conference, 2012.
26. Ishimaru, A., Electromagnetic Propagation, Radiation and Scattering, Prentice Hall, Englewood Cliffs, NJ, 1991.