volume | PIER Journals

Abstract

We present a highly accurate frequency-domain finite-difference algorithm for computing mode field solutions of microwave waveguides with regular and reentrant corners. Based on FBS (Fourier-Bessel series)-derived 3-by-3 compact coefficients, our method allows for a flexible layout of the 2-D uniform grids so that distance from the waveguide boundaries to the adjacent unknowns can be arbitrary. Fourth to sixth-order convergent rates of the proposed coefficients are verified by resonance-frequency error analysis for rectangular microwave waveguides for both TE/TM polarizations. We also study the first four Neumann/Dirichlet eigenvalues of the L-shaped MW-WGs calculated by the flexible scheme, and the Neumann results are reported for the first time. Although our results achieve sixth-order accuracy for analytic modes, the order of accuracy is about one and a third for both fundamental TE and TM modes due to singularity around the reentrant corner.

1. Motz, H., "The treatment of singularities of partial differential equations by relaxation methods," Quart. Appl. Math., Vol. 4, 371-377, 1947.
doi:10.1090/qam/18442

2. Arfken, G. B. and H. J. Weber, Mathematical Methods for Physicists, Elsevier Academic Press, 2005.

3. Antipov, Y. A. and V. V. Sivestrov, "Diffraction of a plane wave by a right-angled penetrable wedge," Radio Science, Vol. 42, RS4006, 2007.
doi:10.1029/2007RS003646

4. 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.
doi:10.1109/JLT.2002.800361

5. 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.
doi:10.2528/PIER10092807

6. 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

7. Magura, S., S. Petropavlovsky, S. Tsynkov, and E. Turkel, "High-order numerical solution of the Helmholtz equation for domains with reentrant corners," Applied Numerical Mathematics, Vol. 118, 87-116, 2017.
doi:10.1016/j.apnum.2017.02.013

8. Pozar, D. M., Microwave Engineering, John Wiley & Sons, Inc., 2012.

9. Fox, L. and R. Sankar, "Boundary singularities in linear elliptic differential equations," J. Inst. Math. Applic., Vol. 5, 340-350, 1969.
doi:10.1093/imamat/5.3.340

10. Singer, I. and E. Turkel, "High-order finite difference methods for the Helmholtz equation," Comput. Methods Appl. Mech. Engrg., Vol. 163, 343-358, 1998.
doi:10.1016/S0045-7825(98)00023-1

11. Nabavi, M., "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

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. Mu, S.-Y. and H.-W. Chang, "Dispersion and Local-Error Analysis of Compact LFE-27 Formula for Obtaining Sixth-Order Accurate Numerical Solutions of 3D Helmholtz Equation," Progress In Electromagnetics Research, Vol. 143, 285-314, 2013.
doi:10.2528/PIER13090103

14. Yuan, Q. and Z. He, "Bounds to eigenvalues of the Laplacian on L-shaped domain by variational methods," Journal of Computational and Applied Mathematics, Vol. 233, 1083-1090, 2009.
doi:10.1016/j.cam.2009.08.114

15. Jones, R. S., "Computing ultra-precise eigenvalues of the Laplacian within polygons," Advances in Computational Mathematics, Vol. 43, 1325-1354, 2017.
doi:10.1007/s10444-017-9527-y

16. Fox, L., P. Henrici, and C. Moler, "Approximations and bounds for eigenvalues of elliptic operators," SIAM J. NUMER. ANAL., Vol. 4, No. 1, 89-102, 1967.
doi:10.1137/0704008

17. Liu, X.-F. and S. Oishi, "Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape," SIAM J. Numer. Anal., Vol. 51, No. 3, 1634-1654, 2013.
doi:10.1137/120878446

Dr. Sin-Yuan Mu

Prof. Hung-Wen Chang