This note is concerned with a semi-analytical method for
the solution of 2-D Helmholtz equation in unit square. The method uses
orthogonal functions to project the problem down to finite dimensional space.
After the projection, the problem simplifies to that of obtaining solutions for
second order constant coefficient differential equations which can be done
analytically. Numerical results indicate that the method is particularly useful
for very high wave numbers.
References
[1]
G. E. Owen, “Introduction to Electromagnetic Theory,” Dover, New York, 2003.
[2]
I. Singer and E. Turkel, “High-Order Finite Difference Methods for the Helmholtz Equation,” Computer Methods in Applied Mechanics and Engineering, Vol. 163, No. 1-4, 1998, pp. 343-358. http://dx.doi.org/10.1016/S0045-7825(98)00023-1
[3]
M. Navabi, 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, No. 3-5, 2007, pp. 972-982. http://dx.doi.org/10.1016/j.jsv.2007.06.070
[4]
P. Nadukandi, E. Onate and J. Garcia, “A Fouth-Order Compact Scheme for the Helmholtz Equation: Alpha-Interpolation and FEM and FDM Stencils,” International Journal for Numerical Methods in Engineering, Vol. 86, No. 1, 2011, pp. 18-46. http://dx.doi.org/10.1002/nme.3043
[5]
H. Dogan, V. Popov and E. Hin Ooi, “The Radial Basis Integral Equation Method for Solving the Helmholtz Equation,” Engineering Analysis with Boundary Elements, Vol. 36, No. 6, 2012, pp. 934-943. http://dx.doi.org/10.1016/j.enganabound.2011.12.003
[6]
X. Feng and H. Wu, “hp-Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number,” Mathematics of Computation, Vol. 80, No. 276, 2011, pp. 1997-2024. http://dx.doi.org/10.1090/S0025-5718-2011-02475-0
[7]
G. F. Dargush and M. M. Grigoiev, “A Multi-Level Multi-Integral Algorithm for the Helmholtz Equation,” Proceedings of IMECE-2005, 5-11 November 2005, Orlando, pp. 1-8.
[8]
K. Otto and E. Larsson, “Iterative Solution of the Helmholtz Equation by a Second-Order Method,” SIAM Journal on Matrix Analysis and Applications, Vol. 21, No. 1, 1999, pp. 209-229. http://dx.doi.org/10.1137/S0895479897316588
[9]
D. Gordon and R. Gordon, “Robust and Highly Scalable Parallel Solution of the Helmholtz Equation with Large Wave Numbers,” Journal of Computational and Applied Mathematics, Vol. 237, No. 1, 2013, pp. 182-196. http://dx.doi.org/10.1016/j.cam.2012.07.024
[10]
Y. Zhuang and X. H. Sun, “A High-Order ADI Method for Separable Generalized Helmholtz Equations,” Advances in Engineering Software, Vol. 31, No. 8-9, 2000, pp. 585-591. http://dx.doi.org/10.1016/S0965-9978(00)00026-0
[11]
B. N. Li, L. Cheng, A. J. Deek and M. Zhao, “A Semi-Analytical Solution Method for Two-Dimensional Helmholtz Equation,” Applied Oscean Research, Vol. 28, No. 3, 2006, pp. 193-207. http://dx.doi.org/10.1016/j.apor.2006.06.003
[12]
G. Bao, G. W. Wei and S. Zhao, “Numerical Solution of the Helmholtz Equation with High Wavenumbers,” International Journal for Numerical Methods in Engineering, Vol. 59, No. 3, 2004, pp. 389-408. http://dx.doi.org/10.1002/nme.883
[13]
Z. C. Li, “The Trefftz Method for the Helmholtz Equation with Degeneracy,” Applied Numerical Mathematics, Vol. 58, No. 2, 2008, pp. 131-159. http://dx.doi.org/10.1016/j.apnum.2006.11.004
