M. TADI

Open Access JAMP

57

ods 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-In-

terpolation 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 Meth-

ods for the Helmholtz Equation with Large Wave Num-

ber,” 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, Or-

lando, pp. 1-8.

[8] K. Otto and E. Larsson, “Iterative Solution of the Helm-

holtz Equation by a Second-Order Method,” SIAM Jour-

nal 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,” Ad-

vances 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 Helm-

holtz 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,” In-

ternational Journal for Numerical Methods in Engineer-

ing, 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

Appendix

In 3-D one can assume a form given by

11

,,sin sin,

ij

ij

uxyzf yixjz

d

(14)

here, after using the orthogonality conditions, the func-

tions are given by

)( yfij

11

00

4,,sinsind

ij .

yuxyzixjzx

z

(15)

Also, for the 3-D Helmholtz equation, one needs to split

the problem into three similar problems.