Modified Compact Central Finite Difference Schemes For The Simulations Of Wave Equation At Any Wave Number

In this paper, we present modified central finite difference (C.F.D.) schemes for solution of the

wave equation. The modified schemes (a) provide highly accurate solutions at nodes of the spatial

grid for all time steps; (b) preserve the compact stencil structure as of standard C.F.D. scheme and

higher order accuracy is achieved without implementation of new code; (c) offer highly accurate

solutions for low as well as high wave numbers without use of fine grid. Finally, in order to display

superiority of modified C.F.D. schemes numerical computations and graphs are presented for

applications such vibrating string and wave propagations compared with standard C.F.D. schemes.

[1] M. Ainsworth and H. A. Wajid, “Optimally blended spectral-finite element scheme for wave

propagation and nonstandard reduced integration,” SIAM Journal on Numerical Analysis,

vol. 48, no. 1, pp. 346–371, 2010.

[2] G. C. Cohen, Higher order numerical methods for transient wave equations. ASA, 2003.

[3] F. Ihlenburg, Finite element analysis of acoustic scattering. Springer Science & Business

Media, 2006, vol. 132.

[4] L. L. Thompson and P. M. Pinsky, “Complex wavenumber fourier analysis of the p-version

finite element method,” Computational Mechanics, vol. 13, no. 4, pp. 255–275, 1994.

[5] H. A.Wajid and S. Ayub, “An optimally blended finite-spectral element scheme with minimal

dispersion for maxwell equations,” Journal of Computational Physics, vol. 231, no. 24, pp.

8176–8187, 2012.

[6] F. Ihlenburg and I. Babuška, “Finite element solution of the helmholtz equation with high

wave number part i: The h-version of the fem,” Computers & Mathematics with Applications,

vol. 30, no. 9, pp. 9–37, 1995.

[7] F. Ihlenburg and I. Babuska, “Finite element solution of the helmholtz equation with high

wave number part ii: the hp version of the fem,” SIAM Journal on Numerical Analysis,

vol. 34, no. 1, pp. 315–358, 1997.

[8] I. Babuška, F. Ihlenburg, E. T. Paik, and S. A. Sauter, “A generalized finite element method

for solving the helmholtz equation in two dimensions with minimal pollution,” Computer

methods in applied mechanics and engineering, vol. 128, no. 3-4, pp. 325–359, 1995.

[9] M. Nabavi, M. 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, pp. 972–982, 2007.

[10] Y. Fu, “Compact fourth-order finite difference schemes for helmholtz equation with high

wave numbers,” Journal of Computational Mathematics, pp. 98–111, 2008.

[11] G. Sutmann, “Compact finite difference schemes of sixth order for the helmholtz equation,”

Journal of Computational and Applied Mathematics, vol. 203, no. 1, pp. 15–31, 2007.

[12] J. W. Nehrbass, 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, pp. 1194–1201, 1998.

[13] Y. S.Wong and G. Li, “Exact finite difference schemes for solving helmholtz equation at any

wavenumber,” International Journal of Numerical Analysis and Modeling, Series B, vol. 2,

no. 1, pp. 91–108, 2011.

[14] H. A. Wajid, N. Ahmed, H. Iqbal, and M. S. Arshad, “Modified finite difference schemes

on uniform grids for simulations of the helmholtz equation at any wave number,” Journal of

Applied Mathematics, vol. 2014, 2014.

[15] H. A. Wajid, “Simulations of the helmholtz equation at any wave number for adaptive grids

using a modified central finite difference scheme,” Turkish Journal of Mathematics, vol. 40,

no. 4, pp. 806–815, 2016.

[16] E. Twizell, Computational methods for partial differential equations. Ellis Horwood, 1984, no. 1.

[17] 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, pp. 343–358, 1998.