Comparison of Finite Difference Schemes for the Wave Equation Based on Dispersion

DOI: 10.4236/jamp.2015.311179   PDF   HTML   XML   3,450 Downloads   4,359 Views   Citations

Abstract

Finite difference techniques are widely used for the numerical simulation of time-dependent partial differential equations. In order to get better accuracy at low computational cost, researchers have attempted to develop higher order methods by improving other lower order methods. However, these types of methods usually suffer from a high degree of numerical dispersion. In this paper, we review three higher order finite difference methods, higher order compact (HOC), compact Padé based (CPD) and non-compact Padé based (NCPD) schemes for the acoustic wave equation. We present the stability analysis of the three schemes and derive dispersion characteristics for these schemes. The effects of Courant Friedrichs Lewy (CFL) number, propagation angle and number of cells per wavelength on dispersion are studied.

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Abdulkadir, Y. (2015) Comparison of Finite Difference Schemes for the Wave Equation Based on Dispersion. Journal of Applied Mathematics and Physics, 3, 1544-1562. doi: 10.4236/jamp.2015.311179.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Canale, R.P. and Chapra, S.C. (1998) Numerical Methods for Engineers with Programming and Software Applications. 3rd Edition, McGraw-Hill, Hoboken.
[2] Zhou, H.M. (Editor) (2013) Computer Modelling for Injection Modelling Simulation Optimization and Control. John Wiley and Sons, Hoboken.
[3] Krumpholz, M. and Katehi, L.P.B. (1996) MRTD: New Time-Domain Schemes Based on Multiresolution Analysis. IEEE Transactions on Microwave Theory and Techniques, 44, 555-571.
[4] Lee, T.-W. and Hagness, S.C. (2004) Pseudospectral Time-Domain Methods for Modeling Optical Wave Propagation in Second-Order Nonlinear Materials. Journal of the Optical Society of America B, 21, 330-342.
http://dx.doi.org/10.1364/JOSAB.21.000330
[5] Das, S., Liao, W.Y. and Gupta, A. (2013) An Efficient Fourth-Order Low Dispersive Finite Difference Scheme for 2-D Acoustic Wave Equation. Journal of computational and Applied Mathematics, 258, 151-167.
[6] Liao, W.Y. (2013) On the Dispersion, Stability and Accuracy of a Compact Higher-Order Finite Difference Scheme for 3D Acoustic Wave Equation. Journal of Computational and Applied Mathematics, 270, 571-583.
[7] Moin, P. (2010) Fundamentals of Engineering Numerical Analysis. 2nd Edition, Cambridge University Press, New York.
http://dx.doi.org/10.1017/CBO9780511781438
[8] Liu, Y. and Sen, M.K. (2009) Advanced Finite-Difference Methods for Seismic Modeling. Geohorizons, 14, 5-16.
[9] Smith, G.D. (1985) Numerical Solutions of Partial Differential Equations: Finite Difference Methods. 3rd Edition, Oxford University Press, New York.
[10] Strikwerda, J.C. (1989) Finite Difference Schemes and Partial Differential Equations. 2nd Edition, Wadsworth and Brooks-Cole.
[11] Bradie, B. (2006) A Friendly Introduction to Numerical Analysis. Pearson International Edition, Pearson Education Inc., New Jersey.
[12] Kowalczyk, K. and Van Walstijn, M. (2010) A Comparison of Nonstaggered Compact FDTD Schemes for the 3D Wave Equation. IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), Dallas, 14-19 March 2010, 197-200.

  
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