Deficiency of finite difference methods for capturing shock waves and wave propagation over uneven bottom seabed


  • Mohammad Fadhli Ahmad
  • Mohd Sofiyan Suliman
  • . .





Numerical schemes, dam break, finite difference method, wave propagation, uneven bottom seabed


The implementation of finite difference method is used to solve shallow water equations under the extreme conditions. The cases such as dam break and wave propagation over uneven bottom seabed are selected to test the ordinary schemes of Lax-Friederichs and Lax-Wendroff numerical schemes. The test cases include the source term for wave propagation and exclude the source term for dam break. The main aim of this paper is to revisit the application of Lax-Friederichs and Lax-Wendroff numerical schemes at simulating dam break and wave propagation over uneven bottom seabed. For the case of the dam break, the two steps of Lax-Friederichs scheme produce non-oscillation numerical results, however, suffering from some of dissipation. Moreover, the two steps of Lax-Wendroff scheme suffers a very bad oscillation. It seems that these numerical schemes cannot solve the problem at discontinuities which leads to oscillation and dissipation. For wave propagation case, those numerical schemes produce inaccurate information of free surface and velocity due to the uneven seabed profile. Therefore, finite difference is unable to model shallow water equations under uneven bottom seabed with high accuracy compared to the analytical solution.




[1] B. S. Halpern et al., "A Global Map of Human Impact on Marine Ecosystems," Science, vol. 319, no. 5865, pp. 948-952, 2008.

[2] K. Klingbeil, F. Lemarié, L. Debreu, and H. Burchard, "The numerics of hydrostatic structured-grid coastal ocean models: State of the art and future perspectives," Ocean Modelling, vol. 125, pp. 80-105, 2018.

[3] M. S. Sulaiman, S. K. Sinnakaudan, S. F. Ng, and K. Strom, "Application of automated grain sizing technique (AGS) for bed load samples at Rasil River: A case study for supply limited channel," CATENA, vol. 121, pp. 330-343, 2014.

[4] M. S. Sulaiman, S. K. Sinnakaudan, S. F. Ng, and K. Strom, "Occurrence of bed load transport in the presence of stable clast," International Journal of Sediment Research, vol. 32, no. 2, pp. 195-209, 2017.

[5] M. S. Sulaiman, S. K. Sinnakaudan, Q. Y. Goh, M. F. Ahmad, and S. Nurhidayu, "Performance Of “Reference†Critical Shields Stress And Bed-Load Formular Using Different Particle Size Representative: A Case Study For Coarse Bedded Streams," International Journal of Civil Engineering and Technology, vol. 3, no. 3, pp. 747-772, 2018.

[6] W.-C. Liu, M.-H. Hsu, and A. Y. Kuo, "Modelling of hydrodynamics and cohesive sediment transport in Tanshui River estuarine system, Taiwan," Marine Pollution Bulletin, vol. 44, no. 10, pp. 1076-1088, 2002.

[7] N. odd and m. owen, "Summary of paper 7517. A two-layer model of mud transport in the thames estuary," Proceedings of the Institution of Civil Engineers, vol. 51, no. 4, p. 714, 1972.

[8] C. Vincenzo and C. R. T., "Semi-implicit finite difference methods for three-dimensional shallow water flow," International Journal for Numerical Methods in Fluids, vol. 15, no. 6, pp. 629-648, 1992.

[9] C. Brehm, M. F. Barad, J. A. Housman, and C. C. Kiris, "A comparison of higher-order finite-difference shock capturing schemes," Computers and Fluids, vol. 122, pp. 184-208, 2015.

[10] A. Tveito, "Convergence and stability of the Lax-Friedrichs scheme for a nonlinear parabolic polymer flooding problem," Advances in Applied Mathematics, vol. 11, no. 2, pp. 220-246, 1990.

[11] L. Peter and W. Burton, "Systems of conservation laws," Communications on Pure and Applied Mathematics, vol. 13, no. 2, pp. 217-237, 1960.

[12] D. Fridrich, R. Liska, and B. Wendroff, "Some cell-centered Lagrangian Lax–Wendroff HLL hybrid schemes," Journal of Computational Physics, vol. 326, pp. 878-892, 2016.

[13] L. F. Shampine, "Two-step Lax–Friedrichs method," Applied Mathematics Letters, vol. 18, no. 10, pp. 1134-1136, 2005.

[14] J. J. Stoker, Water Waves: The Mathematical Theory with Applications. Wiley, 2011.

[15] A. Bermudez and M. E. Vazquez, "Upwind methods for hyperbolic conservation laws with source terms," Computers and Fluids, vol. 23, no. 8, pp. 1049-1071, 1994.

[16] R. Liska and B. Wendroff, "Composite Schemes for Conservation Laws," SIAM Journal on Numerical Analysis, vol. 35, no. 6, pp. 2250-2271, 1998.

[17] J. Hudson and P. K. Sweby, "Formulations for Numerically Approximating Hyperbolic Systems Governing Sediment Transport," Journal of Scientific Computing, vol. 19, no. 1, pp. 225-252, 2003.

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How to Cite

Fadhli Ahmad, M., Sofiyan Suliman, M., & ., . (2018). Deficiency of finite difference methods for capturing shock waves and wave propagation over uneven bottom seabed. International Journal of Engineering & Technology, 7(3.28), 97–101.
Received 2018-10-04
Accepted 2018-10-04
Published 2018-08-17