Numerical Simulation of Nonlinear Surface Gravity Waves Transformation under Shallow-Water Conditions

ABSTRACT

This work considers the problems of numerical simulation of non-linear surface gravity waves transformation under shallow bay conditions. The discrete model is built from non-linear shallow-water equations. Are resulted boundary and initial conditions. The method of splitting into physical processes receives system from three equations. Then we define the approximation order and investigate stability conditions of the discrete model. The sweep method was used to calculate the system of equations. This work presents surface gravity wave profiles for different propagation phases.

This work considers the problems of numerical simulation of non-linear surface gravity waves transformation under shallow bay conditions. The discrete model is built from non-linear shallow-water equations. Are resulted boundary and initial conditions. The method of splitting into physical processes receives system from three equations. Then we define the approximation order and investigate stability conditions of the discrete model. The sweep method was used to calculate the system of equations. This work presents surface gravity wave profiles for different propagation phases.

Cite this paper

I. Abbasov, "Numerical Simulation of Nonlinear Surface Gravity Waves Transformation under Shallow-Water Conditions,"*Applied Mathematics*, Vol. 3 No. 2, 2012, pp. 135-141. doi: 10.4236/am.2012.32021.

I. Abbasov, "Numerical Simulation of Nonlinear Surface Gravity Waves Transformation under Shallow-Water Conditions,"

References

[1] G. Chapalain, R. Cointe and A. Temperville, “Observed and Modeled Resonantly Interacting Progressive WaterWaves,” Coastal Engineering Journal, Vol. 16, No. 3, 1992, pp. 267-300. doi:10.1016/0378-3839(92)90045-V

[2] Y. Eldeberky and P.A. Madsen, “Determenistic and Stochastic Evolution Equations for Fully Dispersive and Weakly Nonlinear Waves,” Coastal Engineering Journal, Vol. 38, No. 1, 1999, pp. 1-24. doi:10.1016/S0378-3839(99)00021-6

[3] V. R. Kogan and V. V. Kuznetsov, “Application of the Theory of Analytical Functions in Numerical Simulation of Unsteady Surface Waves,” Zhurnal Vychislitel’Noi Matematiki i Matematicheskoi Fiziki, Vol. 36. No. 10, 1995, pp. 1448-1456.

[4] S. Elgar, C. Norheim, T. Herbers, “Nonlinear Evolution of Surface Wave Spectra on a Beach,” Journal of Physical Oceanography, Vol. 28. No. 7, 1998, pp. 1534-1551. doi:10.1175/1520-0485(1998)028<1534:NEOSWS>2.0.CO;2

[5] А. А. Litvinenko and G. А. Habahpashev, “Numerical Simulation of Nonlinear Rather Long Two-Dimensional Waves in Basins with Gentle Bottom,” Vytchislitelnye Technologii, Vol. 4, No. 3, 1999, pp. 95-105.

[6] K. Kawasaki, “Numerical Simulation of Breaking and Post-Breaking Wave Deformation Process around a Submerged Breakwater,” Coastal Engineering Journal, Vol. 41. No. 3-4, 1999, pp. 201-223. doi:10.1142/S0578563499000139

[7] T. H. Herbers, S. Elgar, N. A. Sarap and R. T. Guza, “Nonlinear Dispersion of Surface Gravity Waves in Shallow Water,” Journal of Physical Oceanography, Vol. 32. No. 4, 2002, pp. 182-1193. doi:10.1175/1520-0485(2002)032<1181:NDOSGW>2.0.CO;2

[8] T. T. Janssen, T. H. Herbers 2002 and J. A. Battjes, “Generalized Evolution Equations for Nonlinear Surface Gravity Waves over Two-Dimensional Topography,” Journal of Fluid Mechanics, Vol. 552, 2006, pp. 393-418. doi:10.1017/S0022112006008743

[9] G. A. El, R. H. Grimshaw and A. M. Kamchatnov, “Evolution of Solitary Waves and Undular Bores in ShallowWater Ows over a Gradual Slope with Bottom Friction,” Journal of Fluid Mechanics, Vol. 585, 2007, pp. 213-244. doi:10.1017/S0022112007006817

[10] R. Camassa, J. Huang and L. Lee, “On a Completely Integrable Numerical Scheme for a Nonlinear ShallowWater Wave Equation,” Journal of Nonlinear Mathematical Physics, Vol. 12. No. 1, 2005, pp. 146-162. doi:10.2991/jnmp.2005.12.s1.13

[11] H. Lamb, “Hydrodynamics,” Dover Publications, Mineola, 1930, p. 524.

[12] G. B. Whitham, “Linear and Nonlinear Waves,” Wiley, New York, 1974, p. 622.

[13] R. Richtmyer and K. Morton, “Difference Methods for Initial-Value Problems,” 2nd Edition, Wiley, New York, 1967, p. 309.

[14] M. Holt, “Numerical Methods in Fluid Dynamics,” SpringerVerlag, New York, 1977, p. 305.

[15] А. А. Samarskyi, “Introduction to Numerical Methods,” Nauka, Glavnaja Redaktsija Fiziko-Matematicheskoi Literatury, Moscow, 1987, p. 288.

[16] “Hydrometeorology and Hydrochemistry of the Seas in the USSR,” Gidrometeoizdat, Vol. 5, The Azov Sea, Saint-Petersburg, 1991, pp. 75-88.

[17] V. А. Mamykina and Yu. P. Khrustalev, “Coastal Zone of the Azov Sea,” Izdatelstvo Rostov State University, Rostov-on-Don., 1980, p. 176.

[18] Y. Goda and K. Morinobu, “Breaking Wave Heights on Horizontal Bed Affected by Approach Slope,” Coastal Engineering Journal, Vol. 40, No. 4, 1998, pp. 307-326. doi:10.1142/S0578563498000182

[19] I. B. Abbasov, “Study and Simulation of Nonlinear Surface Gravity Waves under Shallow-Water Conditions,” Izvestiya RAN, Atmospheric and Oceanic Physics, Vol. 39, No. 4, 2003, pp. 506-511.

[20] I. B. Abbasov, “Transformation of Nonlinear Surface Gravity Waves under Shallow-Water Conditions,” Applied Mathematics, Vol. 1, No. 4, 2010, pp. 260-264. doi:10.4236/am.2010.14032

[1] G. Chapalain, R. Cointe and A. Temperville, “Observed and Modeled Resonantly Interacting Progressive WaterWaves,” Coastal Engineering Journal, Vol. 16, No. 3, 1992, pp. 267-300. doi:10.1016/0378-3839(92)90045-V

[2] Y. Eldeberky and P.A. Madsen, “Determenistic and Stochastic Evolution Equations for Fully Dispersive and Weakly Nonlinear Waves,” Coastal Engineering Journal, Vol. 38, No. 1, 1999, pp. 1-24. doi:10.1016/S0378-3839(99)00021-6

[3] V. R. Kogan and V. V. Kuznetsov, “Application of the Theory of Analytical Functions in Numerical Simulation of Unsteady Surface Waves,” Zhurnal Vychislitel’Noi Matematiki i Matematicheskoi Fiziki, Vol. 36. No. 10, 1995, pp. 1448-1456.

[4] S. Elgar, C. Norheim, T. Herbers, “Nonlinear Evolution of Surface Wave Spectra on a Beach,” Journal of Physical Oceanography, Vol. 28. No. 7, 1998, pp. 1534-1551. doi:10.1175/1520-0485(1998)028<1534:NEOSWS>2.0.CO;2

[5] А. А. Litvinenko and G. А. Habahpashev, “Numerical Simulation of Nonlinear Rather Long Two-Dimensional Waves in Basins with Gentle Bottom,” Vytchislitelnye Technologii, Vol. 4, No. 3, 1999, pp. 95-105.

[6] K. Kawasaki, “Numerical Simulation of Breaking and Post-Breaking Wave Deformation Process around a Submerged Breakwater,” Coastal Engineering Journal, Vol. 41. No. 3-4, 1999, pp. 201-223. doi:10.1142/S0578563499000139

[7] T. H. Herbers, S. Elgar, N. A. Sarap and R. T. Guza, “Nonlinear Dispersion of Surface Gravity Waves in Shallow Water,” Journal of Physical Oceanography, Vol. 32. No. 4, 2002, pp. 182-1193. doi:10.1175/1520-0485(2002)032<1181:NDOSGW>2.0.CO;2

[8] T. T. Janssen, T. H. Herbers 2002 and J. A. Battjes, “Generalized Evolution Equations for Nonlinear Surface Gravity Waves over Two-Dimensional Topography,” Journal of Fluid Mechanics, Vol. 552, 2006, pp. 393-418. doi:10.1017/S0022112006008743

[9] G. A. El, R. H. Grimshaw and A. M. Kamchatnov, “Evolution of Solitary Waves and Undular Bores in ShallowWater Ows over a Gradual Slope with Bottom Friction,” Journal of Fluid Mechanics, Vol. 585, 2007, pp. 213-244. doi:10.1017/S0022112007006817

[10] R. Camassa, J. Huang and L. Lee, “On a Completely Integrable Numerical Scheme for a Nonlinear ShallowWater Wave Equation,” Journal of Nonlinear Mathematical Physics, Vol. 12. No. 1, 2005, pp. 146-162. doi:10.2991/jnmp.2005.12.s1.13

[11] H. Lamb, “Hydrodynamics,” Dover Publications, Mineola, 1930, p. 524.

[12] G. B. Whitham, “Linear and Nonlinear Waves,” Wiley, New York, 1974, p. 622.

[13] R. Richtmyer and K. Morton, “Difference Methods for Initial-Value Problems,” 2nd Edition, Wiley, New York, 1967, p. 309.

[14] M. Holt, “Numerical Methods in Fluid Dynamics,” SpringerVerlag, New York, 1977, p. 305.

[15] А. А. Samarskyi, “Introduction to Numerical Methods,” Nauka, Glavnaja Redaktsija Fiziko-Matematicheskoi Literatury, Moscow, 1987, p. 288.

[16] “Hydrometeorology and Hydrochemistry of the Seas in the USSR,” Gidrometeoizdat, Vol. 5, The Azov Sea, Saint-Petersburg, 1991, pp. 75-88.

[17] V. А. Mamykina and Yu. P. Khrustalev, “Coastal Zone of the Azov Sea,” Izdatelstvo Rostov State University, Rostov-on-Don., 1980, p. 176.

[18] Y. Goda and K. Morinobu, “Breaking Wave Heights on Horizontal Bed Affected by Approach Slope,” Coastal Engineering Journal, Vol. 40, No. 4, 1998, pp. 307-326. doi:10.1142/S0578563498000182

[19] I. B. Abbasov, “Study and Simulation of Nonlinear Surface Gravity Waves under Shallow-Water Conditions,” Izvestiya RAN, Atmospheric and Oceanic Physics, Vol. 39, No. 4, 2003, pp. 506-511.

[20] I. B. Abbasov, “Transformation of Nonlinear Surface Gravity Waves under Shallow-Water Conditions,” Applied Mathematics, Vol. 1, No. 4, 2010, pp. 260-264. doi:10.4236/am.2010.14032