A finite element method for extended KdV equations

• Autorzy: Karczewska A. , Rozmej P. , Szczeciński M. , Boguniewicz B.

Identyfikatory

DOI

10.1515/amcs-2016-0039

• Języki publikacji: EN

Abstrakty

EN

The finite element method (FEM) is applied to obtain numerical solutions to a recently derived nonlinear equation for the shallow water wave problem. A weak formulation and the Petrov–Galerkin method are used. It is shown that the FEM gives a reasonable description of the wave dynamics of soliton waves governed by extended KdV equations. Some new results for several cases of bottom shapes are presented. The numerical scheme presented here is suitable for taking into account stochastic effects, which will be discussed in a subsequent paper.

Słowa kluczowe

EN

shallow water wave problem; nonlinear equations; second order KdV equations; finite element method; Petrov–Galerkin method

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2016
• Tom: Vol. 26, no. 3
• Strony: 555--567
• Opis fizyczny: Bibliogr. 38 poz., wykr.

Twórcy

autor

Karczewska A.

• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland

autor

Rozmej P.

• Institute of Physics, Faculty of Physics and Astronomy, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland

autor

Szczeciński M.

• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland

autor

Boguniewicz B.

• Institute of Physics, Faculty of Physics and Astronomy, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland

Bibliografia

• [1] Ablowitz, M. and Segur, H. (1979). On the evolution of packets of water waves, Journal of Fluid Mechanics 92(4): 691–715.
• [2] Ali, A. and Kalisch, H. (2014). On the formulation of mass, momentum and energy conservation in the KdV equation, Acta Applicandae Mathematicae 133(1): 113–131.
• [3] Bona, J., Chen, H., Karakashian, O. and Xing, Y. (2013). Conservative, discontinuous-Galerkin methods for the generalized Korteweg–de Vries equation, Mathematics of Computation 82(283): 1401–1432.
• [4] Burde, G. and Sergyeyev, A. (2013). Ordering of two small parameters in the shallow water wave problem, Journal of Physics A 46(7): 075501.
• [5] Cui, Y. and Ma, D. (2007). Numerical method satisfying the first two conservation laws for the Korteweg–de Vries equation, Journal of Computational Physics 227(1): 376–399.
• [6] Debussche, A. and Printems, I. (1999). Numerical simulation of the stochastic Korteweg–de Vries equation, Physica D 134(2): 200–226.
• [7] Dingemans, M. (1997). Water Wave Propagation over Uneven Bottoms, World Scientific, Singapore.
• [8] Drazin, P.G. and Johnson, R.S. (1989). Solitons: An Introduction, Cambridge University Press, Cambridge.
• [9] Fornberg, B. and Whitham, G.B. (1978). A numerical and theoretical study of certain nonlinear wave phenomena, Philosophical Transactions A of the Royal Society 289(1361): 373–404.
• [10] Goda, K. (1975). On instability of some finite difference schemes for the Korteweg–de Vries equation, Journal of the Physical Society of Japan 39(1): 229–236.
• [11] Green, A.E. and Naghdi, P.M. (1976). A derivation of equations for wave propagation in water of variable depth, Journal of Fluid Mechanics 78(2): 237–246.
• [12] Grimshaw, R. (1970). The solitary wave in water of variable depth, Journal of Fluid Mechanics 42(3): 639–656.
• [13] Grimshaw, R.H.J. and Smyth, N.F. (1986). Resonant flow of a stratified fluid over topography, Journal of Fluid Mechanics 169: 429–464.
• [14] Grimshaw, R., Pelinovsky, E. and Talipova, T. (2008). Fission of a weakly nonlinear interfacial solitary wave at a step, Geophysical and Astrophysical Fluid Dynamics 102(2): 179–194.
• [15] Infeld, E. and Rowlands, G. (2000). Nonlinear Waves, Solitons and Chaos, 2nd Edition, Cambridge University Press, Cambridge.
• [16] Kamchatnov, A.M., Kuo, Y.H., Lin, T.C., Horng, T.L., Gou, S.C., Clift, R., El, G.A. and Grimshaw, R.H.J. (2012). Undular bore theory for the Gardner equation, Physical Review E 86: 036605.
• [17] Karczewska, A., Rozmej, P. and Infeld, E. (2014a). Shallow-water soliton dynamics beyond the Korteweg–de Vries equation, Physical Review E 90: 012907.
• [18] Karczewska, A., Rozmej, P. and Rutkowski, L. (2014b). A new nonlinear equation in the shallow water wave problem, Physica Scripta 89(5): 054026.
• [19] Karczewska, A., Rozmej, P. and Infeld, E. (2015). Energy invariant for shallow water waves and the Korteweg–de Vries equation: Doubts about the invariance of energy, Physical Review E 92: 053202.
• [20] Karczewska, A., Szczeciński, M., Rozmej, P., and Boguniewicz, B. (2016). Finite element method for stochastic extended KdV equations, Computational Methods in Science and Technology 22(1): 19–29.
• [21] Kim, J.W., Bai, K.J., Ertekin, R.C. and Webster, W.C. (2001). A derivation of the Green–Naghdi equations for irrotational flows, Journal of Engineering Mathematics 40: 17–42.
• [22] Marchant, T. and Smyth, N. (1990). The extended Korteweg–de Vries equation and the resonant flow of a fluid over topography, Journal of Fluid Mechanics 221(1): 263–288.
• [23] Marchant, T. and Smyth, N. (1996). Soliton interaction for the extended Korteweg–de Vries equation, IMA Journal of Applied Mathematics 56: 157–176.
• [24] Mei, C. and Le Méhauté, B. (1966). Note on the equations of long waves over an uneven bottom, Journal of Geophysical Research 71(2): 393–400.
• [25] Miura, R.M., Gardner, C.S. and Kruskal, M.D. (1968). Korteweg–de Vries equation and generalizations, II: Existence of conservation laws and constants of motion, Journal of Mathematical Physics 9(8): 1204–1209.
• [26] Nadiga, B., Margolin, L. and Smolarkiewicz, P. (1996). Different approximations of shallow fluid flow over an obstacle, Physics of Fluids 8(8): 2066–2077.
• [27] Nakoulima, O., Zahibo, N. Pelinovsky, E., Talipova, T. and Kurkin, A. (2005). Solitary wave dynamics in shallow water over periodic topography, Chaos 15(3): 037107.
• [28] Pelinovsky, E., Choi, B., Talipova, T., Woo, S. and Kim, D. (2010). Solitary wave transformation on the underwater step: Theory and numerical experiments, Applied Mathematics and Computation 217(4): 1704–1718.
• [29] Pudjaprasetya, S.R. and van Greoesen, E. (1996). Uni-directional waves over slowly varying bottom, II: Quasi-homogeneous approximation of distorting waves, Wave Motion 23(1): 23–38.
• [30] Remoissenet, M. (1999). Waves Called Solitons: Concepts and Experiments, Springer, Berlin.
• [31] Skogstad, J. and Kalisch, H. (2009). A boundary value problem for the KdV equation: Comparison of finite difference and Chebyshev methods, Mathematics and Computers in Simulation 80(1): 151–163.
• [32] Smyth, N.F. (1987). Modulation theory solution for resonant flow over topography, Proceedings of the Royal Society of London A 409(1836): 79–97.
• [33] Taha, T.R. and Ablowitz, M.J. (1984). Analytical and numerical aspects of certain nonlinear evolution equations III: Numerical, Korteweg–de Vries equation, Journal of Computational Physics 55(2): 231–253.
• [34] van Greoesen, E. and Pudjaprasetya, S.R. (1993). Uni-directional waves over slowly varying bottom, I: Derivation of a KdV-type of equation, Wave Motion 18(4): 345–370.
• [35] Whitham, G.B. (1974). Linear and Nonlinear Waves, Wiley, New York, NY.
• [36] Yi, N., Huang, Y. and Liu, H. (2013). A direct discontinous Galerkin method for the generalized Korteweg–de Vries equation: Energy conservation and boundary effect, Journal of Computational Physics 242: 351–366.
• [37] Yuan, J.-M., Shen, J. and Wu, J. (2008). A dual Petrov–Galerkin method for the Kawahara-type equations, Journal of Scientific Computing 34: 48–63.
• [38] Zabusky, N.J. and Kruskal, M.D. (1965). Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states, Physical Review Letters 15(6): 240–243.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.