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2016 | 26 | 3 | 555-567
Tytuł artykułu

A finite element method for extended KdV equations

Treść / Zawartość
Warianty tytułu
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.
Rocznik
Tom
26
Numer
3
Strony
555-567
Opis fizyczny
Daty
wydano
2016
otrzymano
2015-06-02
poprawiono
2015-12-20
zaakceptowano
2016-03-08
Twórcy
  • Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland
autor
  • Institute of Physics, Faculty of Physics and Astronomy, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland
  • Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland
  • Institute of Physics, Faculty of Physics and Astronomy, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland
Bibliografia
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  • 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.
  • Burde, G. and Sergyeyev, A. (2013). Ordering of two small parameters in the shallow water wave problem, Journal of Physics A 46(7): 075501.
  • 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.
  • Debussche, A. and Printems, I. (1999). Numerical simulation of the stochastic Korteweg-de Vries equation, Physica D 134(2): 200-226.
  • Dingemans, M. (1997). Water Wave Propagation over Uneven Bottoms, World Scientific, Singapore.
  • Drazin, P.G. and Johnson, R.S. (1989). Solitons: An Introduction, Cambridge University Press, Cambridge.
  • 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.
  • 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.
  • 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.
  • Grimshaw, R. (1970). The solitary wave in water of variable depth, Journal of Fluid Mechanics 42(3): 639-656.
  • Grimshaw, R.H.J. and Smyth, N.F. (1986). Resonant flow of a stratified fluid over topography, Journal of Fluid Mechanics 169: 429-464.
  • 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.
  • Infeld, E. and Rowlands, G. (2000). Nonlinear Waves, Solitons and Chaos, 2nd Edition, Cambridge University Press, Cambridge.
  • 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.
  • Karczewska, A., Rozmej, P. and Infeld, E. (2014a). Shallow-water soliton dynamics beyond the Korteweg-de Vries equation, Physical Review E 90: 012907.
  • Karczewska, A., Rozmej, P. and Rutkowski, L. (2014b). A new nonlinear equation in the shallow water wave problem, Physica Scripta 89(5): 054026.
  • 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.
  • 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.
  • 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.
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  • 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.
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  • 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.
  • 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.
  • 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.
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  • 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.
  • 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.
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Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-amcv26i3p555bwm
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