A finite element method for extended KdV equations
Anna Karczewska; Piotr Rozmej; Maciej Szczeciński; Bartosz Boguniewicz
International Journal of Applied Mathematics and Computer Science (2016)
- Volume: 26, Issue: 3, page 555-567
- ISSN: 1641-876X
Access Full Article
topAbstract
topHow to cite
topReferences
top- Ablowitz, M. and Segur, H. (1979). On the evolution of packets of water waves, Journal of Fluid Mechanics 92(4): 691-715. Zbl0413.76009
- 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. Zbl1310.35206
- 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. Zbl1276.65058
- Burde, G. and Sergyeyev, A. (2013). Ordering of two small parameters in the shallow water wave problem, Journal of Physics A 46(7): 075501. Zbl1311.76011
- 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. Zbl1131.65073
- Debussche, A. and Printems, I. (1999). Numerical simulation of the stochastic Korteweg-de Vries equation, Physica D 134(2): 200-226. Zbl0948.76038
- Dingemans, M. (1997). Water Wave Propagation over Uneven Bottoms, World Scientific, Singapore. Zbl0908.76002
- Drazin, P.G. and Johnson, R.S. (1989). Solitons: An Introduction, Cambridge University Press, Cambridge. Zbl0661.35001
- 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. Zbl0384.65049
- 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. Zbl1337.65136
- 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. Zbl0351.76014
- Grimshaw, R. (1970). The solitary wave in water of variable depth, Journal of Fluid Mechanics 42(3): 639-656. Zbl0193.27304
- Grimshaw, R.H.J. and Smyth, N.F. (1986). Resonant flow of a stratified fluid over topography, Journal of Fluid Mechanics 169: 429-464. Zbl0614.76108
- 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. Zbl0994.76001
- 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. Zbl1053.76012
- 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. Zbl0715.76006
- Marchant, T. and Smyth, N. (1996). Soliton interaction for the extended Korteweg-de Vries equation, IMA Journal of Applied Mathematics 56: 157-176. Zbl0857.35113
- 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.
- 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. Zbl0283.35019
- Nadiga, B., Margolin, L. and Smolarkiewicz, P. (1996). Different approximations of shallow fluid flow over an obstacle, Physics of Fluids 8(8): 2066-2077. Zbl1082.76509
- 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. Zbl1144.37391
- 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. Zbl1222.35037
- 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. Zbl0956.76506
- Remoissenet, M. (1999). Waves Called Solitons: Concepts and Experiments, Springer, Berlin. Zbl0933.35003
- 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. Zbl1177.65133
- Smyth, N.F. (1987). Modulation theory solution for resonant flow over topography, Proceedings of the Royal Society of London A 409(1836): 79-97. Zbl0611.76122
- 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. Zbl0541.65083
- 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. Zbl0819.76012
- Whitham, G.B. (1974). Linear and Nonlinear Waves, Wiley, New York, NY. Zbl0373.76001
- 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. Zbl1297.65122
- 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. Zbl1133.76040
- 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. Zbl1201.35174