Finite Element Method for Stochastic Extended KdV Equations
Karczewska Anna 1*, Szczeciński Maciej 1, Rozmej Piotr 2**, Boguniewicz Bartosz 2
1 Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland
∗E-mail: a.karczewska@wmie.uz.zgora.pl2 Faculty of Physics and Astronomy, Institute of Physics University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra, Poland
∗∗E-mail: p.rozmej@if.uz.zgora.pl
Received:
Received: 12 September 2015; revised: 26 January 2016; accepted: 29 January 2016; published online: 16 March 2016
DOI: 10.12921/cmst.2016.22.01.002
Abstract:
The finite element method is applied to obtain numerical solutions to the recently derived nonlinear equation for shallow water wave problem for several cases of bottom shapes. Results for time evolution of KdV solitons and cnoidal waves under stochastic forces are presented. Though small effects originating from second order dynamics may be obscured by stochastic forces, the main waves, both cnoidal and solitary ones, remain very robust against any distortions.
Key words:
nonlinear equations, second order KdV equations, shallow water wave problem, stochastic forces
References:
[1] A. Karczewska, P. Rozmej and L. Rutkowski, A new nonlinear equation in the shallow water wave problem, Physica Scripta 89, 054026 (2014).
[2] A. Karczewska, P. Rozmej and E. Infeld, Shallow-water soliton dynamics beyond the Korteweg-de Vries equation, Physical Review E 90, 012907 (2014).
[3] G.I. Burde, A. Sergyeyev, Ordering of two small parameters in the shallow water wave problem, J. Phys. A: Math. Theor. 46, 075501 (2013).
[4] A. Karczewska, P. Rozmej and E. Infeld, Energy invariant for shallow water waves and Korteweg – de Vries equation. Is energy always invariant?, arXiv:1503.09089, to be published.
[5] A. Debussche and I. Printems, Numerical simulation of the stochastic Korteweg-de Vries equation, Physica D 134, 200-226 (1999).
[6] A. Karczewska, P. Rozmej, M. Szczeci ́nski and B. Boguniewicz, Finite element method for extended KdV equations, Int. J. Appl. Math. Comp. Sci. (2016), in print.
[7] P.G. Drazin and R.S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, 1989.
[8] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.
[9] T.R. Marchant and N.F. Smyth, The extended Korteweg–de Vries equation and the resonant flow of a fluid over topography, J. Fluid Mech. 221, 263-288 (1990).
[10] E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos, 2nd edition, Cambridge University Press, 2000.
[11] N.J. Zabusky and M.D. Kruskal, Interaction of ‘Solitons’ in a Collisionless Plasma and the Recurrence of Initial States, Phys. Rev. Lett. 15, 240-243 (1965).
[12] A. de Bouard, A. Debussche, On the stochastic Korteveg-de Vries equation, J. Functional Analysis 154, 215-251 (1998).
[13] T. Oh, Periodic stochastic Korteweg-de Vries equation with additive space-time white noise, Anal. PDE 2, 281-304 (2009).
[14] T. Oh, Invariance of the White Noise for KdV, Commun. Math. Phys. 292, 217-236 (2009).
[15] M. Remoissenet, Waves Called Solitons: Concepts and Experiments, Springer, 1999.
[16] T.R. Marchant and N.F. Smyth, Soliton Interaction for the Korteweg-de Vries equation, IMA J. Appl. Math. 56, 157-176 (1996).
[17] T.R. Marchant, Coupled Korteweg-de Vries equations describing, to higher-order, resonant flow of a fluid over topography, Phys. Fluids 11(7), 1797-1804 (1999).
[18] Q. Zou and CH-H. Su, Overtaking collision between two solitary waves, Phys. Fluids 29, No. 7, 2113-2123 (1986).
[19] J. Villegas G., J. Castano B., J. Duarte V., E. Fierro Y., Wavelet-Petrov-Galerkin method for the numerical solution of the KdV equation, Appl. Math. Sci. 6, 3411-3423 (2012).
[20] S.-ul-Islam, S. Haq and A. Ali, A meshfree method for the numerical solution RLW equation, J. Comp. Appl. Math. 223, 997-1012 (2009).
[21] A. Canivar, M. Sari and I. Dag, A Taylor-Galerkin finite element method for the KdV equation, Physica B 405, 3376-3383 (2010).
[22] T.R. Taha, and M.J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations III. Numerical, the Korteweg–de Vries equation, Journal of Computational
[23] M.W. Dingemans, Water wave propagation over uneven bottoms, World Scientific, Singapore, 1997. http://repository.tudelft.nl
The finite element method is applied to obtain numerical solutions to the recently derived nonlinear equation for shallow water wave problem for several cases of bottom shapes. Results for time evolution of KdV solitons and cnoidal waves under stochastic forces are presented. Though small effects originating from second order dynamics may be obscured by stochastic forces, the main waves, both cnoidal and solitary ones, remain very robust against any distortions.
Key words:
nonlinear equations, second order KdV equations, shallow water wave problem, stochastic forces
References:
[1] A. Karczewska, P. Rozmej and L. Rutkowski, A new nonlinear equation in the shallow water wave problem, Physica Scripta 89, 054026 (2014).
[2] A. Karczewska, P. Rozmej and E. Infeld, Shallow-water soliton dynamics beyond the Korteweg-de Vries equation, Physical Review E 90, 012907 (2014).
[3] G.I. Burde, A. Sergyeyev, Ordering of two small parameters in the shallow water wave problem, J. Phys. A: Math. Theor. 46, 075501 (2013).
[4] A. Karczewska, P. Rozmej and E. Infeld, Energy invariant for shallow water waves and Korteweg – de Vries equation. Is energy always invariant?, arXiv:1503.09089, to be published.
[5] A. Debussche and I. Printems, Numerical simulation of the stochastic Korteweg-de Vries equation, Physica D 134, 200-226 (1999).
[6] A. Karczewska, P. Rozmej, M. Szczeci ́nski and B. Boguniewicz, Finite element method for extended KdV equations, Int. J. Appl. Math. Comp. Sci. (2016), in print.
[7] P.G. Drazin and R.S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, 1989.
[8] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.
[9] T.R. Marchant and N.F. Smyth, The extended Korteweg–de Vries equation and the resonant flow of a fluid over topography, J. Fluid Mech. 221, 263-288 (1990).
[10] E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos, 2nd edition, Cambridge University Press, 2000.
[11] N.J. Zabusky and M.D. Kruskal, Interaction of ‘Solitons’ in a Collisionless Plasma and the Recurrence of Initial States, Phys. Rev. Lett. 15, 240-243 (1965).
[12] A. de Bouard, A. Debussche, On the stochastic Korteveg-de Vries equation, J. Functional Analysis 154, 215-251 (1998).
[13] T. Oh, Periodic stochastic Korteweg-de Vries equation with additive space-time white noise, Anal. PDE 2, 281-304 (2009).
[14] T. Oh, Invariance of the White Noise for KdV, Commun. Math. Phys. 292, 217-236 (2009).
[15] M. Remoissenet, Waves Called Solitons: Concepts and Experiments, Springer, 1999.
[16] T.R. Marchant and N.F. Smyth, Soliton Interaction for the Korteweg-de Vries equation, IMA J. Appl. Math. 56, 157-176 (1996).
[17] T.R. Marchant, Coupled Korteweg-de Vries equations describing, to higher-order, resonant flow of a fluid over topography, Phys. Fluids 11(7), 1797-1804 (1999).
[18] Q. Zou and CH-H. Su, Overtaking collision between two solitary waves, Phys. Fluids 29, No. 7, 2113-2123 (1986).
[19] J. Villegas G., J. Castano B., J. Duarte V., E. Fierro Y., Wavelet-Petrov-Galerkin method for the numerical solution of the KdV equation, Appl. Math. Sci. 6, 3411-3423 (2012).
[20] S.-ul-Islam, S. Haq and A. Ali, A meshfree method for the numerical solution RLW equation, J. Comp. Appl. Math. 223, 997-1012 (2009).
[21] A. Canivar, M. Sari and I. Dag, A Taylor-Galerkin finite element method for the KdV equation, Physica B 405, 3376-3383 (2010).
[22] T.R. Taha, and M.J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations III. Numerical, the Korteweg–de Vries equation, Journal of Computational
[23] M.W. Dingemans, Water wave propagation over uneven bottoms, World Scientific, Singapore, 1997. http://repository.tudelft.nl
