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ENERGY PRESERVING INTEGRATION OF KDV-KDV SYSTEMS

Year 2012, Volume 2, Issue 2, 219 - 227, 01.12.2012

Abstract

Coupled Korteweg de Vries KdV equations in Hamiltonian form are integrated by the energy preserving average vector field AVF method. Numerical results confirm long term preservation of the energy and the quadratic invariants. Produced generalized solitary waves are similar to those in the literature for larger mesh sizes and time steps. Numerical and continuous dispersion relations of the linearized equations are compared to analyze the behavior of the traveling waves and the interaction of the solitons.

References

  • Ascher, U. M. and McLachlan, R. I., (2004), Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48, 255–269.
  • Ascher, U. M. and McLachlan, R. I., (2004), On symplectic and multisymplectic schemes for the KdV equation, J. Sci. Comput., 25, 83–104.
  • Aydın, A. and Karas¨ozen, B., (2010), Multisymplectic box schemes for the complex modiŞed Korteweg-de Vries equation, J. Math. Phys., 51, 083511.
  • Bona, J. L, Dougalis, V, A, and Mitsotakis, D. E., (2007), Numerical solution of KdV-KdV systems of Boussinesq equations. I. The numerical scheme and generalized solitary waves, Math. Comput. Simulation, 74, 214-228.
  • Bona, J. L., Dougalis V. A., and Mitsotakis, D. E,, (2008), Numerical solution of Boussinesq systems of KdV-KdV type. II. Evolution of radiating solitary waves, Nonlinearity, 21, 2825-2848.
  • Cohen, D. and Hairer, E., (2011), Linear energy-preserving integrators for Poisson systems, BIT Numerical Mathematics, 51, 91–100.
  • Dahlby, M., (2011), A General Framework for Deriving Integral Preserving Numerical Methods for PDEs, SIAM Journal on ScientiŞc Computing, 33, 2318–2340.
  • Guha, P., (2005), Geodesic flows, bi-Hamiltonian structure and coupled KdV type systems, J. Math. Anal. Appl., 310, 45-56.
  • Hairer, E., (2010), Energy-preserving variant of collocation methods, J. Numer. Anal. Ind. Appl. Math., 5, 73–84.
  • Hairer, E., Lubich, C. and Wanner, G., (2006), Geometric Numerical Integration-Structure- Preserving Algorithms for Ordinary Differential Equations, Springer.
  • Olver, P., (1995), Applications of Lie Groups to Differential Equations, second edition, Springer.

Year 2012, Volume 2, Issue 2, 219 - 227, 01.12.2012

Abstract

References

  • Ascher, U. M. and McLachlan, R. I., (2004), Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48, 255–269.
  • Ascher, U. M. and McLachlan, R. I., (2004), On symplectic and multisymplectic schemes for the KdV equation, J. Sci. Comput., 25, 83–104.
  • Aydın, A. and Karas¨ozen, B., (2010), Multisymplectic box schemes for the complex modiŞed Korteweg-de Vries equation, J. Math. Phys., 51, 083511.
  • Bona, J. L, Dougalis, V, A, and Mitsotakis, D. E., (2007), Numerical solution of KdV-KdV systems of Boussinesq equations. I. The numerical scheme and generalized solitary waves, Math. Comput. Simulation, 74, 214-228.
  • Bona, J. L., Dougalis V. A., and Mitsotakis, D. E,, (2008), Numerical solution of Boussinesq systems of KdV-KdV type. II. Evolution of radiating solitary waves, Nonlinearity, 21, 2825-2848.
  • Cohen, D. and Hairer, E., (2011), Linear energy-preserving integrators for Poisson systems, BIT Numerical Mathematics, 51, 91–100.
  • Dahlby, M., (2011), A General Framework for Deriving Integral Preserving Numerical Methods for PDEs, SIAM Journal on ScientiŞc Computing, 33, 2318–2340.
  • Guha, P., (2005), Geodesic flows, bi-Hamiltonian structure and coupled KdV type systems, J. Math. Anal. Appl., 310, 45-56.
  • Hairer, E., (2010), Energy-preserving variant of collocation methods, J. Numer. Anal. Ind. Appl. Math., 5, 73–84.
  • Hairer, E., Lubich, C. and Wanner, G., (2006), Geometric Numerical Integration-Structure- Preserving Algorithms for Ordinary Differential Equations, Springer.
  • Olver, P., (1995), Applications of Lie Groups to Differential Equations, second edition, Springer.

Details

Primary Language English
Journal Section Research Article
Authors

Bulent KARASOZEN This is me
Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, 06800 Ankara, Turkey


Gorkem SİMSEK This is me
Eindhoven University of Technology, Faculty of Mechanical Engineering, Multiscale Engineering Fluid Dynamics Institute, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

Publication Date December 1, 2012
Published in Issue Year 2012, Volume 2, Issue 2

Cite

Bibtex @ { twmsjaem761713, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2012}, volume = {2}, number = {2}, pages = {219 - 227}, title = {ENERGY PRESERVING INTEGRATION OF KDV-KDV SYSTEMS}, key = {cite}, author = {Karasozen, Bulent and Simsek, Gorkem} }