PDF EndNote BibTex RIS Cite

ENERGY PRESERVING INTEGRATION OF BI-HAMILTONIAN PARTIAL DIFFERENTIAL EQUATIONS

Year 2013, Volume 3, Issue 1, 75 - 86, 01.06.2013

Abstract

The energy preserving average vector field AVF integrator is applied to evolutionary partial differential equations PDEs in bi-Hamiltonian form with nonconstant Poisson structures. Numerical results for the Korteweg de Vries KdV equation and for the Ito type coupled KdV equation confirm the long term preservation of the Hamiltonians and Casimir integrals, which is essential in simulating waves and solitons. Dispersive properties of the AVF integrator are investigated for the linearized equations to examine the nonlinear dynamics after discreization.

References

  • [1] Ascher, U. M. and McLachlan, R. I., (2004), Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48, 255-269.
  • [2] Ascher, U. M., and McLachlan, R.I., (2005) On symplectic and multisymplectic schemes for the KdV equation. J. Sci. Comput., 25, 83-104.
  • [3] Aydın, A., and Karas¨ozen, B., (2008), Symplectic and multisymplectic Lobatto methods for the ”good” Boussinesq equation. J. Math. Phys., 49, 083509.
  • [4] Aydın, A., and Karas¨ozen, B., (2010), Multisymplectic box schemes for the complex modified Korteweg-de Vries equation. J. Math. Phys., 51, 083511
  • [5] Celledoni, E., McLachlan, R. I., McLaren, D. I., Owren, B, Quispel, G. R. W. and Wright, W. M., (2009) Energy-preserving Runge-Kutta methods, M2AN Math. Model. Numer. Anal., 43, 649-645.
  • [6] Cohen, D. and Hairer, E., (2011), Linear energy-preserving integrators for poisson systems, BIT Numerical Mathematics, 51, 91-101.
  • [7] M. Dahlby, M., and Owren, B., (2011) A general framework for deriving integral preserving numerical methods for PDEs. SIAM J. Sci. Comput., 33, 2318–2340.
  • [8] Frank,J., Moore, B.E., and Reich, S., (2006), Linear PDEs and numerical methods that preserve a multisymplectic conservation law. SIAM J. Sci. Comput., 28, 260–277.
  • [9] Ergen¸c, T. and Karas¨ozen, B., (2006), Poisson integrators for Volterra lattice equations, Applied Numerical Mathematics, 56, 879-887.
  • [10] Hairer, E.,(2010), Energy-preserving variant of collocation methods, J. Numer. Anal. Ind. Appl. Math., 5, 73-84.
  • [11] Ito, M., (1980), An extension of nonlinear evolution equations of the K-dV (mK-dV) type to higher orders, J. Phys. Soc. Japan, 49, 771-778.
  • [12] Karas¨ozen, B., (2004), Poisson integrators, Math. Comput. Modelling, 40, 1225-1244.
  • [13] Liu, Q. P., (2000), Hamiltonian structures for Ito’s equation, Phys. Lett. A, 277, 31-34.
  • [14] Magri,F., (1998), A short introduction to Hamiltonian PDEs, Mat. Contemp., 15, 213-230.
  • [15] McLachlan, R. I., (2003), Spatial discretization of partial differential equations with integrals, IMA J. Numer. Anal., 23, 645-664.
  • [16] Olver, P.,(1995), Applications of Lie Groups to Differential Equations, Springer.
  • [17] Olver, P. J. and Rosenau, P., (1996), Tri-Hamiltonian duality between solitons and solutions having compact support, Phys. Rev. E, 53 (3), 1900-1906.
  • [18] G. R. W. Quispel and D. I. McLaren. A new class of energy-preserving numerical integration methods. J. Phys. A, 41:045206, 7, 2008.
  • [19] Schober, C.M., and Wlodarczyk, T.H., (2008), Dispersive properties of multisymplectic integrators. J. Comput. Phys., 227, 5090–5104.
  • [20] Schober, C.M. and Wlodarczyk, T.H., (2009), Dispersion, group velocity, and multisymplectic discretizations. Math. Comput. Simulation, 80, 741–751.
  • [21] Xu, Y. and Shu, C.-W., (2006), Local discontinuous Galerkin methods for the Kuramoto- Sivashinsky equations and the Ito-type coupled KdV equations, Comput. Methods Appl. Mech. Engrg., 195, 3430- 3447.
  • [22] Zhao, P. F. and Qin, M. Z., (2000), Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation, J. Phys. A, 33, 3613-3626.

Year 2013, Volume 3, Issue 1, 75 - 86, 01.06.2013

Abstract

References

  • [1] Ascher, U. M. and McLachlan, R. I., (2004), Multisymplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48, 255-269.
  • [2] Ascher, U. M., and McLachlan, R.I., (2005) On symplectic and multisymplectic schemes for the KdV equation. J. Sci. Comput., 25, 83-104.
  • [3] Aydın, A., and Karas¨ozen, B., (2008), Symplectic and multisymplectic Lobatto methods for the ”good” Boussinesq equation. J. Math. Phys., 49, 083509.
  • [4] Aydın, A., and Karas¨ozen, B., (2010), Multisymplectic box schemes for the complex modified Korteweg-de Vries equation. J. Math. Phys., 51, 083511
  • [5] Celledoni, E., McLachlan, R. I., McLaren, D. I., Owren, B, Quispel, G. R. W. and Wright, W. M., (2009) Energy-preserving Runge-Kutta methods, M2AN Math. Model. Numer. Anal., 43, 649-645.
  • [6] Cohen, D. and Hairer, E., (2011), Linear energy-preserving integrators for poisson systems, BIT Numerical Mathematics, 51, 91-101.
  • [7] M. Dahlby, M., and Owren, B., (2011) A general framework for deriving integral preserving numerical methods for PDEs. SIAM J. Sci. Comput., 33, 2318–2340.
  • [8] Frank,J., Moore, B.E., and Reich, S., (2006), Linear PDEs and numerical methods that preserve a multisymplectic conservation law. SIAM J. Sci. Comput., 28, 260–277.
  • [9] Ergen¸c, T. and Karas¨ozen, B., (2006), Poisson integrators for Volterra lattice equations, Applied Numerical Mathematics, 56, 879-887.
  • [10] Hairer, E.,(2010), Energy-preserving variant of collocation methods, J. Numer. Anal. Ind. Appl. Math., 5, 73-84.
  • [11] Ito, M., (1980), An extension of nonlinear evolution equations of the K-dV (mK-dV) type to higher orders, J. Phys. Soc. Japan, 49, 771-778.
  • [12] Karas¨ozen, B., (2004), Poisson integrators, Math. Comput. Modelling, 40, 1225-1244.
  • [13] Liu, Q. P., (2000), Hamiltonian structures for Ito’s equation, Phys. Lett. A, 277, 31-34.
  • [14] Magri,F., (1998), A short introduction to Hamiltonian PDEs, Mat. Contemp., 15, 213-230.
  • [15] McLachlan, R. I., (2003), Spatial discretization of partial differential equations with integrals, IMA J. Numer. Anal., 23, 645-664.
  • [16] Olver, P.,(1995), Applications of Lie Groups to Differential Equations, Springer.
  • [17] Olver, P. J. and Rosenau, P., (1996), Tri-Hamiltonian duality between solitons and solutions having compact support, Phys. Rev. E, 53 (3), 1900-1906.
  • [18] G. R. W. Quispel and D. I. McLaren. A new class of energy-preserving numerical integration methods. J. Phys. A, 41:045206, 7, 2008.
  • [19] Schober, C.M., and Wlodarczyk, T.H., (2008), Dispersive properties of multisymplectic integrators. J. Comput. Phys., 227, 5090–5104.
  • [20] Schober, C.M. and Wlodarczyk, T.H., (2009), Dispersion, group velocity, and multisymplectic discretizations. Math. Comput. Simulation, 80, 741–751.
  • [21] Xu, Y. and Shu, C.-W., (2006), Local discontinuous Galerkin methods for the Kuramoto- Sivashinsky equations and the Ito-type coupled KdV equations, Comput. Methods Appl. Mech. Engrg., 195, 3430- 3447.
  • [22] Zhao, P. F. and Qin, M. Z., (2000), Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation, J. Phys. A, 33, 3613-3626.

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
Multiscale Engineering Fluid Dynamics, Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Publication Date June 1, 2013
Published in Issue Year 2013, Volume 3, Issue 1

Cite

Bibtex @ { twmsjaem761697, 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 = {2013}, volume = {3}, number = {1}, pages = {75 - 86}, title = {ENERGY PRESERVING INTEGRATION OF BI-HAMILTONIAN PARTIAL DIFFERENTIAL EQUATIONS}, key = {cite}, author = {Karasozen, Bulent and Simsek, Gorkem} }