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ENERGY PRESERVING METHODS FOR VOLTERRA LATTICE EQUATION

Year 2011, Volume 01, Issue 2, 192 - 202, 01.12.2011

Abstract

We investigate linear energy preserving methods for the Volterra lattice equation as non-canonical Hamiltonian system. The averaged vector field method was applied to the Volterra lattice equation in bi-Hamiltonian form with quadratic and cubic Poisson brackets. Numerical results confirm the excellent long time preservation of the Hamiltonians and the polynomial integrals.

References

  • Chartier, P., Faou, E. and Murua, A., (2006), An algebraic approach to invariant preserving integra- tors: the case of quadratic and Hamiltonian invariants, Numer. Math., 103, 575–590.
  • Celledoni, E., McLachlan, R. I., Owren, B. and Quispel, G. R. W., (2010), Structure of B-series for Some Classes of Geometric Integrators, Found. Comput. Math., 10, 673–693.
  • Celledoni, E., McLachlan, R. I., McLaren, D. I., Owren, B., Quispel, G. R. W. and Wright, W.M., (2009), Energy-preserving Runge-Kutta methods, ESAIM: Mathematical Modelling and Numerical Analysis 43, 645-649.
  • Celledoni, E., Grimm, V., McLahlan, R. I., McLaren, D. I., O’Neale, D. R. J., Owren, B., and Quispel, G. R. W., (2009), Preserving energy resp. dissipation in numerical pdes, using the Average Vector Field method. Technical Report 7/2009, Norwegian University of Science and Technology Trondheim, Norway.
  • Cohen, D. and Hairer, E., (2011), Linear energy-preserving integrators for Poisson systems, BIT, 51, 91-101.
  • Courant, R., Friedrichs, K., and Lewy, H., (1967), On the partial difference equations of mathematical physics. IBM J., 11:215-234.
  • Dahlby, M. and Owren, B., (2010), A general framework for deriving integral preserving numerical methods for PDEs, SIAM J. Sci. Comput., 33, 2318-2340.
  • Ergen¸c, T. and Karas¨ozen, B., (2006), Poisson integrators for Volterra lattice equations, Applied Numerical Mathematics, 56, 879-887.
  • Faou, E., Hairer, E., and Pham, T. L., (2004), Energy conservation with non-symplectic methods: examples and counter-examples, BIT, 44, 699-709.
  • Furihata, D. and Matsuo, T., (2010), Discrete Variational Derivative Method: A Structure- Preserving Numerical Method for Partial Differential Equations, Chapman and Hall.
  • Gonzalez, O., (1996), Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6, 449-467 Hairer, E., Lubich, C., and Wanner, G., (2006), Geometric Numerical Integration. Structure- Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathe- matics 31, Springer-Verlag, Berlin, 2nd edition.
  • Hairer, E., (2010), Energy-preserving variant of collocation methods, J. Numer. Anal. Ind. Appl. Math., 5, 73-84.
  • Iavernaro, F. and Pace, B., (2007), s-stage trapezoidal methods for the conservation of Hamiltonian functions of polynomial type, AIP Conf. Proc., 936, 603–606.
  • Iavernaro F. and Trigiante, D., (2009), High-order symmetric schemes for the energy conservation of polynomial Hamiltonian problems, J. Numer. Anal. Ind. Appl. Math., 4, 87-101.
  • Kac, M. and van Moerbeke, P., (1975), On explicit soluble system of nonlinear differential equations related to certain Toda lattices, Advances in Mathematics, 16, 160–169.
  • Karas¨ozen, B., (2004), Poisson integrators, Mathematical Modelling and Computation 40, 1225–1244.
  • Leimkuhler, B. and Reich, S., (2004), Simulating Hamiltonian Dynamics, Cambridge Univesity Press.
  • McLachlan, R. I., Quispel, G. R. W. and Robidoux, N., (1999), Geometric integration using discrete gradients, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357, 1021-1045.
  • McLachlan, R. I., (2007), A new implementation of symplectic Runge-Kutta methods, SIAM J. Sci. Comput., 29, 1637–1649.
  • McLaren, D. I. and Quispel, G. R. W., (2004), Integral-preserving integrators, J. Phys. A: Math. Gen. 37, 489-495.
  • Quispel, G. R. W. and McLaren, D. I., (2008), A new class of energy-preserving numerical integration methods, J. Phys. A: Math. Theor. 41, 045206.
  • Quispel, G. R. W. and Capel, H. W., (1996), Solving ODEs numerically while preserving a first integral. Phys. Lett. A, 218, 223-228.
  • Sanz-Serna, J. M. and Calvo, M. P., (1994), Numerical Hamiltonian Problems, Chapman and Hall.
  • Suris, Y. B., (1997), A note on an integrable discretization of the nonlinear Schr¨odinger equation, Inverse Problems, 13, 1211–1236.

Year 2011, Volume 01, Issue 2, 192 - 202, 01.12.2011

Abstract

References

  • Chartier, P., Faou, E. and Murua, A., (2006), An algebraic approach to invariant preserving integra- tors: the case of quadratic and Hamiltonian invariants, Numer. Math., 103, 575–590.
  • Celledoni, E., McLachlan, R. I., Owren, B. and Quispel, G. R. W., (2010), Structure of B-series for Some Classes of Geometric Integrators, Found. Comput. Math., 10, 673–693.
  • Celledoni, E., McLachlan, R. I., McLaren, D. I., Owren, B., Quispel, G. R. W. and Wright, W.M., (2009), Energy-preserving Runge-Kutta methods, ESAIM: Mathematical Modelling and Numerical Analysis 43, 645-649.
  • Celledoni, E., Grimm, V., McLahlan, R. I., McLaren, D. I., O’Neale, D. R. J., Owren, B., and Quispel, G. R. W., (2009), Preserving energy resp. dissipation in numerical pdes, using the Average Vector Field method. Technical Report 7/2009, Norwegian University of Science and Technology Trondheim, Norway.
  • Cohen, D. and Hairer, E., (2011), Linear energy-preserving integrators for Poisson systems, BIT, 51, 91-101.
  • Courant, R., Friedrichs, K., and Lewy, H., (1967), On the partial difference equations of mathematical physics. IBM J., 11:215-234.
  • Dahlby, M. and Owren, B., (2010), A general framework for deriving integral preserving numerical methods for PDEs, SIAM J. Sci. Comput., 33, 2318-2340.
  • Ergen¸c, T. and Karas¨ozen, B., (2006), Poisson integrators for Volterra lattice equations, Applied Numerical Mathematics, 56, 879-887.
  • Faou, E., Hairer, E., and Pham, T. L., (2004), Energy conservation with non-symplectic methods: examples and counter-examples, BIT, 44, 699-709.
  • Furihata, D. and Matsuo, T., (2010), Discrete Variational Derivative Method: A Structure- Preserving Numerical Method for Partial Differential Equations, Chapman and Hall.
  • Gonzalez, O., (1996), Time integration and discrete Hamiltonian systems, J. Nonlinear Sci., 6, 449-467 Hairer, E., Lubich, C., and Wanner, G., (2006), Geometric Numerical Integration. Structure- Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathe- matics 31, Springer-Verlag, Berlin, 2nd edition.
  • Hairer, E., (2010), Energy-preserving variant of collocation methods, J. Numer. Anal. Ind. Appl. Math., 5, 73-84.
  • Iavernaro, F. and Pace, B., (2007), s-stage trapezoidal methods for the conservation of Hamiltonian functions of polynomial type, AIP Conf. Proc., 936, 603–606.
  • Iavernaro F. and Trigiante, D., (2009), High-order symmetric schemes for the energy conservation of polynomial Hamiltonian problems, J. Numer. Anal. Ind. Appl. Math., 4, 87-101.
  • Kac, M. and van Moerbeke, P., (1975), On explicit soluble system of nonlinear differential equations related to certain Toda lattices, Advances in Mathematics, 16, 160–169.
  • Karas¨ozen, B., (2004), Poisson integrators, Mathematical Modelling and Computation 40, 1225–1244.
  • Leimkuhler, B. and Reich, S., (2004), Simulating Hamiltonian Dynamics, Cambridge Univesity Press.
  • McLachlan, R. I., Quispel, G. R. W. and Robidoux, N., (1999), Geometric integration using discrete gradients, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357, 1021-1045.
  • McLachlan, R. I., (2007), A new implementation of symplectic Runge-Kutta methods, SIAM J. Sci. Comput., 29, 1637–1649.
  • McLaren, D. I. and Quispel, G. R. W., (2004), Integral-preserving integrators, J. Phys. A: Math. Gen. 37, 489-495.
  • Quispel, G. R. W. and McLaren, D. I., (2008), A new class of energy-preserving numerical integration methods, J. Phys. A: Math. Theor. 41, 045206.
  • Quispel, G. R. W. and Capel, H. W., (1996), Solving ODEs numerically while preserving a first integral. Phys. Lett. A, 218, 223-228.
  • Sanz-Serna, J. M. and Calvo, M. P., (1994), Numerical Hamiltonian Problems, Chapman and Hall.
  • Suris, Y. B., (1997), A note on an integrable discretization of the nonlinear Schr¨odinger equation, Inverse Problems, 13, 1211–1236.

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


Ozge ERDEM This is me
Max-Planck Institute, Mathematics in the Sciences, 04103 Leipzig, Germany.

Publication Date December 1, 2011
Published in Issue Year 2011, Volume 01, Issue 2

Cite

Bibtex @ { twmsjaem761798, 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 = {2011}, volume = {01}, number = {2}, pages = {192 - 202}, title = {ENERGY PRESERVING METHODS FOR VOLTERRA LATTICE EQUATION}, key = {cite}, author = {Karasozen, Bulent and Erdem, Ozge} }