A symmetric nearly preserving general linear method for Hamiltonian problems

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2015, 2015(special): 330-339. doi: 10.3934/proc.2015.0330

A symmetric nearly preserving general linear method for Hamiltonian problems

Raffaele D’Ambrosio ^1, , Giuseppe De Martino ^1, and Beatrice Paternoster ^1,

1.	Department of Mathematics - University of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy, Italy, Italy

Received August 2014 Revised September 2015 Published November 2015

Abstract
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This paper is concerned with the numerical solution of Hamiltonian problems, by means of nearly conservative multivalue numerical methods. In particular, the method we propose is symmetric, G-symplectic, diagonally implicit and generates bounded parasitic components over suitable time intervals. Numerical experiments on a selection of separable Hamiltonian problems are reported, also based on real data provided by Nasa Horizons System.

Keywords: Symmetry, Parasitic components, Multivalue methods, G-symplecticity, Celestial Mechanics., Hamiltonian problems.

Mathematics Subject Classification: Primary: 65L0.

Citation: Raffaele D’Ambrosio, Giuseppe De Martino, Beatrice Paternoster. A symmetric nearly preserving general linear method for Hamiltonian problems. Conference Publications, 2015, 2015 (special) : 330-339. doi: 10.3934/proc.2015.0330

References:

[1]	J.C. Butcher, General Linear Methods,, Acta Numer., 15 (2006), 157.
[2]	J.C. Butcher, Numerical methods for Ordinary Differential Equations,, Second Edition, (2008).
[3]	J. C. Butcher, Y. Habib, A. T. Hill, and T. J. T. Norton, The control of parasitism in $G$-symplectic methods,, SIAM J. Numer. Anal., 52 (2014), 2440.
[4]	J. C. Butcher and R. D'Ambrosio, Partitioned general linear methods for separable Hamiltonian problems,, in preparation., ().
[5]	J. C. Butcher and L. L. Hewitt, The existence of symplectic general linear methods,, Numer. Algor., 51 (2009), 77.
[6]	R. D'Ambrosio, On the G-symplecticity of two-step Runge-Kutta methods,, Commun. Appl. Ind. Math., 3 (2012).
[7]	R. D'Ambrosio, Multi-value numerical methods for hamiltonian systems,, Numerical Mathematics and Advanced Applications - ENUMATH 2013, 103 (2015), 185.
[8]	R. D'Ambrosio, G. De Martino and B. Paternoster, Construction of nearly conservative multivalue numerical methods for Hamiltonian problems,, Commun. Appl. Ind. Math., 3 (2012).
[9]	R. D'Ambrosio, G. De Martino and B. Paternoster, Numerical integration of Hamiltonian problems by G-symplectic methods,, Adv. Comput. Math., 40 (2014), 553.
[10]	R. D'Ambrosio, E. Esposito and B. Paternoster, General Linear Methods for $y''=f(y(t))$,, Numer. Algorithms, 61 (2012), 331.
[11]	R. D'Ambrosio and E. Hairer, Long-term stability of multi-value methods for ordinary differential equations,, J. Sci. Comput., 60 (2014), 627.
[12]	R. D'Ambrosio, E. Hairer and C. Zbinden, G-symplecticity implies conjugate-symplecticity of the underlying one-step method,, BIT, 53 (2013), 867.
[13]	E. Hairer and P. Leone, Order barriers for symplectic multi-value methods, Numerical analysis 1997,, Proc. of the 17th Dundee Biennial Conference 1997, (1997).
[14]	E. Hairer and C. Lubich, Symmetric multistep methods over long times ,, Numer. Math., 97 (2004), 699.
[15]	E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations,, Second edition, (2006).
[16]	E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations,, Second edition, (2008).
[17]	Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations,, John Wiley & Sons, (2009).
[18]	P. Leone, Symplecticity and symmetry of general integration methods,, Ph.D. thesis, (2000).
[19]	R. I McLachlan, and G. R. W. Quispel, Geometric Integrators for ODEs,, J. Phys. A: Math. Gen. 39 (2006), 39 (2006), 5251.
[20]	K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,, Series: Applied Mathematical Sciences, (2009).
[21]	J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems,, Chapman & Hall, (1994).
[22]	D. Okunbor and R. D. Skeel, Explicit Canonical Methods for Hamiltonian Systems,, Math. Comput., 59 (1992), 439.
[23]	Y. F. Tang, The simplecticity of multistep methods,, Comput. Math. Appl., 25 (1993), 83.

show all references

References:

[1]	J.C. Butcher, General Linear Methods,, Acta Numer., 15 (2006), 157.
[2]	J.C. Butcher, Numerical methods for Ordinary Differential Equations,, Second Edition, (2008).
[3]	J. C. Butcher, Y. Habib, A. T. Hill, and T. J. T. Norton, The control of parasitism in $G$-symplectic methods,, SIAM J. Numer. Anal., 52 (2014), 2440.
[4]	J. C. Butcher and R. D'Ambrosio, Partitioned general linear methods for separable Hamiltonian problems,, in preparation., ().
[5]	J. C. Butcher and L. L. Hewitt, The existence of symplectic general linear methods,, Numer. Algor., 51 (2009), 77.
[6]	R. D'Ambrosio, On the G-symplecticity of two-step Runge-Kutta methods,, Commun. Appl. Ind. Math., 3 (2012).
[7]	R. D'Ambrosio, Multi-value numerical methods for hamiltonian systems,, Numerical Mathematics and Advanced Applications - ENUMATH 2013, 103 (2015), 185.
[8]	R. D'Ambrosio, G. De Martino and B. Paternoster, Construction of nearly conservative multivalue numerical methods for Hamiltonian problems,, Commun. Appl. Ind. Math., 3 (2012).
[9]	R. D'Ambrosio, G. De Martino and B. Paternoster, Numerical integration of Hamiltonian problems by G-symplectic methods,, Adv. Comput. Math., 40 (2014), 553.
[10]	R. D'Ambrosio, E. Esposito and B. Paternoster, General Linear Methods for $y''=f(y(t))$,, Numer. Algorithms, 61 (2012), 331.
[11]	R. D'Ambrosio and E. Hairer, Long-term stability of multi-value methods for ordinary differential equations,, J. Sci. Comput., 60 (2014), 627.
[12]	R. D'Ambrosio, E. Hairer and C. Zbinden, G-symplecticity implies conjugate-symplecticity of the underlying one-step method,, BIT, 53 (2013), 867.
[13]	E. Hairer and P. Leone, Order barriers for symplectic multi-value methods, Numerical analysis 1997,, Proc. of the 17th Dundee Biennial Conference 1997, (1997).
[14]	E. Hairer and C. Lubich, Symmetric multistep methods over long times ,, Numer. Math., 97 (2004), 699.
[15]	E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations,, Second edition, (2006).
[16]	E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations,, Second edition, (2008).
[17]	Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations,, John Wiley & Sons, (2009).
[18]	P. Leone, Symplecticity and symmetry of general integration methods,, Ph.D. thesis, (2000).
[19]	R. I McLachlan, and G. R. W. Quispel, Geometric Integrators for ODEs,, J. Phys. A: Math. Gen. 39 (2006), 39 (2006), 5251.
[20]	K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,, Series: Applied Mathematical Sciences, (2009).
[21]	J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems,, Chapman & Hall, (1994).
[22]	D. Okunbor and R. D. Skeel, Explicit Canonical Methods for Hamiltonian Systems,, Math. Comput., 59 (1992), 439.
[23]	Y. F. Tang, The simplecticity of multistep methods,, Comput. Math. Appl., 25 (1993), 83.

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