American Institute of Mathematical Sciences

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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

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), 157-256.

[2]

J.C. Butcher, Numerical methods for Ordinary Differential Equations, Second Edition, Wiley, Chichester, 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(5) (2014), 2440-2465.

[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), 77-84.

[6]

R. D'Ambrosio, On the G-symplecticity of two-step Runge-Kutta methods, Commun. Appl. Ind. Math., 3(1) (2012), e-403, DOI: 10.1685/journal.caim.403.

[7]

R. D'Ambrosio, Multi-value numerical methods for hamiltonian systems, Numerical Mathematics and Advanced Applications - ENUMATH 2013, Lecture Notes in Computational Science and Engineering 103 (2015), 185-193.

[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(2) (2012), e-412, DOI: 10.1685/journal.caim.412.

[9]

R. D'Ambrosio, G. De Martino and B. Paternoster, Numerical integration of Hamiltonian problems by G-symplectic methods, Adv. Comput. Math., 40(2) (2014), 553-575.

[10]

R. D'Ambrosio, E. Esposito and B. Paternoster, General Linear Methods for $y''=f(y(t))$, Numer. Algorithms, 61(2) (2012), 331-349.

[11]

R. D'Ambrosio and E. Hairer, Long-term stability of multi-value methods for ordinary differential equations, J. Sci. Comput., 60(3) (2014), 627-640.

[12]

R. D'Ambrosio, E. Hairer and C. Zbinden, G-symplecticity implies conjugate-symplecticity of the underlying one-step method, BIT, 53(4) (2013), 867-872.

[13]

E. Hairer and P. Leone, Order barriers for symplectic multi-value methods, Numerical analysis 1997, Proc. of the 17th Dundee Biennial Conference 1997, (eds. D. F. Griffiths, D. J. Higham, G. A. Watson), Pitman Research Notes in Mathematics Series 380, 1998.

[14]

E. Hairer and C. Lubich, Symmetric multistep methods over long times , Numer. Math., 97 (2004), 699-723.

[15]

E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, Second edition, Springer Series in Computational Mathematics 31, Springer-Verlag, Berlin, 2006.

[16]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations, Second edition, Springer Series in Computational Mathematics 8, Springer-Verlag, Berlin, 2008.

[17]

Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations, John Wiley & Sons, Hoboken, New Jersey, 2009.

[18]

P. Leone, Symplecticity and symmetry of general integration methods, Ph.D. thesis, Universite de Geneve, Section de Mathematiques, 2000.

[19]

R. I McLachlan, and G. R. W. Quispel, Geometric Integrators for ODEs, J. Phys. A: Math. Gen. 39 (2006), 5251-5285.

[20]

K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Series: Applied Mathematical Sciences, Vol. 90, XIII, 2009.

[21]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Chapman & Hall, London, England, 1994.

[22]

D. Okunbor and R. D. Skeel, Explicit Canonical Methods for Hamiltonian Systems, Math. Comput., 59(200) (1992), 439-455.

[23]

Y. F. Tang, The simplecticity of multistep methods, Comput. Math. Appl., 25(3) (1993), 83-90.

show all references

References:
[1]

J.C. Butcher, General Linear Methods, Acta Numer., 15 (2006), 157-256.

[2]

J.C. Butcher, Numerical methods for Ordinary Differential Equations, Second Edition, Wiley, Chichester, 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(5) (2014), 2440-2465.

[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), 77-84.

[6]

R. D'Ambrosio, On the G-symplecticity of two-step Runge-Kutta methods, Commun. Appl. Ind. Math., 3(1) (2012), e-403, DOI: 10.1685/journal.caim.403.

[7]

R. D'Ambrosio, Multi-value numerical methods for hamiltonian systems, Numerical Mathematics and Advanced Applications - ENUMATH 2013, Lecture Notes in Computational Science and Engineering 103 (2015), 185-193.

[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(2) (2012), e-412, DOI: 10.1685/journal.caim.412.

[9]

R. D'Ambrosio, G. De Martino and B. Paternoster, Numerical integration of Hamiltonian problems by G-symplectic methods, Adv. Comput. Math., 40(2) (2014), 553-575.

[10]

R. D'Ambrosio, E. Esposito and B. Paternoster, General Linear Methods for $y''=f(y(t))$, Numer. Algorithms, 61(2) (2012), 331-349.

[11]

R. D'Ambrosio and E. Hairer, Long-term stability of multi-value methods for ordinary differential equations, J. Sci. Comput., 60(3) (2014), 627-640.

[12]

R. D'Ambrosio, E. Hairer and C. Zbinden, G-symplecticity implies conjugate-symplecticity of the underlying one-step method, BIT, 53(4) (2013), 867-872.

[13]

E. Hairer and P. Leone, Order barriers for symplectic multi-value methods, Numerical analysis 1997, Proc. of the 17th Dundee Biennial Conference 1997, (eds. D. F. Griffiths, D. J. Higham, G. A. Watson), Pitman Research Notes in Mathematics Series 380, 1998.

[14]

E. Hairer and C. Lubich, Symmetric multistep methods over long times , Numer. Math., 97 (2004), 699-723.

[15]

E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, Second edition, Springer Series in Computational Mathematics 31, Springer-Verlag, Berlin, 2006.

[16]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations, Second edition, Springer Series in Computational Mathematics 8, Springer-Verlag, Berlin, 2008.

[17]

Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations, John Wiley & Sons, Hoboken, New Jersey, 2009.

[18]

P. Leone, Symplecticity and symmetry of general integration methods, Ph.D. thesis, Universite de Geneve, Section de Mathematiques, 2000.

[19]

R. I McLachlan, and G. R. W. Quispel, Geometric Integrators for ODEs, J. Phys. A: Math. Gen. 39 (2006), 5251-5285.

[20]

K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Series: Applied Mathematical Sciences, Vol. 90, XIII, 2009.

[21]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Chapman & Hall, London, England, 1994.

[22]

D. Okunbor and R. D. Skeel, Explicit Canonical Methods for Hamiltonian Systems, Math. Comput., 59(200) (1992), 439-455.

[23]

Y. F. Tang, The simplecticity of multistep methods, Comput. Math. Appl., 25(3) (1993), 83-90.

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