American Institute of Mathematical Sciences

Advanced
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.   Google Scholar

[2]

J.C. Butcher, Numerical methods for Ordinary Differential Equations,, Second Edition, (2008).   Google Scholar

[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.   Google Scholar

[4]

J. C. Butcher and R. D'Ambrosio, Partitioned general linear methods for separable Hamiltonian problems,, in preparation., ().   Google Scholar

[5]

J. C. Butcher and L. L. Hewitt, The existence of symplectic general linear methods,, Numer. Algor., 51 (2009), 77.   Google Scholar

[6]

R. D'Ambrosio, On the G-symplecticity of two-step Runge-Kutta methods,, Commun. Appl. Ind. Math., 3 (2012).   Google Scholar

[7]

R. D'Ambrosio, Multi-value numerical methods for hamiltonian systems,, Numerical Mathematics and Advanced Applications - ENUMATH 2013, 103 (2015), 185.   Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[10]

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

[11]

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

[12]

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

[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).   Google Scholar

[14]

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

[15]

E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations,, Second edition, (2006).   Google Scholar

[16]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations,, Second edition, (2008).   Google Scholar

[17]

Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations,, John Wiley & Sons, (2009).   Google Scholar

[18]

P. Leone, Symplecticity and symmetry of general integration methods,, Ph.D. thesis, (2000).   Google Scholar

[19]

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

[20]

K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,, Series: Applied Mathematical Sciences, (2009).   Google Scholar

[21]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems,, Chapman & Hall, (1994).   Google Scholar

[22]

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

[23]

Y. F. Tang, The simplecticity of multistep methods,, Comput. Math. Appl., 25 (1993), 83.   Google Scholar

show all references

References:
[1]

J.C. Butcher, General Linear Methods,, Acta Numer., 15 (2006), 157.   Google Scholar

[2]

J.C. Butcher, Numerical methods for Ordinary Differential Equations,, Second Edition, (2008).   Google Scholar

[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.   Google Scholar

[4]

J. C. Butcher and R. D'Ambrosio, Partitioned general linear methods for separable Hamiltonian problems,, in preparation., ().   Google Scholar

[5]

J. C. Butcher and L. L. Hewitt, The existence of symplectic general linear methods,, Numer. Algor., 51 (2009), 77.   Google Scholar

[6]

R. D'Ambrosio, On the G-symplecticity of two-step Runge-Kutta methods,, Commun. Appl. Ind. Math., 3 (2012).   Google Scholar

[7]

R. D'Ambrosio, Multi-value numerical methods for hamiltonian systems,, Numerical Mathematics and Advanced Applications - ENUMATH 2013, 103 (2015), 185.   Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[10]

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

[11]

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

[12]

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

[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).   Google Scholar

[14]

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

[15]

E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations,, Second edition, (2006).   Google Scholar

[16]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations,, Second edition, (2008).   Google Scholar

[17]

Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations,, John Wiley & Sons, (2009).   Google Scholar

[18]

P. Leone, Symplecticity and symmetry of general integration methods,, Ph.D. thesis, (2000).   Google Scholar

[19]

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

[20]

K. Meyer, G. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,, Series: Applied Mathematical Sciences, (2009).   Google Scholar

[21]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems,, Chapman & Hall, (1994).   Google Scholar

[22]

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

[23]

Y. F. Tang, The simplecticity of multistep methods,, Comput. Math. Appl., 25 (1993), 83.   Google Scholar

[1]

Alessandra Celletti. Some KAM applications to Celestial Mechanics. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 533-544. doi: 10.3934/dcdss.2010.3.533
[2]

Luca Biasco, Luigi Chierchia. Exponential stability for the resonant D'Alembert model of celestial mechanics. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 569-594. doi: 10.3934/dcds.2005.12.569
[3]

Alexander Mielke. Weak-convergence methods for Hamiltonian multiscale problems. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 53-79. doi: 10.3934/dcds.2008.20.53
[4]

Jan J. Sławianowski, Vasyl Kovalchuk, Agnieszka Martens, Barbara Gołubowska, Ewa E. Rożko. Essential nonlinearity implied by symmetry group. Problems of affine invariance in mechanics and physics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 699-733. doi: 10.3934/dcdsb.2012.17.699
[5]

P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1
[6]

Rasmus Dalgas Kongskov, Yiqiu Dong. Tomographic reconstruction methods for decomposing directional components. Inverse Problems & Imaging, 2018, 12 (6) : 1429-1442. doi: 10.3934/ipi.2018060
[7]

Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869
[8]

Piotr Gwiazda, Piotr Minakowski, Agnieszka Świerczewska-Gwiazda. On the anisotropic Orlicz spaces applied in the problems of continuum mechanics. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1291-1306. doi: 10.3934/dcdss.2013.6.1291
[9]

Tran Ninh Hoa, Ta Duy Phuong, Nguyen Dong Yen. Linear fractional vector optimization problems with many components in the solution sets. Journal of Industrial & Management Optimization, 2005, 1 (4) : 477-486. doi: 10.3934/jimo.2005.1.477
[10]

Colin J. Cotter, Darryl D. Holm. Geodesic boundary value problems with symmetry. Journal of Geometric Mechanics, 2010, 2 (1) : 51-68. doi: 10.3934/jgm.2010.2.51
[11]

Marian Gidea, Rafael De La Llave. Topological methods in the instability problem of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 295-328. doi: 10.3934/dcds.2006.14.295
[12]

Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042
[13]

James Montaldi. Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry. Journal of Geometric Mechanics, 2014, 6 (2) : 237-260. doi: 10.3934/jgm.2014.6.237
[14]

Pedro Freitas. The linear damped wave equation, Hamiltonian symmetry, and the importance of being odd. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 635-640. doi: 10.3934/dcds.1998.4.635
[15]

Sanjay Dharmavaram, Timothy J. Healey. Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1669-1684. doi: 10.3934/dcdss.2019112
[16]

Miguel Rodríguez-Olmos. Book review: Geometric mechanics and symmetry, by Darryl D. Holm, Tanya Schmah and Cristina Stoica. Journal of Geometric Mechanics, 2009, 1 (4) : 483-488. doi: 10.3934/jgm.2009.1.483
[17]

Lucio Cadeddu, Giovanni Porru. Symmetry breaking in problems involving semilinear equations. Conference Publications, 2011, 2011 (Special) : 219-228. doi: 10.3934/proc.2011.2011.219
[18]

Claudia Anedda, Giovanni Porru. Symmetry breaking and other features for Eigenvalue problems. Conference Publications, 2011, 2011 (Special) : 61-70. doi: 10.3934/proc.2011.2011.61
[19]

Jana Kopfová. Nonlinear semigroup methods in problems with hysteresis. Conference Publications, 2007, 2007 (Special) : 580-589. doi: 10.3934/proc.2007.2007.580
[20]

Assyr Abdulle. Multiscale methods for advection-diffusion problems. Conference Publications, 2005, 2005 (Special) : 11-21. doi: 10.3934/proc.2005.2005.11

 Impact Factor: 

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences