American Institute of Mathematical Sciences

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March  2014, 34(3): 1105-1120. doi: 10.3934/dcds.2014.34.1105

On an asymptotic method for computing the modified energy for symplectic methods

Per Christian Moan 1, and  Jitse Niesen 2,

1. 

Centre of Mathematics for Applications, University of Oslo, Norway
2. 

School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom

Received  January 2013 Revised  April 2013 Published  August 2013

We revisit an algorithm by Skeel et al. [5,16] for computing the modified, or shadow, energy associated with symplectic discretizations of Hamiltonian systems. We amend the algorithm to use Richardson extrapolation in order to obtain arbitrarily high order of accuracy. Error estimates show that the new method captures the exponentially small drift associated with such discretizations. Several numerical examples illustrate the theory.
Keywords: symplectic integration, Modified energy, Hamiltonian systems, Richardson extrapolation..
Mathematics Subject Classification: Primary: 65P10; Secondary: 37J40, 37M1.
Citation: Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1105-1120. doi: 10.3934/dcds.2014.34.1105
References:
[1]

G. Benettin and F. Fasso, From Hamiltonian perturbation theory to symplectic integrators and back, Appl. Numer. Math., 29 (1999), 73-87. doi: 10.1016/S0168-9274(98)00074-9.  Google Scholar

[2]

G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms, J. Stat. Phys., 74 (1994), 1117-1143. doi: 10.1007/BF02188219.  Google Scholar

[3]

S. Blanes and P. C. Moan, Practical symplectic Runge-Kutta and Runge-Kutta-Nyström methods, J. Comput. Appl. Math, 142 (2002), 313-330. doi: 10.1016/S0377-0427(01)00492-7.  Google Scholar

[4]

M. P. Calvo, A. Murua and J. M. Sanz-Serna, Modified equations for ODEs, in "Chaotic Numerics" 172 of Contemp. Math., 63C-74. Amer. Math. Soc., Providence, RI, (1994). doi: 10.1090/conm/172/01798.  Google Scholar

[5]

R. D. Engle, R. D. Skeel and M. Drees, Monitoring energy drift with shadow Hamiltonians, J. Comput. Phys., 206 (2005), 432-452. doi: 10.1016/j.jcp.2004.12.009.  Google Scholar

[6]

E. Hairer and C. Lubich, The life-span of backward error analysis for numerical integrators, Numer. Math., 76 (1997), 441-462. doi: 10.1007/s002110050271.  Google Scholar

[7]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations," 31 of Springer Series in Computational Mathematics. Springer, Berlin, 2002.  Google Scholar

[8]

L. O. Jay, Beyond conventional Runge-Kutta methods in numerical integration of ODEs and DAEs by use of structures and local models, J. Comput. Appl. Math, 204 (2007), 56-76. doi: 10.1016/j.cam.2006.04.028.  Google Scholar

[9]

B. Leimkuhler and S. Reich, "Simulating Hamiltonian Dynamics," Cambridge University Press, 2005. doi: 10.1017/CBO9780511614118.  Google Scholar

[10]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X.  Google Scholar

[11]

P. C. Moan, On the KAM and Nekhoroshev theorems for symplectic integrators and implications for error growth, Nonlinearity, 17 (2004), 67-83. doi: 10.1088/0951-7715/17/1/005.  Google Scholar

[12]

P. C. Moan, On rigorous modified equations for discretizations of ODEs, Technical Report 2005-3, Geometric Integration Preprint Server, 2005. Available from http://www.focm.net/gi/gips/2005/3.html. doi: 10.1088/0305-4470/39/19/S13.  Google Scholar

[13]

P. C. Moan, On modified equations for discretizations of ODEs, J. Phys. A, 39 (2006), 5545-5561. doi: 10.1088/0305-4470/39/19/S13.  Google Scholar

[14]

S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), 1549-1570. doi: 10.1137/S0036142997329797.  Google Scholar

[15]

Z. Shang, KAM theorem of symplectic algorithms for Hamiltonian systems, Numer. Math., 83 (1999), 477-496. doi: 10.1007/s002110050460.  Google Scholar

[16]

R. D. Skeel and D. J. Hardy, Practical construction of modified Hamiltonians, SIAM J. Sci. Comput., 23 (2001), 1172-1188. doi: 10.1137/S106482750138318X.  Google Scholar

[17]

R. D. Skeel, G. Zhang and T. Schlick, A family of symplectic integrators: Stability, accuracy, and molecular dynamics applications, SIAM J. Sci. Comput., 18 (1997), 203-222. doi: 10.1137/S1064827595282350.  Google Scholar

[18]

P. F. Tupper, Ergodicity and the numerical simulation of Hamiltonian systems, SIAM J. Appl. Dynam. Systems, 4 (2005), 563-587. doi: 10.1137/040603802.  Google Scholar

[19]

J. Wisdom and M. Holman, Symplectic maps for the $n$-body problem: Stability analysis, Astron. J., 104 (1992), 2022-2029. doi: 10.1086/116378.  Google Scholar

[20]

H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268. doi: 10.1016/0375-9601(90)90092-3.  Google Scholar

show all references

References:
[1]

G. Benettin and F. Fasso, From Hamiltonian perturbation theory to symplectic integrators and back, Appl. Numer. Math., 29 (1999), 73-87. doi: 10.1016/S0168-9274(98)00074-9.  Google Scholar

[2]

G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms, J. Stat. Phys., 74 (1994), 1117-1143. doi: 10.1007/BF02188219.  Google Scholar

[3]

S. Blanes and P. C. Moan, Practical symplectic Runge-Kutta and Runge-Kutta-Nyström methods, J. Comput. Appl. Math, 142 (2002), 313-330. doi: 10.1016/S0377-0427(01)00492-7.  Google Scholar

[4]

M. P. Calvo, A. Murua and J. M. Sanz-Serna, Modified equations for ODEs, in "Chaotic Numerics" 172 of Contemp. Math., 63C-74. Amer. Math. Soc., Providence, RI, (1994). doi: 10.1090/conm/172/01798.  Google Scholar

[5]

R. D. Engle, R. D. Skeel and M. Drees, Monitoring energy drift with shadow Hamiltonians, J. Comput. Phys., 206 (2005), 432-452. doi: 10.1016/j.jcp.2004.12.009.  Google Scholar

[6]

E. Hairer and C. Lubich, The life-span of backward error analysis for numerical integrators, Numer. Math., 76 (1997), 441-462. doi: 10.1007/s002110050271.  Google Scholar

[7]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations," 31 of Springer Series in Computational Mathematics. Springer, Berlin, 2002.  Google Scholar

[8]

L. O. Jay, Beyond conventional Runge-Kutta methods in numerical integration of ODEs and DAEs by use of structures and local models, J. Comput. Appl. Math, 204 (2007), 56-76. doi: 10.1016/j.cam.2006.04.028.  Google Scholar

[9]

B. Leimkuhler and S. Reich, "Simulating Hamiltonian Dynamics," Cambridge University Press, 2005. doi: 10.1017/CBO9780511614118.  Google Scholar

[10]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numerica, 10 (2001), 357-514. doi: 10.1017/S096249290100006X.  Google Scholar

[11]

P. C. Moan, On the KAM and Nekhoroshev theorems for symplectic integrators and implications for error growth, Nonlinearity, 17 (2004), 67-83. doi: 10.1088/0951-7715/17/1/005.  Google Scholar

[12]

P. C. Moan, On rigorous modified equations for discretizations of ODEs, Technical Report 2005-3, Geometric Integration Preprint Server, 2005. Available from http://www.focm.net/gi/gips/2005/3.html. doi: 10.1088/0305-4470/39/19/S13.  Google Scholar

[13]

P. C. Moan, On modified equations for discretizations of ODEs, J. Phys. A, 39 (2006), 5545-5561. doi: 10.1088/0305-4470/39/19/S13.  Google Scholar

[14]

S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), 1549-1570. doi: 10.1137/S0036142997329797.  Google Scholar

[15]

Z. Shang, KAM theorem of symplectic algorithms for Hamiltonian systems, Numer. Math., 83 (1999), 477-496. doi: 10.1007/s002110050460.  Google Scholar

[16]

R. D. Skeel and D. J. Hardy, Practical construction of modified Hamiltonians, SIAM J. Sci. Comput., 23 (2001), 1172-1188. doi: 10.1137/S106482750138318X.  Google Scholar

[17]

R. D. Skeel, G. Zhang and T. Schlick, A family of symplectic integrators: Stability, accuracy, and molecular dynamics applications, SIAM J. Sci. Comput., 18 (1997), 203-222. doi: 10.1137/S1064827595282350.  Google Scholar

[18]

P. F. Tupper, Ergodicity and the numerical simulation of Hamiltonian systems, SIAM J. Appl. Dynam. Systems, 4 (2005), 563-587. doi: 10.1137/040603802.  Google Scholar

[19]

J. Wisdom and M. Holman, Symplectic maps for the $n$-body problem: Stability analysis, Astron. J., 104 (1992), 2022-2029. doi: 10.1086/116378.  Google Scholar

[20]

H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), 262-268. doi: 10.1016/0375-9601(90)90092-3.  Google Scholar

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