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

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.
Citation: Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[20]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[20]

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

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