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 & Continuous Dynamical Systems - A, 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. 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. 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. 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, 172 (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. 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. 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, 31 (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. 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. 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. 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, (2005), 2005. 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. 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. doi: 10.1137/S0036142997329797. Google Scholar

[15]

Z. Shang, KAM theorem of symplectic algorithms for Hamiltonian systems,, Numer. Math., 83 (1999), 477. 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. 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. 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. 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. doi: 10.1086/116378. Google Scholar

[20]

H. Yoshida, Construction of higher order symplectic integrators,, Phys. Lett. A, 150 (1990), 262. 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. 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. 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. 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, 172 (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. 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. 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, 31 (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. 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. 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. 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, (2005), 2005. 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. 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. doi: 10.1137/S0036142997329797. Google Scholar

[15]

Z. Shang, KAM theorem of symplectic algorithms for Hamiltonian systems,, Numer. Math., 83 (1999), 477. 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. 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. 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. 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. doi: 10.1086/116378. Google Scholar

[20]

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

[1]

Robert I McLachlan, Christian Offen, Benjamin K Tapley. Symplectic integration of PDEs using Clebsch variables. Journal of Computational Dynamics, 2019, 6 (1) : 111-130. doi: 10.3934/jcd.2019005

[2]

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

[3]

Morched Boughariou. Closed orbits of Hamiltonian systems on non-compact prescribed energy surfaces. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 603-616. doi: 10.3934/dcds.2003.9.603

[4]

Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2705-2715. doi: 10.3934/dcds.2017116

[5]

Liang Ding, Rongrong Tian, Jinlong Wei. Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1617-1625. doi: 10.3934/dcdsb.2018222

[6]

Guillermo Dávila-Rascón, Yuri Vorobiev. Hamiltonian structures for projectable dynamics on symplectic fiber bundles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1077-1088. doi: 10.3934/dcds.2013.33.1077

[7]

Patrick Cummings, C. Eugene Wayne. Modified energy functionals and the NLS approximation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1295-1321. doi: 10.3934/dcds.2017054

[8]

Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431

[9]

Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325

[10]

Wei Wang. Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 679-701. doi: 10.3934/dcds.2012.32.679

[11]

Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201

[12]

Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855

[13]

Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194

[14]

Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2019-2034. doi: 10.3934/dcdss.2019130

[15]

C. D. Ahlbrandt, A. C. Peterson. A general reduction of order theorem for discrete linear symplectic systems. Conference Publications, 1998, 1998 (Special) : 7-18. doi: 10.3934/proc.1998.1998.7

[16]

Shuang Liu, Xinfeng Liu. Krylov implicit integration factor method for a class of stiff reaction-diffusion systems with moving boundaries. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-19. doi: 10.3934/dcdsb.2019176

[17]

K. Tintarev. Critical values and minimal periods for autonomous Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 389-400. doi: 10.3934/dcds.1995.1.389

[18]

Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249

[19]

Tianqing An, Zhi-Qiang Wang. Periodic solutions of Hamiltonian systems with anisotropic growth. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1069-1082. doi: 10.3934/cpaa.2010.9.1069

[20]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]