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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 |
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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