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On the exact discretization of the classical harmonic oscillator equation. (English) Zbl 1232.65175

The author presents the exact discretization of the classical harmonic oscillator equation with a special stress on the energy integral. Numerical applications are also discussed.

MSC:

65P10 Numerical methods for Hamiltonian systems including symplectic integrators
65L12 Finite difference and finite volume methods for ordinary differential equations
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
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[1] Agarwal R.P., Difference Equations and Inequalities (2000) · Zbl 0952.39001
[2] DOI: 10.1002/num.1025 · Zbl 0988.65055
[3] DOI: 10.1007/BF02163028 · Zbl 0198.49601
[4] DOI: 10.1007/BF01235122 · Zbl 0257.65062
[5] Bohner M., Dynamic Equations on Time Scales: An Introduction with Applications (2001) · Zbl 0978.39001
[6] DOI: 10.1023/A:1008369619073
[7] DOI: 10.1088/0305-4470/39/19/S02 · Zbl 1090.65138
[8] Castro A., Adv. Difference Equ. 2009 pp 290625– (2009)
[9] DOI: 10.1016/j.physleta.2007.05.020 · Zbl 1209.70003
[10] J.L. Cieśliński, Comment on ’conservative discretizations of the Kepler motion’, J. Phys. A: Math. Theor. 43 (2010) 228001 (4 pp) · Zbl 1241.70018
[11] J.L. Cieśliński, Locally exact modifications of numerical integrators, preprint arXiv:1101.0578 [math.NA]
[12] Cieśliński J.L., Adv. Difference Equ. 2006 pp 40171– (2006)
[13] DOI: 10.1088/1751-8113/42/10/105204 · Zbl 1160.65064
[14] J.L. Cieśliński and B. Ratkiewicz, How to improve the accuracy of the discrete gradient method in the one-dimensional case, preprint. Available at arXiv, 0901.1906 [cs.NA] (2009)
[15] DOI: 10.1103/PhysRevE.81.016704
[16] Elaydi S.N., An Introduction to Difference Equations, 3. ed. (2005) · Zbl 1071.39001
[17] DOI: 10.1007/BF01386037 · Zbl 0163.39002
[18] DOI: 10.1090/S0002-9904-1973-13197-0 · Zbl 0266.70004
[19] Hairer E., Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2. ed. (2006) · Zbl 1094.65125
[20] Hildebrand F.B., Finite Difference Equations and Simulations (1968) · Zbl 0157.22702
[21] Hilger S., Results Math. 18 pp 18– (1990) · Zbl 0722.39001
[22] DOI: 10.1007/s002110050456 · Zbl 0937.65077
[23] DOI: 10.1088/1751-8113/40/17/009 · Zbl 1113.70003
[24] DOI: 10.1515/crll.1965.218.204 · Zbl 0151.34901
[25] DOI: 10.1016/0021-9991(74)90081-3 · Zbl 0301.70006
[26] DOI: 10.1137/0704033 · Zbl 0223.65030
[27] DOI: 10.1098/rsta.1999.0363 · Zbl 0933.65143
[28] Mickens R.E., Nonstandard Finite Difference Models of Differential Equations (1994) · Zbl 0810.65083
[29] DOI: 10.1080/1023619021000000807
[30] DOI: 10.1023/A:1008217427749 · Zbl 0898.70003
[31] B.V. Minchev and W.M. Wright, A review of exponential integrators for first order semi-linear problems, preprint. Available at NTNU/Numerics/N2/2005, Trondheim, 2005
[32] DOI: 10.1016/j.physleta.2004.02.059 · Zbl 1123.70302
[33] DOI: 10.1145/366707.367592 · Zbl 0117.11204
[34] Rubinowicz W., Mechanika Teoretyczna (Theoretical Mechanics) (1978)
[35] DOI: 10.2307/2321656 · Zbl 0498.34049
[36] DOI: 10.1007/BF02163234 · Zbl 0219.65062
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