Convergence analysis of the formal energies of symplectic methods for Hamiltonian systems. (English) Zbl 1350.65138
The authors present a convergence analysis of the formal energies of symplectic methods for Hamiltonian systems. They first introduce some basic facts about B-series, formal vector field and backward error analysis. They prove that in contrast to the cases for the four types trees discussed in [Y.-F. Tang et al., Comput. Math. Appl. 43, No. 8–9, 1171–1181 (2002; Zbl 1050.65128)], in the tree expansion of the formal energy of the midpoint, the coefficient sequence of bushy trees of height \( m\geq 2\) goes to \(\infty\) at great speed. The conclusion is extended to other Runge-Kutta methods.
Reviewer: Seenith Sivasundaram (Daytona Beach)
MSC:
| 65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |
| 37M15 | Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 70H05 | Hamilton’s equations |
Keywords:
convergence analysis; formal energy; symplectic method; Hamiltonian system; bushy tree; B-series; formal vector field; backward error analysis; Runge-Kutta methodsCitations:
Zbl 1050.65128References:
| [1] | Benettin G, Giorgilli A. On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms. J Statist Phys, 1994, 74: 1117-1143 · Zbl 0842.58020 · doi:10.1007/BF02188219 |
| [2] | Calvo M P, Murua A, Sanz-Serna J M. Modified equations for ODEs. Contemp Math, 1994, 172: 63-74 · Zbl 0807.65074 · doi:10.1090/conm/172/01798 |
| [3] | Channell P J, Scovel J C. Symplectic integration of Hamiltonian systems. Nonlinearity, 1990, 3: 231-259 · Zbl 0704.65052 · doi:10.1088/0951-7715/3/2/001 |
| [4] | Feng, K., Formal power series and numerical algorithms for dynamical systems, 28-35 (1992), Singapore |
| [5] | Feng K. The calculus of generating functions and the formal energy for Hamiltonian algorithms. Collected Works of Feng Kang (II). Beijing: National Defence Industry Press, 1995, 284-302 |
| [6] | Field C M, Nijhoff F W. A note on modified Hamiltonians for numerical integrations admitting an exact invariant. Nonlinearity, 2003, 16: 1673-1683 · Zbl 1046.65109 · doi:10.1088/0951-7715/16/5/308 |
| [7] | Hairer E, Wanner G. On the Butcher group and general multi-value methods. Computing, 1974, 13: 1-15 · Zbl 0293.65050 · doi:10.1007/BF02268387 |
| [8] | Hairer E. Backward analysis of numerical integrators and symplectic methods. Ann Numer Math, 1994, 1: 107-132 · Zbl 0828.65097 |
| [9] | Hairer E, Lubich C, Wanner G. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. New York: Springer, 2002 · Zbl 0994.65135 · doi:10.1007/978-3-662-05018-7 |
| [10] | Kuang J C. Applied Inequalities. Shandong: Shandong Science and Technology Press, 2010 |
| [11] | Moser J K. Lectures on Hamiltonian systems. Mem Amer Math, 1968, 81: 1-60 · Zbl 0172.11401 |
| [12] | Murua A. Formal series and numerical integrators, part i: Systems of ODEs and symplectic integrators. Appl Numer Math, 1999, 29: 221-251 · Zbl 0929.65126 · doi:10.1016/S0168-9274(98)00064-6 |
| [13] | Neishtadt A I. The separation of motions in systems with rapidly rotating phase. J Appl Math Mech, 1984, 48: 133-139 · Zbl 0571.70022 · doi:10.1016/0021-8928(84)90078-9 |
| [14] | Sanz-Serna J M, Calvo M P. Numerical Hamiltonian Problems. London: Chapman and Hall, 1994 · Zbl 0816.65042 · doi:10.1007/978-1-4899-3093-4 |
| [15] | Tang Y F. Formal energy of a symplectic scheme for Hamiltonian systems and its application (I). Comput Math Appl, 1994, 27: 31-39 · Zbl 0822.70012 |
| [16] | Tang Y F, Vázquez L, Zhang F, et al. Symplectic methods for the nonlinear schrödinger Equation. Comput Math Appl, 1996, 32: 73-83 · Zbl 0858.65124 · doi:10.1016/0898-1221(96)00136-8 |
| [17] | Tang Y F, Xiao A G, Chen J B. Is the formal energy of the mid-point rule convergent? Comput Math Appl, 2002, 43: 1171-1181 · Zbl 1050.65128 · doi:10.1016/S0898-1221(02)80021-9 |
| [18] | Wang Z X, Guo D R. Introduction to special function (in Chinese). Beijing: Peking University Press, 2000 |
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