Energy-Preserving Integrators and the Structure of B-series

Foundations of Computational Mathematics

Volumn 10, Issue 6, 2010, Pages 673-693

  Celledoni, Elena ^a   McLachlan, Robert I ^b   Owren, Brynjulf ^a   Quispel, G R W ^c  

^a NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

^b MASSEY UNIVERSITY   (New Zealand)

^c LA TROBE UNIVERSITY   (Australia)

Author keywords

B series methods; Conjugate methods; Energy preservation; Symplectic integration; Trees

Indexed keywords

ALGEBRAIC STRUCTURES; B-SERIES; CONJUGATE METHODS; HAMILTONIAN VECTOR FIELDS; LINEAR SUBSPACE; NUMERICAL INTEGRATOR; SYMPLECTIC INTEGRATION; TREES;

NUMERICAL METHODS; RUNGE KUTTA METHODS;

HAMILTONIANS;

EID: 77957968010     PISSN: 16153375     EISSN: 16153383     Source Type: Journal    
DOI: 10.1007/s10208-010-9073-1     Document Type: Article

References (26)

1. J. C. Butcher, Numerical Methods for Ordinary Differential Equations, 2nd edn. (Wiley, New York, 2008).
2. M. P. Calvo, J. M. Sanz-Serna, Canonical B-series, Numer. Math. 67, 161-175 (1994).
3. E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren, G. R. W. Quispel, W. M. Wright, Energy-preserving Runge-Kutta methods, Math. Model. Numer. Anal. 43, 645-649 (2009).
4. P. Chartier, A. Murua, Preserving first integrals and volume forms of additively split systems, IMA J. Numer. Anal. 27, 381-405 (2007).
5. P. Chartier, E. Faou, A. Murua, An algebraic approach to invariant preserving integrators: the case of quadratic and Hamiltonian invariants, Numer. Math. 103, 575-590 (2006).
6. G. J. Cooper, Stability of Runge-Kutta methods for trajectory problems, IMA J. Numer. Anal. 7, 1-13 (1987).
7. B. A. Davey, H. A. Priestley, Introduction to Lattices and Order (Cambridge University Press, Cambridge, 2002).
8. K. Ebrahimi-Fard, D. Manchon, A Magnus- and Fer-type formula in dendriform algebras, Found. Comput. Math. 9, 295-316 (2009).
9. E. Faou, E. Hairer, T.-L. Pham, Energy conservation with non-symplectic methods: examples and counter-examples, BIT Numer. Math. 44, 699-709 (2004).
10. Z. Ge, J. E. Marsden, Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators, Phys. Lett. A133, 134-139 (1988).
11. E. Hairer, S. P. Nørsett, G. Wanner, Solving Ordinary Differential Equations: I. Nonstiff Problems, 2nd edn. (Springer, Berlin, 1993).
12. E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. (Springer, Berlin, 2006).
13. E. Hairer, R. I. McLachlan, R. D. Skeel, On energy conservation of the simplified Takahashi-Imada method, Math. Model. Numer. Anal. 43, 631-644 (2009).
14. F. Iavernaro, B. Pace, s-Stage trapezoidal methods for the conservation of Hamiltonian functions of polynomial type, AIP Conf. Proc. 936, 603-606 (2007).
15. F. Iavernaro, D. Trigiante, High-order symmetric schemes for energy conservation of polynomial Hamiltonian problems, J. Numer. Anal. Ind. Appl. Math. 4, 87-101 (2009).
16. A. Iserles, G. R. W. Quispel, P. S. P. Tse, B-series methods cannot be volume-preserving, BIT Numer. Math. 47, 351-378 (2007).
17. R. I. McLachlan, The structure of a set of vector fields on Poisson manifolds, J. Phys. A42, 142001 (2009), 3 pp.
18. R. I. McLachlan, G. R. W. Quispel, G. S. Turner, Numerical integrators that preserve symmetries and reversing symmetries, SIAM J. Numer. Anal. 35, 586-599 (1998).
19. R. I. McLachlan, G. R. W. Quispel, N. Robidoux, Geometric integration using discrete gradients, Philos. Trans. R. Soc. A357, 1021-1046 (1999).
20. R. I. McLachlan, G. R. W. Quispel, P. S. P. Tse, Linearization-preserving self-adjoint and symplectic integrators, BIT Numer. Math. 49, 177-197 (2009).
21. A. Murua, Formal series and numerical integrators, part I: Systems of ODEs and symplectic integrators, Appl. Numer. Math. 29, 221-251 (1999).
22. A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Found. Comput. Math. 6, 387-426 (2006).
23. R. Otter, The number of trees, Ann. Math. 49(3), 583-599 (1948).
24. G. R. W. Quispel, D. I. McLaren, A new class of energy-preserving numerical integration methods. J. Phys. A41, 045206 (2008), 7 pp.
25. J. E. Scully, A search for improved numerical integration methods using rooted trees and splitting. M. Sc. Thesis, La Trobe University, 2002.
26. L. F. Shampine, Conservation laws and the numerical solution of ODEs, Comput. Math. Appl. B12, 1287-1296 (1986).