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A formal series approach to averaging: Exponentially small error estimates
1. | INRIA Rennes and ENS Cachan Bretagne, Campus Ker-Lann, av. Robert Schumann, F-35170 Bruz, France |
2. | Konputazio Zientziak eta A. A. Saila, Informatika Fakultatea, UPV/EHU, E-20018 Donostia-San Sebastián, Spain |
3. | Departamento de Matemática Aplicada e IMUVA, Facultad de Ciencias, Universidad de Valladolid, Valladolid, Spain |
References:
[1] |
V. I. Arnol'd, "Geometrical Methods in the Theory of Ordinary Differential Equations," 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 250, Springer-Verlag, New York, 1988. |
[2] |
V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," 2nd edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. |
[3] |
S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep., 470 (2009), 151-238.
doi: 10.1016/j.physrep.2008.11.001. |
[4] |
M. P. Calvo, Ph. Chartier, A. Murua and J. M. Sanz-Serna, A stroboscopic method for highly oscillatory problems, in "Numerical Analysis of Multiscale Computations" (eds. B. Engquist, O. Runborg and R. Tsai), Springer-Verlag, (2011), 73-87. |
[5] |
M. P. Calvo, Ph. Chartier, A. Murua and J. M. Sanz-Serna, Numerical stroboscopic averaging for ODEs and DAEs, Appl. Numer. Math., 61 (2011), 1077-1095.
doi: 10.1016/j.apnum.2011.06.007. |
[6] |
F. Casas, J. A. Oteo and J. Ros, Floquet theory: Exponential perturbative treatment, J. Phys. A, 34 (2001), 3379-3388.
doi: 10.1088/0305-4470/34/16/305. |
[7] |
P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration I: B-series, Found. Comput. Math., 10 (2010), 695-727.
doi: 10.1007/s10208-010-9074-0. |
[8] |
P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration II: The quasi-periodic case,, Found. Comput. Math., ().
|
[9] |
K.-T. Chen, Integration of paths, geometric invariants and a generalized Baker-Haussdorff formula, Annals Math. (2), 65 (1957), 163-178.
doi: 10.2307/1969671. |
[10] |
M. Fliess, Fonctionelles causales nonlinéaires et indeterminées non commutatives, Bull. Soc. Math. France, 109 (1981), 3-40. |
[11] |
E. Hairer, Ch. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations," 2nd edition, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. |
[12] |
E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I. Nonstiff Problems," 2nd edition, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1993. |
[13] |
P. Lochak and C. Meunier, "Multiphase Averaging for Classical Systems. With Applications to Adiabatic Theorems," Applied Mathematical Sciences, 72, Springer-Verlag, New York, 1988. |
[14] |
A. Murua, Formal series and numerical integrators. I. Systems of ODEs and symplectic integrators, Appl. Numer. Math., 29 (1999), 221-251.
doi: 10.1016/S0168-9274(98)00064-6. |
[15] |
A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Found. Comput. Math., 6 (2006), 387-426.
doi: 10.1007/s10208-003-0111-0. |
[16] |
A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48 (1984), 133-139.
doi: 10.1016/0021-8928(84)90078-9. |
[17] |
J.-P. Ramis and R. Schäfke, Gevrey separation of fast and slow variables, Nonlinearity, 9 (1996), 353-384.
doi: 10.1088/0951-7715/9/2/004. |
[18] |
J. A. Sanders, F. Verhulst and J. Murdock, "Averaging Methods in Nonlinear Dynamical Systems," 2nd edition, Applied Mathematical Sciences, 59, Springer-Verlag, New York, 2007. |
[19] |
J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Applied Mathematics and Mathematical Computation, 7, Chapman & Hall, London, 1994. |
[20] |
C. Simó, Averaging under fast quasiperiodic forcing, in "Hamiltonian Mechanics" (Toruń, 1993) (ed. I. Seimenis), NATO Adv. Sci. Inst. Ser. B Phys., 331, Plenum, New York, (1994), 13-34. |
[21] |
H. Sussman, A product expansion of the Chen series, in "Theory and Applications of Nonlinear Control Systems" (Stockholm, 1985) (eds. C. Byrnes and A. Linquist), North Holland, Amsterdam, (1986), 325-335. |
show all references
References:
[1] |
V. I. Arnol'd, "Geometrical Methods in the Theory of Ordinary Differential Equations," 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 250, Springer-Verlag, New York, 1988. |
[2] |
V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," 2nd edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. |
[3] |
S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep., 470 (2009), 151-238.
doi: 10.1016/j.physrep.2008.11.001. |
[4] |
M. P. Calvo, Ph. Chartier, A. Murua and J. M. Sanz-Serna, A stroboscopic method for highly oscillatory problems, in "Numerical Analysis of Multiscale Computations" (eds. B. Engquist, O. Runborg and R. Tsai), Springer-Verlag, (2011), 73-87. |
[5] |
M. P. Calvo, Ph. Chartier, A. Murua and J. M. Sanz-Serna, Numerical stroboscopic averaging for ODEs and DAEs, Appl. Numer. Math., 61 (2011), 1077-1095.
doi: 10.1016/j.apnum.2011.06.007. |
[6] |
F. Casas, J. A. Oteo and J. Ros, Floquet theory: Exponential perturbative treatment, J. Phys. A, 34 (2001), 3379-3388.
doi: 10.1088/0305-4470/34/16/305. |
[7] |
P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration I: B-series, Found. Comput. Math., 10 (2010), 695-727.
doi: 10.1007/s10208-010-9074-0. |
[8] |
P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration II: The quasi-periodic case,, Found. Comput. Math., ().
|
[9] |
K.-T. Chen, Integration of paths, geometric invariants and a generalized Baker-Haussdorff formula, Annals Math. (2), 65 (1957), 163-178.
doi: 10.2307/1969671. |
[10] |
M. Fliess, Fonctionelles causales nonlinéaires et indeterminées non commutatives, Bull. Soc. Math. France, 109 (1981), 3-40. |
[11] |
E. Hairer, Ch. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations," 2nd edition, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. |
[12] |
E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I. Nonstiff Problems," 2nd edition, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1993. |
[13] |
P. Lochak and C. Meunier, "Multiphase Averaging for Classical Systems. With Applications to Adiabatic Theorems," Applied Mathematical Sciences, 72, Springer-Verlag, New York, 1988. |
[14] |
A. Murua, Formal series and numerical integrators. I. Systems of ODEs and symplectic integrators, Appl. Numer. Math., 29 (1999), 221-251.
doi: 10.1016/S0168-9274(98)00064-6. |
[15] |
A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Found. Comput. Math., 6 (2006), 387-426.
doi: 10.1007/s10208-003-0111-0. |
[16] |
A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48 (1984), 133-139.
doi: 10.1016/0021-8928(84)90078-9. |
[17] |
J.-P. Ramis and R. Schäfke, Gevrey separation of fast and slow variables, Nonlinearity, 9 (1996), 353-384.
doi: 10.1088/0951-7715/9/2/004. |
[18] |
J. A. Sanders, F. Verhulst and J. Murdock, "Averaging Methods in Nonlinear Dynamical Systems," 2nd edition, Applied Mathematical Sciences, 59, Springer-Verlag, New York, 2007. |
[19] |
J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Applied Mathematics and Mathematical Computation, 7, Chapman & Hall, London, 1994. |
[20] |
C. Simó, Averaging under fast quasiperiodic forcing, in "Hamiltonian Mechanics" (Toruń, 1993) (ed. I. Seimenis), NATO Adv. Sci. Inst. Ser. B Phys., 331, Plenum, New York, (1994), 13-34. |
[21] |
H. Sussman, A product expansion of the Chen series, in "Theory and Applications of Nonlinear Control Systems" (Stockholm, 1985) (eds. C. Byrnes and A. Linquist), North Holland, Amsterdam, (1986), 325-335. |
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