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A theoretical framework for backward error analysis on manifolds
1. | Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WA, United Kingdom |
References:
[1] |
R. Abraham, J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications," volume 75 of "Applied Mathematical Sciences,", Springer-Verlag, (1988).
|
[2] |
V. I. Arnold, "Mathematical Methods of Classical Mechanics," volume 60 of "Graduate Texts in Mathematics,", Springer-Verlag, (1989).
|
[3] |
G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms,, J. Statist. Phys., 74 (1994), 1117.
doi: 10.1007/BF02188219. |
[4] |
M. P. Calvo, A. Murua and J. M. Sanz-Serna, Modified equations for ODEs,, In, 172 (1993), 63.
|
[5] |
E. Cartan, Les groupes de transformations continus, infinis, simples,, Ann. Sci. École Norm. Sup. (3), 26 (1909), 93.
|
[6] |
D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid,, Ann. of Math. (2), 92 (1970), 102.
doi: 10.2307/1970699. |
[7] |
O. Gonzalez, D. J. Higham and A. M. Stuart, Qualitative properties of modified equations,, IMA J. Numer. Anal., 19 (1999), 169.
doi: 10.1093/imanum/19.2.169. |
[8] |
E. Hairer, Global modified Hamiltonian for constrained symplectic integrators,, Numer. Math., 95 (2003), 325.
doi: 10.1007/s00211-002-0428-7. |
[9] |
E. Hairer and C. Lubich, The life-span of backward error analysis for numerical integrators,, Numer. Math., 76 (1997), 441.
doi: 10.1007/s002110050271. |
[10] |
E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," volume 31 of "Springer Series in Computational Mathematics,", Springer Series in Computational Mathematics, 31 (2002).
|
[11] |
E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I," volume 8 of "Springer Series in Computational Mathematics,", Springer Series in Computational Mathematics, 8 (1993).
|
[12] |
A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods,, Acta Numerica, 2000, 9 (2000), 215.
|
[13] |
J. M. Lee, "Introduction to Smooth Manifolds," volume 218 of "Graduate Texts in Mathematics,", Graduate Texts in Mathematics, 218 (2003).
|
[14] |
R. I. McLachlan and G. R. W. Quispel, Splitting methods,, Acta Numer., 11 (2002), 341.
doi: 10.1017/S0962492902000053. |
[15] |
H. Omori, "Infinite-dimensional Lie Groups," volume 158 of "Translations of Mathematical Monographs,", Translations of Mathematical Monographs, 158 (1997).
|
[16] |
R. S. Palais, "Foundations of Global Non-linear Analysis,", Foundations of Global Non-linear Analysis, (1968).
|
[17] |
S. Reich, "Numerical Integration of the Generatized Euler Equations,", Technical report, (1993). Google Scholar |
[18] |
S. Reich, On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems,, Numer. Math., 76 (1997), 231.
doi: 10.1007/s002110050261. |
[19] |
S. Reich, Backward error analysis for numerical integrators,, SIAM J. Numer. Anal., 36 (1999), 1549.
doi: 10.1137/S0036142997329797. |
[20] |
R. Schmid, Infinite-dimensional Lie groups with applications to mathematical physics,, J. Geom. Symmetry Phys., 1 (2004), 54.
|
show all references
References:
[1] |
R. Abraham, J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications," volume 75 of "Applied Mathematical Sciences,", Springer-Verlag, (1988).
|
[2] |
V. I. Arnold, "Mathematical Methods of Classical Mechanics," volume 60 of "Graduate Texts in Mathematics,", Springer-Verlag, (1989).
|
[3] |
G. Benettin and A. Giorgilli, On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms,, J. Statist. Phys., 74 (1994), 1117.
doi: 10.1007/BF02188219. |
[4] |
M. P. Calvo, A. Murua and J. M. Sanz-Serna, Modified equations for ODEs,, In, 172 (1993), 63.
|
[5] |
E. Cartan, Les groupes de transformations continus, infinis, simples,, Ann. Sci. École Norm. Sup. (3), 26 (1909), 93.
|
[6] |
D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid,, Ann. of Math. (2), 92 (1970), 102.
doi: 10.2307/1970699. |
[7] |
O. Gonzalez, D. J. Higham and A. M. Stuart, Qualitative properties of modified equations,, IMA J. Numer. Anal., 19 (1999), 169.
doi: 10.1093/imanum/19.2.169. |
[8] |
E. Hairer, Global modified Hamiltonian for constrained symplectic integrators,, Numer. Math., 95 (2003), 325.
doi: 10.1007/s00211-002-0428-7. |
[9] |
E. Hairer and C. Lubich, The life-span of backward error analysis for numerical integrators,, Numer. Math., 76 (1997), 441.
doi: 10.1007/s002110050271. |
[10] |
E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," volume 31 of "Springer Series in Computational Mathematics,", Springer Series in Computational Mathematics, 31 (2002).
|
[11] |
E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I," volume 8 of "Springer Series in Computational Mathematics,", Springer Series in Computational Mathematics, 8 (1993).
|
[12] |
A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods,, Acta Numerica, 2000, 9 (2000), 215.
|
[13] |
J. M. Lee, "Introduction to Smooth Manifolds," volume 218 of "Graduate Texts in Mathematics,", Graduate Texts in Mathematics, 218 (2003).
|
[14] |
R. I. McLachlan and G. R. W. Quispel, Splitting methods,, Acta Numer., 11 (2002), 341.
doi: 10.1017/S0962492902000053. |
[15] |
H. Omori, "Infinite-dimensional Lie Groups," volume 158 of "Translations of Mathematical Monographs,", Translations of Mathematical Monographs, 158 (1997).
|
[16] |
R. S. Palais, "Foundations of Global Non-linear Analysis,", Foundations of Global Non-linear Analysis, (1968).
|
[17] |
S. Reich, "Numerical Integration of the Generatized Euler Equations,", Technical report, (1993). Google Scholar |
[18] |
S. Reich, On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems,, Numer. Math., 76 (1997), 231.
doi: 10.1007/s002110050261. |
[19] |
S. Reich, Backward error analysis for numerical integrators,, SIAM J. Numer. Anal., 36 (1999), 1549.
doi: 10.1137/S0036142997329797. |
[20] |
R. Schmid, Infinite-dimensional Lie groups with applications to mathematical physics,, J. Geom. Symmetry Phys., 1 (2004), 54.
|
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