# American Institute of Mathematical Sciences

March  2011, 3(1): 81-111. doi: 10.3934/jgm.2011.3.81

## 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

Received  September 2010 Revised  March 2011 Published  April 2011

Backward Error Analysis (BEA) has been a crucial tool when analyzing long-time behavior of numerical integrators, in particular, one is interested in the geometric properties of the perturbed vector field that a numerical integrator generates. In this article we present a new framework for BEA on manifolds. We extend the previously known "exponentially close" estimates from $\mathbb{R}^n$ to smooth manifolds and also provide an abstract theory for classifications of numerical integrators in terms of their geometric properties. Classification theorems of type "symplectic integrators generate symplectic perturbed vector fields" are known to be true in $\mathbb{R}^n.$ We present a general theory for proving such theorems on manifolds by looking at the preservation of smooth $k$-forms on manifolds by the pull-back of a numerical integrator. This theory is related to classification theory of subgroups of diffeomorphisms. We also look at other subsets of diffeomorphisms that occur in the classification theory of numerical integrators. Typically these subsets are anti-fixed points of group homomorphisms.
Citation: Anders C. Hansen. A theoretical framework for backward error analysis on manifolds. Journal of Geometric Mechanics, 2011, 3 (1) : 81-111. doi: 10.3934/jgm.2011.3.81
##### References:
 [1] R. Abraham, J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications," volume 75 of "Applied Mathematical Sciences," Springer-Verlag, New York, second edition, 1988. [2] V. I. Arnold, "Mathematical Methods of Classical Mechanics," volume 60 of "Graduate Texts in Mathematics," Springer-Verlag, New York, second edition, 1989, Translated from the Russian by K. Vogtmann and A. Weinstein. [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), 1117-1143. doi: 10.1007/BF02188219. [4] M. P. Calvo, A. Murua and J. M. Sanz-Serna, Modified equations for ODEs, In "Chaotic Numerics (Geelong, 1993)," volume 172 of "Contemp. Math.," pages 63-74, Amer. Math. Soc., Providence, RI, 1994. [5] E. Cartan, Les groupes de transformations continus, infinis, simples, Ann. Sci. École Norm. Sup. (3), 26 (1909), 93-161. [6] D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math. (2), 92 (1970), 102-163. 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), 169-190. doi: 10.1093/imanum/19.2.169. [8] E. Hairer, Global modified Hamiltonian for constrained symplectic integrators, Numer. Math., 95 (2003), 325-336. 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), 441-462. doi: 10.1007/s002110050271. [10] E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," volume 31 of "Springer Series in Computational Mathematics," Springer-Verlag, Berlin, 2002. Structure-preserving algorithms for ordinary differential equations. [11] E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I," volume 8 of "Springer Series in Computational Mathematics," Springer-Verlag, Berlin, second edition, 1993. Nonstiff problems. [12] A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, In "Acta Numerica, 2000," volume 9 of "Acta Numer.," pages 215-365, Cambridge Univ. Press, Cambridge, 2000. [13] J. M. Lee, "Introduction to Smooth Manifolds," volume 218 of "Graduate Texts in Mathematics," Springer-Verlag, New York, 2003. [14] R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434. doi: 10.1017/S0962492902000053. [15] H. Omori, "Infinite-dimensional Lie Groups," volume 158 of "Translations of Mathematical Monographs," American Mathematical Society, Providence, RI, 1997, Translated from the 1979 Japanese original and revised by the author. [16] R. S. Palais, "Foundations of Global Non-linear Analysis," W. A. Benjamin, Inc., New York-Amsterdam, 1968. [17] S. Reich, "Numerical Integration of the Generatized Euler Equations," Technical report, Vancouver, BC, Canada, Canada, 1993. [18] S. Reich, On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems, Numer. Math., 76 (1997), 231-247. doi: 10.1007/s002110050261. [19] S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), 1549-1570 (electronic). doi: 10.1137/S0036142997329797. [20] R. Schmid, Infinite-dimensional Lie groups with applications to mathematical physics, J. Geom. Symmetry Phys., 1 (2004), 54-120.

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##### References:
 [1] R. Abraham, J. E. Marsden and T. Ratiu, "Manifolds, Tensor Analysis, and Applications," volume 75 of "Applied Mathematical Sciences," Springer-Verlag, New York, second edition, 1988. [2] V. I. Arnold, "Mathematical Methods of Classical Mechanics," volume 60 of "Graduate Texts in Mathematics," Springer-Verlag, New York, second edition, 1989, Translated from the Russian by K. Vogtmann and A. Weinstein. [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), 1117-1143. doi: 10.1007/BF02188219. [4] M. P. Calvo, A. Murua and J. M. Sanz-Serna, Modified equations for ODEs, In "Chaotic Numerics (Geelong, 1993)," volume 172 of "Contemp. Math.," pages 63-74, Amer. Math. Soc., Providence, RI, 1994. [5] E. Cartan, Les groupes de transformations continus, infinis, simples, Ann. Sci. École Norm. Sup. (3), 26 (1909), 93-161. [6] D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the notion of an incompressible fluid, Ann. of Math. (2), 92 (1970), 102-163. 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), 169-190. doi: 10.1093/imanum/19.2.169. [8] E. Hairer, Global modified Hamiltonian for constrained symplectic integrators, Numer. Math., 95 (2003), 325-336. 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), 441-462. doi: 10.1007/s002110050271. [10] E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration," volume 31 of "Springer Series in Computational Mathematics," Springer-Verlag, Berlin, 2002. Structure-preserving algorithms for ordinary differential equations. [11] E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I," volume 8 of "Springer Series in Computational Mathematics," Springer-Verlag, Berlin, second edition, 1993. Nonstiff problems. [12] A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, In "Acta Numerica, 2000," volume 9 of "Acta Numer.," pages 215-365, Cambridge Univ. Press, Cambridge, 2000. [13] J. M. Lee, "Introduction to Smooth Manifolds," volume 218 of "Graduate Texts in Mathematics," Springer-Verlag, New York, 2003. [14] R. I. McLachlan and G. R. W. Quispel, Splitting methods, Acta Numer., 11 (2002), 341-434. doi: 10.1017/S0962492902000053. [15] H. Omori, "Infinite-dimensional Lie Groups," volume 158 of "Translations of Mathematical Monographs," American Mathematical Society, Providence, RI, 1997, Translated from the 1979 Japanese original and revised by the author. [16] R. S. Palais, "Foundations of Global Non-linear Analysis," W. A. Benjamin, Inc., New York-Amsterdam, 1968. [17] S. Reich, "Numerical Integration of the Generatized Euler Equations," Technical report, Vancouver, BC, Canada, Canada, 1993. [18] S. Reich, On higher-order semi-explicit symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems, Numer. Math., 76 (1997), 231-247. doi: 10.1007/s002110050261. [19] S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), 1549-1570 (electronic). doi: 10.1137/S0036142997329797. [20] R. Schmid, Infinite-dimensional Lie groups with applications to mathematical physics, J. Geom. Symmetry Phys., 1 (2004), 54-120.
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