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

A theoretical framework for backward error analysis on manifolds

Anders C. Hansen 1,

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.
Keywords: infinite-dimensional manifolds, diffeomorphism groups, Lie groups., Backward error analysis.
Mathematics Subject Classification: 34A26, 65L05, 65J99, 53A35, 53A4.
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, (1988).   Google Scholar

[2]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," volume 60 of "Graduate Texts in Mathematics,", Springer-Verlag, (1989).   Google Scholar

[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.  Google Scholar

[4]

M. P. Calvo, A. Murua and J. M. Sanz-Serna, Modified equations for ODEs,, In, 172 (1993), 63.   Google Scholar

[5]

E. Cartan, Les groupes de transformations continus, infinis, simples,, Ann. Sci. École Norm. Sup. (3), 26 (1909), 93.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[8]

E. Hairer, Global modified Hamiltonian for constrained symplectic integrators,, Numer. Math., 95 (2003), 325.  doi: 10.1007/s00211-002-0428-7.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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).   Google Scholar

[12]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods,, Acta Numerica, 2000, 9 (2000), 215.   Google Scholar

[13]

J. M. Lee, "Introduction to Smooth Manifolds," volume 218 of "Graduate Texts in Mathematics,", Graduate Texts in Mathematics, 218 (2003).   Google Scholar

[14]

R. I. McLachlan and G. R. W. Quispel, Splitting methods,, Acta Numer., 11 (2002), 341.  doi: 10.1017/S0962492902000053.  Google Scholar

[15]

H. Omori, "Infinite-dimensional Lie Groups," volume 158 of "Translations of Mathematical Monographs,", Translations of Mathematical Monographs, 158 (1997).   Google Scholar

[16]

R. S. Palais, "Foundations of Global Non-linear Analysis,", Foundations of Global Non-linear Analysis, (1968).   Google Scholar

[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.  Google Scholar

[19]

S. Reich, Backward error analysis for numerical integrators,, SIAM J. Numer. Anal., 36 (1999), 1549.  doi: 10.1137/S0036142997329797.  Google Scholar

[20]

R. Schmid, Infinite-dimensional Lie groups with applications to mathematical physics,, J. Geom. Symmetry Phys., 1 (2004), 54.   Google Scholar

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).   Google Scholar

[2]

V. I. Arnold, "Mathematical Methods of Classical Mechanics," volume 60 of "Graduate Texts in Mathematics,", Springer-Verlag, (1989).   Google Scholar

[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.  Google Scholar

[4]

M. P. Calvo, A. Murua and J. M. Sanz-Serna, Modified equations for ODEs,, In, 172 (1993), 63.   Google Scholar

[5]

E. Cartan, Les groupes de transformations continus, infinis, simples,, Ann. Sci. École Norm. Sup. (3), 26 (1909), 93.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. Hairer, Global modified Hamiltonian for constrained symplectic integrators,, Numer. Math., 95 (2003), 325.  doi: 10.1007/s00211-002-0428-7.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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).   Google Scholar

[12]

A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods,, Acta Numerica, 2000, 9 (2000), 215.   Google Scholar

[13]

J. M. Lee, "Introduction to Smooth Manifolds," volume 218 of "Graduate Texts in Mathematics,", Graduate Texts in Mathematics, 218 (2003).   Google Scholar

[14]

R. I. McLachlan and G. R. W. Quispel, Splitting methods,, Acta Numer., 11 (2002), 341.  doi: 10.1017/S0962492902000053.  Google Scholar

[15]

H. Omori, "Infinite-dimensional Lie Groups," volume 158 of "Translations of Mathematical Monographs,", Translations of Mathematical Monographs, 158 (1997).   Google Scholar

[16]

R. S. Palais, "Foundations of Global Non-linear Analysis,", Foundations of Global Non-linear Analysis, (1968).   Google Scholar

[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.  Google Scholar

[19]

S. Reich, Backward error analysis for numerical integrators,, SIAM J. Numer. Anal., 36 (1999), 1549.  doi: 10.1137/S0036142997329797.  Google Scholar

[20]

R. Schmid, Infinite-dimensional Lie groups with applications to mathematical physics,, J. Geom. Symmetry Phys., 1 (2004), 54.   Google Scholar

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