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....Therefore, the fact that we are solving a nearby reversible problem, together with the fact that the solution is constrained on a manifold (the sphere) justifies the observed behaviour of the Hamiltonian error. A more rigourous explanation follows from backward error analysis for Lie group methods [6]. 4.2 The heavy top We will next consider a symmetric heavy top modelled on se(3) the dual space of the Lie algebra se(3) We recall that se(3) is isomorphic to the space R 3 Theta R 3 , therefore we will use a pair of 3 vectors ( Pi; Gamma) to represent the state of the heavy top. ....

S. Faltinsen. Backward error analysis for Lie-group methods. Technical Report NA1998/12, DAMTP, University of Cambridge, 1998.

....particular valid for 1 h . Hence, exp(hX) m) 1 h (m) 1) p 1 Y [ 1 h ] exp(hX) m) h p 1 O(h p 2 ) which can be rearranged as ( h (m) exp(hX) m) 1) p Y [ 1 h ] exp(hX) m) h p 1 O(h p 2 ) 11) Assuming backward error analysis [6], one can argue that 1 h = h 1 h 2 2 2 : 12) Expanding Y [ 1 h ] exp(hX) m) into its Lie series Y [ 1 h ] m) O(h) and substituting this and (12) into (11) yields the desired result (10) We conclude this section with stressing the following fact: ....

S. FALTINSEN, Backward Error Analysis for Lie--Group Methods, Tech. Rep. 1998/NA12, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England, 1998.

....of generality that tB is in the neighbourhood of the origin. The reason for this is that global results on the asymptotic behaviour of approximate solutions of Lie group integrators do not rely on global proprieties of exp, the local diffeomorphism from g to G. Such results, have been obtained in (Faltinsen 1998) using backward error analysis techniques. ffl Polynomial approximants are typically based on the construction of approximations of e tB v for some v 2 R n that lie in the m dimensional Krylov subspace Km (B; v) Spfv; Bv; B m Gamma1 vg: Such methods are based on the idea of ....

Faltinsen, S. (1998), Backward error analysis for Lie-group methods, Technical Report 1998/NA12, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England.

....Therefore, the fact that we are solving a nearby reversible problem, together with the fact that the solution is constrained on a manifold (the sphere) justifies the observed behaviour of the Hamiltonian error. A more rigourous explanation follows from backward error analysis for Lie group methods [6]. 4.2 The heavy top We will next consider a symmetric heavy top modelled on se(3) the dual space of the Lie algebra se(3) We recall that se(3) is isomorphic to the space R 3 Theta R 3 , therefore we will use a pair of 3 vectors ( Pi; Gamma) to represent the state of the heavy ....

S. Faltinsen. Backward error analysis for Lie-group methods. Technical report, DAMTP, University of Cambridge, 1998.

No context found.

S. Faltinsen, Backward Error Analysis for Lie-Group Methods, Tech. Rep.

....fi Oy 0 (y) where H j Oy 0 is the restriction of the function H to the coadjoint orbit. Backward error analysis tells us that a Lie group type numerical method is exponentially close in the step size to a nearby perturbed differential equation with solutions on the same coadjoint orbit O y0 [7]. Denote the solution of the perturbed differential equation by y(t) Since the numerical integrator is assumed to be Lie Poisson and preserve Casimirs, the nearby differential equation has the form y 0 (t) J( y)r H( y) y(t 0 ) y 0 ; 3.16) where H : O y0 R is a Hamiltonian ....

....In a majority of the methods the energy drifts. This is the case for instance for the methods based on forward and backward Euler. However, for the selfadjoint Lie group methods presented in [26] the error in the Hamiltonian energy is bounded. Applying the backward error analysis results of [7], the Lie group methods are solving a nearby differential equation with solutions on the same coadjoint orbit. Moreover, the results by Hairer and Stoffer [11] and Reich in [21] combined with backward error analysis on Lie groups, imply that if the original problem is reversible, and the ....

[Article contains additional citation context not shown here]

S. Faltinsen, Backward Error Analysis for Lie-Group Methods, Tech. Rep. 1998/NA12, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England, 1998.

....# Oy 0 (y) where H Oy 0 is the restriction of the function H to the coadjoint orbit. Backward error analysis tells us that a Lie group type numerical method is exponentially close in the step size to a nearby perturbed differential equation with solutions on the same coadjoint orbit O y0 [7]. Denote the solution of the perturbed differential equation by y(t) Since the numerical integrator is assumed to be Lie Poisson and preserve Casimirs, the nearby differential equation has the form y # (t) J(y)# H(y) y(t 0 ) y 0 , 3.16) where H : O y0 # R is a Hamiltonian ....

....In a majority of the methods the energy drifts. This is the case for instance for the methods based on forward and backward Euler. However, for the selfadjoint Lie group methods presented in [26] the error in the Hamiltonian energy is bounded. Applying the backward error analysis results of [7], the Lie group methods are solving a nearby differential equation with solutions on the same coadjoint orbit. Moreover, the results by Hairer and Stoffer [11] and Reich in [21] combined with backward error analysis on Lie groups, imply that if the original problem is reversible, and the ....

[Article contains additional citation context not shown here]

S. FALTINSEN, Backward Error Analysis for Lie-Group Methods, Tech. Rep. 1998/NA12, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England, 1998. 3.2, 5.1, 5.1 22

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S. Faltinsen. Backward error analysis for Lie-group methods. BIT, 40(4):652--670, 2000.

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