A. Zanna, The Fer expansion and time symmetry: a Strang-type approach, Tech. Rep. NA  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... when it evolves in Lie groups or a homogeneous space (see [11] for a review) In particular, the classical Magnus [13] and Fer [7] analytic expansions have been turned into very effective numerical methods, even for nonlinear matrix differential equations in Lie groups and homogeneous spaces [8, 18, 19], whereas other schemes based on the use of the Cayley transform [10, 14] and (conveniently modified) Runge Kutta methods [15] have also been proposed. Although some preliminary analysis of complexity issues related to this class of integration algorithms are already available [5] it is clear ....

.... e S 2 (h) e S 1 (h) X 0 Symmetric Fer I (4) Delta Delta Delta e T2 (h) e T1 (h) e T2 (h) Delta Delta Delta X 0 Symmetric Fer II (5) I Gamma 1 2 C(h) Gamma1 I 1 2 C(h) X 0 Cayley (6) and these expansions are convergent for sufficiently small values of h [2, 10, 19]. Observe that if A(t) belongs to a Lie algebra g then schemes (2) 5) and the numerical integrators based on them provide approximate solutions staying in the corresponding Lie group G if X 0 2 G, whereas this is true for the Cayley transform (6) only when G is a J orthogonal (also called ....

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A. Zanna, The Fer expansion and time-symmetry: a Strang-type approach, DAMTP tech. report 1998/NA14, University of Cambridge (1998). 22

....however, they are related to the the Fer [7] and Magnus [19] expansions: in fact, both expansions are based on the same product integral as we used above. Recently, the use of the Fer and Magnus expansions for numerically solving differential equations has been investigated by a number of people [15, 16, 17, 27, 36]. 27] discusses the use of graded Lie algebras and implementation issues for deriving Lie bracket identities in computational software (Matlab) 6 Experimental results We ran a number of experiments to compare the performance of the various simulation methods described above. All the ....

A. Zanna, The Fer expansion and time symmetry: a Strang-type approach, Tech. Rep. NA1998/14, DAMTP, Cambridge, 1998. 37

....Magnus expansion, or methods of the Munthe Kaas type) are not time symmetric for nonlinear problems, while the implicit midpoint appears to be so. That higher order Gaussian type methods are not time symmetric for nonlinear problem is also easily verified by numerical experiments (see for instance [24]) 1.2 Time symmetry of Lie group implicit midpoint Let us return to the Lie group implicit midpoint scheme (1:5) We can write yn 1 = exp(oe 1 )y n = exp(hfl 1 )y n = exp( h 2 fl 1 h 2 fl 1 )y n = exp( h 2 fl 1 )y n 1 2 ; 4 whereby we have denoted y n 1 2 = exp( h 2 fl 1 ....

A. Zanna. The Fer expansion and time symmetry: a Strang-type approach. Technical Report NA1998/14, DAMTP, 1998.

....Magnus expansion, or methods of the Munthe Kaas type) are not time symmetric for nonlinear problems, while the implicit midpoint appears to be so. That higher order Gaussian type methods are not time symmetric for nonlinear problem is also easily verified by numerical experiments (see for instance [25]) 1.2 Time symmetry of Lie group implicit midpoint Let us return to the Lie group implicit midpoint scheme (1:5) We can write yn 1 = exp(oe 1 )y n = exp(hfl 1 )y n = exp( h 2 fl 1 h 2 fl 1 )y n = exp( h 2 fl 1 )y n 1 2 ; whereby we have denoted y n 1 2 = exp( h 2 fl 1 ....

A. Zanna. The Fer expansion and time symmetry: a Strang-type approach. Technical Report NA1998/14, DAMTP, 1998.

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A. Zanna, The Fer expansion and time symmetry: a Strang-type approach, Tech. Rep. NA

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