The Discrete Moser?Veselov Algorithm for the Free Rigid Body, Revisited

Abstract

In this paper we revisit the Moser-Veselov description for the free Rigid Body, which, in the 3×3 case, can be implemented as an explicit, second order, integrable approximation of the continuous solution. By backward error analysis, we study the modified vector field which is integrated exactly by the discrete algorithm. We deduce that the discrete Moser-Veselov (DMV) is well approximated to higher order by time-reparametrizations of the continuous equations (modified vector field). We use the modified vector field to preprocess the initial data to the DMV and show the equivalence of the DMV algorithm and the RATTLE algorithm. Numerical integration with these preprocessed initial data is several order of magnitude more accurate of the original DMV and RATTLE approach. IntroductionIn 1991 Moser and Veselov published a memorable paper (Moser & Veselov 1991) in which they described discrete versions of several classical integrable systems, among which, the Rigid Body (RB). The integrability of these discrete maps was shown with the help of a Lax-pair representation corresponding to a factorization of certain matrix polynomials.There is a large interest of the computational community towards good methods for the integration of the RB equations, since they arise naturally in a number of applications, for instance celestial mechanics and molecular dynamics. RB equations are used here to follow particles (planets, atoms, molecules, etc.) in between other body-body interactions. It is of fundamental importance that some qualitative features of the system under consideration are preserved under integration, for instance energy. However, in the recent years it has become clear that numerical preservation of energy alone is not enough to obtain 'good' qualitative descriptions of the system. Other properties like symplecticity and time-reversibility have been shown to produce favourable propagation of errors, hence better numerical methods, especially when interested over long time behaviour.Another interesting aspect of the work of Moser and Veselov is that their approach is based on the theory of discrete Lagrangians, which has become emergent in the computational mechanics community (see for instance (Marsden & West 2001) and references therein) because the methods derived by this approach naturally inherit the symplectic structure of the system under consideration, they preserve the discrete Lagrangian, preserve momentum and preserve energy. However, this approach has also some major drawbacks: i) it is difficult to obtain numerical methods of order higher than two, ii) it is usually difficult to estimate the numerical error, and iii) the derived methods (with some few exceptions) are highly implicit, hence computationally expensive. In facts, the DMV algorithm for the 3 × 3 RB is an exception, since, as we shall see in the sequel, it can be implemented as an explicit numerical method. * Institute of Fundamental Sciences, Massey University, Private Bag 11-222, Palmerston North, New Zealand. Email: r.mclachlan...

Efficient time-symmetric simulation of torqued rigid bodies using Jacobi elliptic functions

“…The splitting method proposed in this paper is compared with the symplectic method of [2] and [16] which we denote in short by MR, with the Discrete Moser-Veselov methods of [13] (DMV), and with the classical fourth order Runge-Kutta method (RK4). We also refer to the second order symmetric splitting method, described in the previous section, as SEJ.…”

“…The appropriate composition of the corresponding flows produces a symplectic approximation of the problem. This symplectic method seems to outperform most of the known and previously proposed strategies of symplectic integration of the FRB problem, [4], [13]. We use this splitting for comparison in our numerical experiments.…”

“…Symplectic integration methods for the Euler equations have been constructed by various authors, [6], [15], [10], [13], see also [9] and references therein. Many of these methods cannot be straightforwardly generalized to the broader class of non canonical Hamiltonian problems, thus their use is limited to the numerical approximation of the FRB equations.…”

“…Accurate, symplectic, energy and momentum preserving approximations for the solution of the FRB equations, have been recently addressed in [13]. In this work the authors propose a new implementation of the Discrete Moser-Veselov algorithm of [15], and achieve order four and six in the integration by applying appropriate rescaling of the initial condition.…”

“…Some technical issues for the implementation of this approach are discussed in section 3. In section 4 we report some numerical experiments comparing the proposed approach to the Discrete Moser Veselov approach of [13] and the symplectic splitting of [16], [2].…”