Semi-Implicit Reversible Algorithms for Rigid Body Rotational Dynamics

Abstract

This paper presents two semi-implicit algorithms based on splitting methodology for rigid body rotational dynamics. The first algorithm is a variation of partitioned Runge-Kutta (PRK) methodology that can be formulated as a splitting method. The second algorithm is akin to a multiple time stepping scheme and is based on modified Crouch-Grossman (MCG) methodology, which can also be expressed as a splitting algorithm. These algorithms are second-order accurate and time-reversible; however, they are not Poisson integrators, i.e., non-symplectic. These algorithms conserve some of the first integrals of motion, but some others are not conserved; however, the fluctuations in these invariants are bounded over exponentially long time intervals. These algorithms exhibit excellent long-term behavior because of their reversibility property and their (approximate) Poisson structure preserving property. The numerical results indicate that the proposed algorithms exhibit superior performance compared to some of the currently well known algorithms such as the Simo-Wong algorithm, Newmark algorithm, discrete Moser-Veselov algorithm, Lewis-Simo algorithm, and the LIEMID[EA] algorithm.