Topics on geometric integration

In many different engineering branches, computer-based simulations for analyzing systems’ behavior hold a growing importance for the development of new products and new technical solutions. Pursuing this aim requires two stages: building a mathematical model of the system and solving it employing a computer. Algorithms for calculating the numerical solution of differential equations, called numerical integrators, become thus extremely decisive as they influence the reliability of the simulation results. Many dynamical systems exhibit properties that are preserved by the flow, e.g., energy conservation, symmetry, momentum, symplecticity, configuration manifold. A conventional numerical integrator approximates the flow of the continuous-time equations using only the information about the vector field, ignoring the physical laws and the properties of the original trajectory. In this way, small inaccuracies accumulated over long times will significantly diminish the operational lifespan of such discrete solutions. Geometric integrators, on the other hand, are built in a way that preserve the structure of continuous dynamics, so maintaining the qualitative behavior of the exact flow even for long-time integrations. In this thesis, two different issues related to geometric integration are investigated. The first one describes the design of a numerical test which can be employed to assess in a easy way the long-time behavior of a rigid body integrator, in terms of energy preservation. The second one presents a straightforward approach to use off-the-shelf Lie methods for the integration of Hamiltonian systems evolving on a product of two spheres. - A numerical test of long-time stability for rigid body integrators The continuous-time flow of a Hamiltonian system (like a rigid body immersed in a static potential field) is symplectic, that is, it preserves the symplectic form. If the chosen integration method is symplectic or conjugate-symplectic then backward error analysis can be used to prove that (under some technical conditions) the method has an excellent long-time behavior. Moreover, symplecticity is often regarded as a key property for the preservation of the structure and the properties of the continuous-time flow. A large number of geometric algorithms for time integration of rigid body rotational dynamics have been proposed in the last 30 years. Some of these algorithms preserve by construction the canonical symplectic form, and therefore their good long-time behavior is assured from backward error analysis. Some other algorithms have been obtained approximating the continuous-time dynamics using ad hoc methodologies, with the declared aim of obtaining computationally fast algorithms with small error constant. In this latter case, the long-time behavior has been assessed using numerical experiments on a series of test cases, monitoring the behavior of the energy and known first integrals, but without any guarantee of good performance over long-time in a generic situation. Our contribution in this context is to propose a simple numerical test that has shown to be able to spot an energy drift in many rigid body algorithms that were reported in the literature to possess good energy behavior over long-time. The test consists in integrating the rotation dynamics of a rigid body in a suitable static field which is the sum of a bounded and an unbounded attractive terms. The presence of an energy drift implies that the energy error is not bounded over exponentially long times, and this allows us to conclude that these methods are not symplectic neither conjugate-symplectic. Among them, we cite the Lie-Newmark family of methods, whose flat space counterpart is instead known to be conjugate-symplectic. - Lie methods for integration of dynamical systems on two-spheres The two-sphere S^2 is defined as the set of all points in R^3 which have a unit length from the origin. Many classical and important dynamical systems evolve on two-sphere or on a product of two-spheres. In these cases, the configuration is usually described using 2 angles or a constraint enforcing unit length (on (S^2)^n, 2n angles or n constraints); these representations should be however avoided, since they yield additional complexity in the computation. The geometric approach to this problem exploits the fact that the special orthogonal Lie group of rotation matrices SO(3) acts transitively on the two-sphere; in this way, it is assured that the discrete trajectory belongs to the configuration space at any time without enforcing any constraints. One can therefore move the problem to the group space, searching for the trajectory in SO(3) that, through the group action, generates the flow in S2. Our contribution to this issue consists in a simple and straightforward method to adapt off-the-shelf Lie methods to solve for the dynamics of a system whose configuration space is the product of two-spheres. This approach is based on the Euler-Lagrange equations written on two-sphere, and has been tested on several significative numerical examples, proving to be accurate and computationally efficient. The biggest advantage of the proposed approach is represented by the possibility of effortlessly employing well-known Lie group methods, thus easily obtaining a high-order accuracy in the integration.