Due to the increasing demands for modeling large-scale and complex systems, designing optimal controls, and conducting optimization tasks, many real-world applications require sophisticated models, in which the traditional dynamic system setup using continuous dynamics given by differential equations alone is reconsidered. Geometric methods are designed to capture the underlying ”structure” and to preserve the global qualitative or geometric properties of the flow, such as symplecticity, volume preservation and symmetry. As such, they are named structure-preserving methods. The following key advantages appear to be well adapted in building algorithms with high reliability and robustness:

- geometrical methods preserve fundamental (physical) invariants;

- methods are build within an intrinsic formulation;

- produce integrators over much less computational cost;

- stability and long time integration are guaranteed by physical coherence consistency;

- covariant formulation for which space and time are treated at an equal footing.

It is today well established that, for finite dimensional mechanical systems, integration algorithms based on variational principles give a unified treatment of many symplectic numerical schemes. In this context, the discrete Noether theorem [1]
allow for a numerical formulation that preserve symmetries and conservation laws. It is also often advantageous (e.g. in non linear elasticity) to variationally discretize the equations simultaneously in space and time to give rise to the notion of multi-symplectic integrators [2].
 
Beyond these helpful intrinsic invariance property some drawbacks may appears in the practical applications: it is difficult to define methods with order higher than 2, estimating the numerical error is usually non trivial, and the constructed method are often implicit, which may increase the computational cost.

In the case of homogeneous spaces (smooth manifold on which acts a Lie group), previous variational methods are compared to other geometrical approaches. For the so-called Lie group integrators, comprising Runge-Kutta-Munthe-Kaas [3] and Crouch-Grossman methods, the main preoccupation is to ensure that discrete solutions are guaranted to stay on the given manifold. In that setting, well known and powerful tools (i.e. Runge-Kutta, Butcher series) can be applied to equations expressed on the Lie algebra to design higher order methods as well as to determine their numerical convergence. However in this case no particular preservation of symmetries is obtained without further constraints.

Testing the validity of such simulations will be achieved by exhibiting analytically solvable models and comparing the result of simulations with their exact behavior.

[1] J.E. Marsden et M. West, "Discrete mechanics and variational integrators", Acta Numerica, 2001
[2] F. Demoures et al., "Multisymplectic Lie group variational integrator for a geometrically exact beam in R3", Communications in Nonlinear Science and Numerical Simulation, 2014
[3] H. Munthe-Kaas, "High order Runge-Kutta methods on manifolds", Applied Numerical Mathematics, 1999