The underlying philosophy is that numerical methods methods for the integration of differential equations should incorporate as much information and geometry as possible from the original problem. In many instances, geometric integration methods have been proved to perform superiorly with respect to classical methods.

1.  Splitting methods for divergence-free vector fields. Divergence free vector fields give rise to differential equations for which the volume of a subregion does not change under the time evolution. A numerical integrator is said to be volume preserving if has the same property of conserving (to machine precision) the volume of any subregion under time evolution. Volume preservation is a hard task numerically, as ordinary methods will either shrink or increase the volume. We achieve volume preservation via splitting methods, meaning that the original vector field is divided in simpler pieces that are easier to integrate. These problems arise mostly in fluid dynamics, however newer fancier applications include simulation of muscles deformations.
2.  Numerical integration of the time-dependent Schoedinger equation (CODY project). The Schrödinger equation is the fundamental equation of physics for describing quantum mechanical behavior. It is also often called the Schrödinger wave equation, and is a partial differential equation that describes how the wave function of a physical system evolves over time. It is a computationally demanding problem, and it is very difficult to model the behavior of more of a particle for more than a few femtoseconds. The potential may be highly oscillatory, rendering the numerical approximation even more challenging.
3.  Highly oscillatory differential equations. High oscillations are present in many practical applications, from molecular dynamics to chemical reactions, weather models, etc. Classically, high oscillations imply that the numerical method must use a very small integration stepsize. In a problem with both long scale and short scale behavior, this limitation on the stepsize is very severe, as it means that the entire problem becomes very expensive to solve. Recently, new numerical methods based on Filon quadrature, have been proven to be very promising, as their error decreases the faster the oscillation.  At present this is a very hot topic of research, because of the relevance in many applications.
4.  Geometric methods for Hamiltonian/Poisson problems/Variational integrators. Occur in many applications, from molecular dynamics, celestial mechanics, weather forecasting, waves, etc. Any Hamiltonian problem has an underlying symplectic form that corresponds to the conservation of the sum of areas in 2D-phase subspaces (the spaces (q_i, p_i)). A symplectic method is a method that preserves the same symplectic form. Symplectic methods are better than usual methods as it can be proved that their numerical approximation solves exactly a nearby Hamiltonian problem, for exponentially long time.  To-day, many symplectic methods are known, however, in many cases, better and more efficient methods are needed. The Poisson case is much less understood and there is no `universal’ approach. Here there is much work to be done. Variational Integrators apply to problems in the Lagrangian formulation. Here, the Lagrangian is discretized and then discrete variations are taken instead of continuous ones. These methods have the same good properties as symplectic methods for Hamiltonian problems.