For many decades, researchers agree that designing physics-conserving numerical solvers should be done through the preservation over time of the geometrical structure of the equations at the discrete scale. It is in this spirit that the most popular geometry-based integrators were developped. They include, for instance, symplectic integrators, variational integrators and Lie-symmetry invariant schemes which preserve, respectively, the symplectic two-form, the Lagrangian structure, and the Lie symmetry group of the equations numerically.

In this communication, we present the discrete exterior calculus (DEC) which is a more general geometry-based integrator. DEC aims to be a discrete theory of differential geometry, and especially exterior calculus, which preserves exactly the cohomology structure provided by the exterior derivative operator d. One of the advantage of this approach is the naturally consistent discretisations of the tensorial derivation operators such as divergence, gradient and curl. DEC also guarantees that Stoke's theorem holds at discrete scale.

The introduction of DEC in mechanics is new, but has already been used, for example, to build a solver for Euler's equations which respects Kelvin's theorem on the preservation of circulation. In this communication, other applications to Navier-Stokes equations with temperature and passive scalars will be presented.