Geometric integrators in fluid dynamics and elasticity – GEOMFLUID

The proposal entitled GEOMFLUID is for work on the development and applications of geometric numerical integrators for partial differential equations arising in atmospheric and oceanic circulation, and in nonlinear continuum mechanics.
Geometric integrators form a particular class of numerical schemes for evolutionary equations that are
especially constructed in order to preserve the geometric properties of the equation they discretize such as energy conservation, and conservation laws associated to symmetries. Such properties are of primordial importance for the applications we are focusing on in this proposal, and rely on fundamental mathematical structures from symplectic and Poisson geometry. Whereas our applications are on the two different topics atmospheric/oceanic circulations and nonlinear continuum mechanics, the mathematical approach we will use to derive the numerical scheme is common in both situations, namely, we use the geometric variational principles of continuum mechanics underlying the dynamics for both cases. This provides us with a unique point of view that allows transfer of ideas and technologies from one area to the other. This fact also illustrates the versatility of geometric integration schemes.
-In the case of ordinary differential equations (ODEs), geometric integrators have been developed and
exploited in various directions and form today a well established tool in designing numerical schemes in various area of applications in dynamical systems and engineering. Among them we find the well-known variational and symplectic numerical integrators for Hamiltonian systems.
-For partial differential equations (PDEs), the situation is much more involved and there is still no well established development that fully parallels the case of ODEs. It is the goal of this project to start a systematic construction, implementation, and validation of geometric numerical integrators for certain class of Hamiltonian PDEs used in dynamical cores of weather and climate prediction models, and in contact and friction problems in continuum mechanics. As it will be highlighted below, these are two areas where the specific properties of geometric integrators are of crucial importance. Our proposal is organized in three topics.
Topic A: Geometric integrators for oceanic and atmospheric dynamics (OAD) ;
Topic B: Casimir dissipation in Lie-Poisson systems – energy preserving damping in OAD;
Topic C: Geometric multisymplectic integrators in nonlinear elasticity – contact and friction;
These topics are related, not only from the point of view of the unifying differential geometric techniques that we use, but also in the sense that the advances made in each of the topic will have direct implications on the development of the other topics.
- Topics A consists in the development of structure preserving numerical schemes for OAD, based on the geometric Lie group formulation of theses models as Euler-Poincaré equations associated to diffeomorphism groups, extending the original idea of V. Arnold. The approach we use is to consistently reformulate the fluid models as Euler-Poincaré equations on a finite dimensional Lie group approximation of the diffeomorphism group. This leads to a symplectic integrator that allows for discrete versions of circulation theorems and is valid on unstructured meshes.
- Topics B consists in the development and study of a new type of Casimir dissipation for Euler-Poincaré systems recently proposed by F. Gay-Balmaz and D. Holm. When applied to diffeomorphism groups, this type of dissipation yields new classes of OAD models that allow for a parametrization of multiscale interaction. We also study the associated geometric integrators for these models.
- Topics C consists in the development of geometric integrators in nonlinear elasticity, with main focus on beams and plates, and problems with contact and friction. This is done, as above, by making use of the geometric variational formulation of these problems.