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Numerical analysis

GEOMFLUID

Development of numerical integrators for fluid dynamics and nonlinear elasticity

This project initiated the systematic construction, implementation and validation of geometric numerical schemes for the equations of atmospheric and ocean dynamics and for collision and friction problems in continuum mechanics.

In both these fields, respecting the geometric properties of the equations in numerical terms is crucial, and this is reflected directly in the preservation of conservation laws and the overall behaviour of the dynamical system. These integrators are based on the variational formulation of equations on Lie groups together with tools from symplectic and multisymplectic geometry.

A major advance achieved by the project is the creation and evaluation of a set of geometric discretisation methods that preserve the main conservation laws of dynamics on unstructured meshes for geophysical fluids. Similar progress has been made for a class of non-linear elasticity models, including the treatment of collisions and loss of regularity.

The project

ANR programme: Generic Call for Proposals

Edition, project duration: 2014, 48 months

Subvention ANR: € 285,512

Coordinator:

Project region: Île-de-France

Main publication or contribution:

  • Bauer and Gay-Balmaz [2019], Towards a geometric variational discretization of compressible fluids: the rotating shallow water equations, J. Comp. Dyn., 16(1), 1–37. 
  • Demoures, Gay-Balmaz, Desbrun, Ratiu, Aragon [2017], A multisymplectic integrator for elastodynamic frictionless impact problems, Computer Methods in Applied Mechanics and Engineering, 315, 1025–1052.
geomfluid
Schematic representation of the project tasks, underlying mathematical structures and perspectives. © 1. Bauer and Gay-Balmaz [2019] 2. Demoures, Gay-Balmaz, Desbrun, Ratiu, Aragon [2017]
perspectives
perspectives

The results of GEOMFLUID will contribute significantly to constructing the next generation of general circulation models used for weather and climate predictions. In parallel, the project has opened new research prospects for the modelling and numerical simulation of irreversible processes using abstract geometric variational structures.

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