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Center for Nonlinear Studies |
Geometric numerical methods seek to transfer powerful theories in geometric mechanics to computational continuum dynamics. They preserve geometric structure in the flow field leading to excellent conservative properties. This property makes them attractive for climate modeling. For example, one can derive a geometric numerical method for the Lagrangian description of rotating shallow water equations which conserves mass, energy, potential vorticity and enstrophy. Lagrangian methods for hydrodynamics which describe the velocity field from particle positions and the density field with a moving mesh (with fixed connectivity) are not suited to long-time simulation of the climate because the mesh tangles. Moreover, rezoning and remapping techniques such as the alternating Eulerian-Lagrangian (ALE) method destroys the geometric structure of the flow field leading to poorer conservation properties. In this talk, I shall present my summer research project on the design of a new Lagrangian method for rotating shallow water with bottom topography which has excellent conservative properties and is suitable for long-time simulation. The method is derived from a semi-discrete Hamilton's action principle to ensure preservation of geometric structure. The novel part is the use of a Voronoi diagram to represent the density field. This is dynamically reconstructed at each time step to avoid the tangling problem. Preliminary numerical results of long-time simulations confirm that this method conserves mass (locally) and energy. We close the talk with a discussion of potential vorticity conservation.
Host: Natali Gulbahce and Praveen Ramaprabhu