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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 and weather prediction. 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. Weather and climate prediction practioners prefer Eulerian (i.e. grid based) methods but the Geometric numerical methods which exist are Lagrangian (i.e. particle based). In this talk, I shall discuss my summer project which is to develop a semi-lagrangian method which combines some of the conservative properties exhibited by geometric numerical methods with the convenience of a grid. I will present numerical results of a new locally mass conserving semi-lagrangian approximation for rotating shallow water which is adapted from a geometric numerical method. I will finally discuss ongoing work on developing this approach to conserve other quantities.