The geometric nature of Euler fluids has been clearly identified andextensively studied over the years, culminating with Lagrangian and Hamiltoniandescriptions of fluid dynamics where the configuration space is defined as thevolume-preserving diffeomorphisms, and Kelvin's circulation theorem is viewedas a consequence of Noether's theorem associated with the particle relabelingsymmetry of fluid mechanics. However computational approaches to fluidmechanics have been largely derived from a numerical-analytic point of view,and are rarely designed with structure preservation in mind, and often sufferfrom spurious numerical artifacts such as energy and circulation drift. Incontrast, this paper geometrically derives discrete equations of motion forfluid dynamics from first principles in a purely Eulerian form. Our approachapproximates the group of volume-preserving diffeomorphisms using a finitedimensional Lie group, and associated discrete Euler equations are derived froma variational principle with non-holonomic constraints. The resulting discreteequations of motion yield a structure-preserving time integrator with goodlong-term energy behavior and for which an exact discrete Kelvin's circulationtheorem holds.
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