Computational Mathematics and Scientific Computing Seminar

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Simulating incompressible fluid motion remains a central topic in physics-based simulation, where the goal is to construct discrete dynamics that faithfully reproduce salient features of the continuous motion.  We introduce a novel discretization of the incompressible Euler equations based on their interpretation as geodesic equations on the Lie group of volume-preserving diffeomorphisms.  It is well known that encoding diffeomorphisms and their infinitesimal generators through a truncated Koopman representation places a nonholonomic constraint on discrete velocities, for which there is no consensus on a variational treatment.  We show that taking the vakonomic perspective, as opposed to the usual perspective of Lagrange-d'Alembert, yields discrete fluid trajectories that remain geodesics on a (sub-)Riemannian manifold.  In particular, the resulting vakonomic dynamics are Lie-Poisson and their solutions admit a discrete relabeling symmetry, leading to machine-precision satisfaction of Casimir invariants along with a discrete analogue of Kelvin's Circulation Theorem.  Using an efficient momentum map representation based on low-rank Clebsch variables, we show that these vakonomic fluids behave stably and consistently even at low grid resolutions, leading to increased robustness and physical realism in the long term.