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Session I.5 - Geometric Integration and Computational Mechanics

Wednesday, June 14, 16:30 ~ 17:00

Eulerian and Lagrangian stability in Zeitlin's model of hydrodynamics

Klas Modin

Chalmers and University of Gothenburg, Sweden   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The 2-D Euler equations of a perfect fluid possess a beautiful geometric description: they are reduced geodesic equations on the infinite-dimensional Lie group of symplectomorphims with respect to a right-invariant Riemannian metric. This structure enables insights to Eulerian and Lagrangian stability via sectional curvature and Jacobi equations.

The Zeitlin model is a finite-dimensional analog of the 2-D Euler equations; the only known discretization that preserves the rich geometric structure. Theoretical and numerical studies indicate that Zeitlin's model provides consistent long-time behaviour on large scales, but to which extent it truly reflects the Euler equations is mainly open. Towards progress, I present here two new results. First, convergence of the sectional curvature in the Euler--Zeitlin equations on the Lie algebra $\mathfrak{su}(N)$ to that of the Euler equations on the sphere. Second, $L^2$-convergence of the corresponding Jacobi equations for Lagrangian and Eulerian stability. The new results allow geometric conclusions about Zeitlin's model to be transferred to Euler's equations and vice versa.

Joint work with Manolis Perrot (Univ. Grenoble Alpes, France).

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