An Adaptive Low-Rank Method for the Navier Stokes Equations

Presenter: Amit Rotem, Mathematics

Authors: A. Rotem

Abstract: For many time-dependent problems in fluid dynamics, snapshots of the solution are often well approximated by low rank representations. Such approximations are natural when the dynamics are characterized by coherent structures, but such approximations are also (eventually) valid for chaotic flows owing to the effects of diffusion. The low rank property of these dynamics motivates the use of reduced order models and compression techniques. We present a novel low rank method for approximately solving the Navier Stokes equations that dynamically updates the rank of the solution and never evaluates the solution on the entire grid.  The method is based on a pseudo-spectral discretization in space and Runge-Kutta time-stepping. The incompressibility condition is preserved by solving an elliptic problem for which we present a direct low rank solver based on the fast Fourier transform and the CrossDEIM algorithm. The nonlinear terms in the Navier Stokes equations are also approximated with CrossDEIM. Finally, we show the effectiveness of the solver for a few classical test cases.