Accelerating and Enabling Convergence of Nonlinear Solvers for Navier-Stokes Equations by Continuous Data Assimilation

Abstract: This talk considers improving the Picard and Newton iterative solvers for the Navier-Stokes equations in the setting where data measurements or solution observations are available.  We construct adapted iterations that use continuous data assimilation (CDA) style nudging to incorporate the known solution data into the solvers.  For CDA-Picard, we prove the method has an improved convergence rate compared to usual Picard, and the rate improves as more measurement data is incorporated.  We also prove that CDA-Picard is contractive for larger Reynolds numbers than usual Picard, and the more measurement data that is incorporated the larger the Reynolds number can be with CDA- Picard still being contractive.  For CDA-Newton, we prove that the domain of convergence, with respect to both the initial guess and the Reynolds number, increases as the amount of measurement data is increased.  Additionally, for both methods we show that CDA can be implemented as direct enforcement of measurement data into the solution.  Numerical results for common benchmark Navier-Stokes tests illustrate the theory.

Speaker’s Bio: Duygu Vargun is a postdoctoral research associate in the Multiscale Methods and Dynamics Group at Oak Ridge National Laboratory (ORNL).  She received her Ph.D. degree in Mathematics from Clemson University in 2023, where she worked on acceleration methods for nonlinear solvers and application to fluid flow simulations. At ORNL, she will be working on efficient methods incorporating data into partial differential eqation models of physical problems and some improvement techniques in finite element solvers aiming to simulate multiphysics problems.