Acceleration Methods for Nonlinear Solvers and Application to Fluid Flow Simulations
This thesis studies nonlinear iterative solvers for the simulation of Newtonian and non-Newtonian fluid models with two different approaches: Anderson acceleration (AA), an extrapolation technique that accelerates the convergence rate and improves the robustness of fixed-point iterations schemes, and continuous data assimilation (CDA) which drives the approximate solution towards coarse data measurements or observables by adding a penalty term.
We analyze the properties of nonlinear solvers to apply the AA technique. We consider the Picard iteration for the Bingham equation which models the motion of viscoplastic materials, and the classical iterated penalty Picard and Arrow-Hurwicz iterations for the incompressible Navier–Stokes equations (NSE) which model the Newtonian fluid flows. All these nonlinear solvers have some drawbacks. They lack robustness, and the required number of iterations for convergence could be large. In this thesis, we show that AA significantly improves the convergence properties of these nonlinear solvers, makes them more robust, and significantly reduces the number of iterations for convergence. We support our accelerated convergence analysis with various numerical tests.
We also consider CDA, which is used to improve the convergence of Picard iterations for the first time in literature. We analyze the improved contraction property of CDA applied to the Picard scheme for steady NSE. We give results for several numerical experiments of CDA applied to the Picard iteration to solve 1D, 2D and 3D nonlinear partial differential equations and show that significant reduction in the required number of iterations thanks to CDA.