Quasi-convergence of Optimal Balance by Nudging in Two-scale Dynamical Systems

Explore a 43-minute lecture on "Quasi-convergence of optimal balance by nudging" presented by Marcel Oliver at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delivered as part of the Thematic Programme on "The Dynamics of Planetary-scale Fluid Flows," this talk delves into the intricacies of optimal balance, a non-asymptotic numerical method for computing points on elliptic slow manifolds in two-scale dynamical systems with strong gyroscopic forces. Discover how the method works by solving a modified differential equation as a boundary value problem in time, with nonlinear terms adiabatically ramped up from zero to fully nonlinear dynamics. Learn about the challenges of implementing dedicated boundary value solvers and the alternative nudging solver approach, which involves repeated forward and backward solving with boundary condition restoration. Gain insights into the quasi-convergence of this scheme, where the termination residual of the nudging iteration is shown to be as small as the method's asymptotic error. Understand the implications of this finding, confirming the effectiveness of optimal balance in its nudging formulation and demonstrating the well-posedness of the boundary value problem formulation. Explore the key proof step involving a careful two-component Gronwall inequality, and grasp the significance of these findings in the context of planetary-scale fluid flow dynamics.