Stabilization of the Adjoint for Turbulent Flows

On the Application of Optimal Control Techniques to the Shadowing Approach for Time Averaged Sensitivity Analysis of Chaotic Systems

“…While the Shadowing approaches utilise the Shadowing lemma the numerical viscosity methods either add an additional control term to the adjoint equations, Refs. [23,49,120,121], or use the dissipation of the time stepping scheme,…”

“…In this section a method for keeping the two trajectories uniformly close is to apply a control term that damps excess energy, Refs. [23,49,120,121], resulting in accurate sensitivity values. These methods are referred to, in this thesis, as numerical viscosity (NV) methods.…”

“…Ref. [49]. The main benefit of these NV methods is that one non-linear and one linear solution is required.…”

“…Using this approach removes the numerical inconsistency between the adjoint and non-linear systems when utilising the control term, Ref. [49].…”

“…The second class of methods for time averaged sensitivity analysis of chaotic systems is the numerical viscosity (NV) methods, Refs. [12,23,49,120,121]. These methods have been primarily developed on the adjoint equation.…”