Discrete Adjoint Based Time-Step Adaptation and Error Reduction in Unsteady Flow Problems

The paper presents an adjoint based approach for determining global error in the time domain that is relevant to scalar outputs computed as functions of the unsteady flow solution. The algorithm is derived for the unsteady Euler equations and takes into account the effect of dynamic meshes. Two primary components of the total error are specifically identified, namely, the error due to temporal resolution and the error due to partial convergence of the governing equations at each implicit time step. The primary error components are further decomposed into individual contributions arising from the flow equations and the mesh motion equations. The distribution of the global error from these various components is then used as the criterion for adaptation. The developed method is applied to a simple unsteady test case involving a sinusoidally pitching airfoil in order to demonstrate its strength and features.