Unsteady combined entropy and output-based adjoint approach for mesh refinement and error estimation

This paper presents a strategy for estimating temporal and spatial numerical errors in scalar outputs of unsteady fluid dynamics simulations using a combination of two error indicators: one based on a user-specified engineering output, and the other based on entropy variables. This work is an extension of a combined error indicator approach for steady-state simulations introduced in our previous work. Using an entropy-based approach to calculate error estimates is computationally advantageous as it does not require the solution of an auxiliary adjoint equation. This advantage is significant for unsteady problems, where the reverse adjoint solve is eliminated, as the entropy variables are computed directly from the state. Unfortunately, unlike the output-based approach that specifically targets regions of the spatial domain that are critical to an output, the entropy-based spatial error indicator is prone to target all regions of the domain where spurious entropy is generated. Through a combined approach, in which both spatial and temporal errors are combined, the limitations of the entropy-based approach are mitigated, leading to much better output error estimates that in many instances are comparable to those obtained using only the output-based approach. The main approach for combining the indicators investigated in this work is one that uses a coarser output-based adjoint that, while less accurate by itself, is far less computationally expensive. In addition, the use of masks to help improve the spatial error estimates using the entropy variables is also considered. This work focuses predominantly on ways of combining the error indicators for a simulation that uses the compressible Navier-Stokes equations with mesh optimization via error sampling and synthesis (MOESS).