Entropy-Based Mesh Refinement, I: The Entropy Adjoint Approach

This work presents a mesh refinement indicator based on entropy-variables, with an application to the compressible Navier-Stokes equations. The entropy variables are shown to satisfy an adjoint equation, an observation that allows recent work in adjoint-based error estimation to be leveraged in constructing a relatively cheap but effective adaptation indicator. The output associated with the entropy-variable adjoint is shown to be the net entropy generation in the computational domain, including physical viscous dissipation when present. Adaptation using entropy variables thus targets areas of the domain responsible for numerical, or spurious, entropy generation. Adaptive results for inviscid and viscous aerodynamic examples demonstrate performance efficiency on par with output-based adaptation, as measured by errors in various engineering quantities of interest, with the comparative advantage that no adjoint equations need to be solved.