Error Estimation for High Speed Flows Using Continuous and Discrete Adjoints

Abstract

In this work, the utility of the adjoint equations in error estimation of functional outputs and goal-oriented mesh adaptation is investigated with specific emphasis towards application to high speed flows with strong shocks. Continuous and discrete adjoint formulations are developed for the compressible Euler equations and the accuracy and robustness of the implementation is assessed by evaluating adjoint-computed sensitivities. The two-grid approach of Venditti and Darmofal 1 - where the flow and adjoint solutions on a baseline mesh are processed to estimate the functional on a finer mesh - is used for error estimation and mesh adaptation. Using a carefully designed set of test cases for the quasi one dimensional Euler equations, it is shown that discrete adjoints can be used to estimate the fine grid functional more accurately, whereas continuous adjoints are marginally better at estimating the analytical value of the functional when the flow and adjoint solutions are well resolved. These observations appear to be true in multi-dimensional inviscid flows, but the distinction is not as clear. The discrete adjoint is shown to be robust when applied to goal oriented mesh adaptation in a flow with multiple shocks, and hence offers promise as a viable strategy to control the numerical error in Hypersonic flow applications.