Recovering high-order accuracy in WENO computations of steady-state hyperbolic systems

In this note we consider the application of the WENO scheme to simulations of steady-state flow in a converging diverging nozzle. We demonstrate the recovery of design accuracy through Gegenbauer postprocessing, despite the degradation of the order of accuracy for the numerical solution of the Euler equations to first-order in regions where the characteristics passed through the shock. We have shown a case in which the Gegenbauer postprocessing can recover the order of accuracy right up to the shock location. This suggests that high-order accurate information which crosses through the shock may not be irretrievably lost, and we can strive to recover it through various types of postprocessing.