A procedure for obtaining steady-state solutions with WENO through sensitivity tuning and dynamic control

WENO schemes are known to struggle to obtain steady-state solutions in the presence of shocks. In this work, the 2D compressible Euler equations are solved using 5th-order finite difference WENO to obtain steady-state solutions by modifying the WENO parameters and implementing a dynamic weight controller. A descaling procedure is introduced to consistently compare and apply WENO across formulations and implementations. The response of WENO schemes to step inputs is studied to understand the effect of WENO parameters on weight activation and flux reconstruction. The effects on steady-state convergence and non-oscillatory behavior are then studied when the WENO parameters are varied. A PID control algorithm is introduced on the WENO weights that is capable of extending the ability of WENO to obtains teady solutions without introducing spatial oscillations.