Stabilisation of discrete steady adjoint solvers

A new implicit time-stepping scheme which uses Runge-Kutta time-stepping and Krylov methods as a smoother inside FAS-cycle multigrid acceleration is proposed to stabilise the flow solver and its discrete adjoint counterpart. The algorithm can fully converge the discrete adjoint solver in a wide range of cases where conventional point-implicit methods fail due to either physical or numerical instability. This enables the discrete adjoint to be applied to a much wider range of flow regimes. In addition, the new algorithm offers improved efficiency when applied to stable cases for which the conventional Block-Jacobi solver can fully converge. Both stable and unstable cases are presented to demonstrate the improved robustness and performance of the new scheme. Eigen-analysis is presented to outline the mechanism of the adjoint stabilisation effect.