  Iterative steady-state solvers are widely used in computational fluid
dynamics. Unfortunately, it is difficult to obtain steady-state solution for
unstable problem caused by physical instability and numerical instability.
Optimization is a better choice for solving unstable problem because
steady-state solution is always the extreme point of optimization regardless of
whether the problem is unstable or ill-conditioned, but it is difficult to
solve partial differential equations (PDEs) due to too many optimization
variables. In this study, we propose an Online Dimension Reduction Optimization
(ODRO) method to enhance the convergence of the traditional iterative method to
obtain the steady-state solution of unstable problem. This method performs
proper orthogonal decomposition (POD) on the snapshots collected from a few
iteration steps, optimizes PDE residual in the POD subspace to get a solution
with lower residual, and then continues to iterate with the optimized solution
as the initial value, repeating the above three steps until the residual
converges. Several typical cases show that the proposed method can efficiently
calculate the steady-state solution of unstable problem with both the high
efficiency and robustness of the iterative method and the good convergence of
the optimization method. In addition, this method is easy to implement in
almost any iterative solver with minimal code modification.
