A novel convergence enhancement method based on online dimension reduction optimization

Abstract

Iterative steady-state solvers are widely used in computational fluid dynamics. Unfortunately, it is difficult to obtain steady-state solutions for unstable problems caused by physical instability and numerical instability. Optimization is a better choice for solving unstable problems because the 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 method to enhance the convergence of the traditional iterative method to obtain the steady-state solutions of unstable problems. This method performs proper orthogonal decomposition (POD) on the snapshots collected from a few iteration steps of computational fluid dynamics (CFD) simulation, optimizes the POD mode coefficients to minimize the PDE residual to obtain a solution with a lower residual in the POD subspace, 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 problems with both the high efficiency and robustness of the iterative method and the good convergence of the optimization method. In addition, this method avoids specific knowledge about the underlying numerical scheme of the CFD code and is easy to implement in almost any iterative solver with minimal code modification.