A novel convergence enhancement method based on Online Dimension Reduction Optimization

TL;DR

This paper introduces the Online Dimension Reduction Optimization (ODRO) method, which enhances convergence in iterative solvers for unstable problems in computational fluid dynamics by combining POD with residual optimization.

Contribution

The paper presents a novel ODRO method that integrates proper orthogonal decomposition with residual optimization to improve convergence and robustness in solving unstable PDE problems.

Findings

Efficiently computes steady-state solutions for unstable problems.

Combines POD with residual optimization for better convergence.

Easy to implement with minimal code changes.

Abstract

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.