Solutions of Polynomial Systems Derived from the Steady Cavity Flow Problem
For researchers in polynomial optimization and fluid dynamics, this work provides a novel enumeration method for polynomial systems, though its application is domain-specific and convergence to the original PDE problem remains conjectural.
The paper proposes an algorithm to enumerate all solutions of zero-dimensional polynomial systems using sparse semidefinite programming relaxations, applied to the steady cavity flow problem. The algorithm successfully finds the k smallest kinetic energy solutions for first- and second-order relaxations, with solutions converging to minimal kinetic energy solutions as relaxation order increases.
We propose a general algorithm to enumerate all solutions of a zero-dimensional polynomial system with respect to a given cost function. The algorithm is developed and is used to study a polynomial system obtained by discretizing the steady cavity flow problem in two dimensions. The key technique on which our algorithm is based is to solve polynomial optimization problems via sparse semidefinite programming relaxations (SDPR), which has been adopted successfully to solve reaction-diffusion boundary value problems recently. The cost function to be minimized is derived from discretizing the fluid's kinetic energy. The enumeration algorithm's solutions are shown to converge to the minimal kinetic energy solutions for SDPR of increasing order. We demonstrate the algorithm with SDPR of first and second order on polynomial systems for different scenarios of the cavity flow problem and succeed in deriving the $k$ smallest kinetic energy solutions. The question whether these solutions converge to solutions of the steady cavity flow problem is discussed, and we pose a conjecture for the minimal energy solution for increasing Reynolds number.